0.07/0.15 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.16 % Command : tptp2X_and_run_prover9 %d %s 0.15/0.37 % Computer : n024.cluster.edu 0.15/0.37 % Model : x86_64 x86_64 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.15/0.37 % Memory : 8042.1875MB 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64 0.15/0.37 % CPULimit : 960 0.15/0.37 % DateTime : Thu Jul 2 12:12:10 EDT 2020 0.15/0.37 % CPUTime : 8.61/8.63 ============================== Prover9 =============================== 8.61/8.63 Prover9 (32) version 2009-11A, November 2009. 8.61/8.63 Process 32621 was started by sandbox2 on n024.cluster.edu, 8.61/8.63 Thu Jul 2 12:12:18 2020 8.61/8.63 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_32446_n024.cluster.edu". 8.61/8.63 ============================== end of head =========================== 8.61/8.63 8.61/8.63 ============================== INPUT ================================= 8.61/8.63 8.61/8.63 % Reading from file /tmp/Prover9_32446_n024.cluster.edu 8.61/8.63 8.61/8.63 set(prolog_style_variables). 8.61/8.63 set(auto2). 8.61/8.63 % set(auto2) -> set(auto). 8.61/8.63 % set(auto) -> set(auto_inference). 8.61/8.63 % set(auto) -> set(auto_setup). 8.61/8.63 % set(auto_setup) -> set(predicate_elim). 8.61/8.63 % set(auto_setup) -> assign(eq_defs, unfold). 8.61/8.63 % set(auto) -> set(auto_limits). 8.61/8.63 % set(auto_limits) -> assign(max_weight, "100.000"). 8.61/8.63 % set(auto_limits) -> assign(sos_limit, 20000). 8.61/8.63 % set(auto) -> set(auto_denials). 8.61/8.63 % set(auto) -> set(auto_process). 8.61/8.63 % set(auto2) -> assign(new_constants, 1). 8.61/8.63 % set(auto2) -> assign(fold_denial_max, 3). 8.61/8.63 % set(auto2) -> assign(max_weight, "200.000"). 8.61/8.63 % set(auto2) -> assign(max_hours, 1). 8.61/8.63 % assign(max_hours, 1) -> assign(max_seconds, 3600). 8.61/8.63 % set(auto2) -> assign(max_seconds, 0). 8.61/8.63 % set(auto2) -> assign(max_minutes, 5). 8.61/8.63 % assign(max_minutes, 5) -> assign(max_seconds, 300). 8.61/8.63 % set(auto2) -> set(sort_initial_sos). 8.61/8.63 % set(auto2) -> assign(sos_limit, -1). 8.61/8.63 % set(auto2) -> assign(lrs_ticks, 3000). 8.61/8.63 % set(auto2) -> assign(max_megs, 400). 8.61/8.63 % set(auto2) -> assign(stats, some). 8.61/8.63 % set(auto2) -> clear(echo_input). 8.61/8.63 % set(auto2) -> set(quiet). 8.61/8.63 % set(auto2) -> clear(print_initial_clauses). 8.61/8.63 % set(auto2) -> clear(print_given). 8.61/8.63 assign(lrs_ticks,-1). 8.61/8.63 assign(sos_limit,10000). 8.61/8.63 assign(order,kbo). 8.61/8.63 set(lex_order_vars). 8.61/8.63 clear(print_given). 8.61/8.63 8.61/8.63 % formulas(sos). % not echoed (5228 formulas) 8.61/8.63 8.61/8.63 ============================== end of input ========================== 8.61/8.63 8.61/8.63 % From the command line: assign(max_seconds, 960). 8.61/8.63 8.61/8.63 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 8.61/8.63 8.61/8.63 % Formulas that are not ordinary clauses: 8.61/8.63 1 (all V_b_2 all V_a_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_equiv__class__eq) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 2 (all V_x all V_y all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))))) # label(fact_inf__sup__ord_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 3 (all V_w_2 all V_c_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2))))))) # label(fact_divide__less__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 4 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_a))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_finite__UN__I) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 5 (all V_z_2 all T_b (class_Rings_Osemiring__1(T_b) -> (c_Int_Oiszero(T_b,V_z_2) <-> c_Groups_Ozero__class_Ozero(T_b) = V_z_2))) # label(fact_iszero__def) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 6 (all V_B_2 all V_A_2 all V_I_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_B_2))) # label(fact_INT__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 7 (all T_1 class_Nat_Osize(tc_Lazy__Sequence_Olazy__sequence(T_1))) # label(arity_Lazy____Sequence__Olazy____sequence__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 8 (all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_finite__UNIV__card__ge__0) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 9 (all V_P_2 all V_f_2 all T_b all T_c c_New__Random__Sequence_Oneg__bind(T_c,T_b,V_P_2,hAPP(c_Fun_Ocomp(T_b,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b)))))),T_c,c_New__Random__Sequence_Oneg__single(T_b)),V_f_2)) = c_New__Random__Sequence_Oneg__map(T_c,T_b,V_f_2,V_P_2)) # label(fact_neg__map__def) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 10 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_B_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_B_2))))) # label(fact_Inter__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 11 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_y)))) # label(fact_inf__sup__ord_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 12 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_k) = c_Int_OBit0(V_k)) # label(fact_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 13 (all V_N_2 ((exists B_m all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_x),B_m)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2)))) # label(fact_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 14 (all V_xs_2 all V_p_2 all T_b c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Ofilter(T_b,V_p_2)),V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_p_2),c_List_Oconcat(T_b,V_xs_2))) # label(fact_filter__concat) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 15 (all V_r_2 all T_b c_Relation_Orefl__on(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))) # label(fact_refl__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 16 (all V_n all V_z all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)),V_n) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)))) # label(fact_of__int__power) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 17 (all V_z_2 all V_w_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 18 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_j)))))) # label(fact_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 19 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))))) # label(fact_equiv__type) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 20 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_z -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x)),V_y)),V_z)))) # label(fact_add__divide__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 21 (all V_x all T_a hAPP(c_List_OremoveAll(T_a,V_x),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_removeAll_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 22 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 23 (all V_n_2 all V_k_2 all V_m_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2) <-> V_m_2 = V_n_2)) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 24 (all V_z_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__graph__fold) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 25 (all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(V_A_2,V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 26 (all V_xs_2 all T_b all V_x_2 all V_P_2 ((hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)) = c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) & (-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = c_List_Olist_ONil(T_b)))) # label(fact_takeWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 27 (all V_a_2 all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_2)),V_r_2))))) # label(fact_refl__onD) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 28 (all V_P_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> c_List_Olinorder__class_Osorted(T_b,c_List_OtakeWhile(T_b,V_P_2,V_xs_2))))) # label(fact_sorted__takeWhile) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 29 (all V_z_2 all V_w_2 (V_z_2 = V_w_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))))) # label(fact_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 30 (all V_b all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 31 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n_2)),hAPP(c_Code__Numeral_Oof__nat,V_m_2))))) # label(fact_less__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 32 (all T_b all V_B_2 all V_A_2 (V_A_2 = V_B_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 33 (all V_n all V_k all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k))) # label(fact_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 34 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_b_2),V_a_2)) -> c_SetInterval_Oord_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OatLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.61/8.63 35 (all V_r_H_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_H_2),V_r_2)))) # label(fact_wf__Int2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 36 (all V_z_2 hAPP(c_Int_OAbs__Integ,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(c_Int_ORep__Integ,V_z_2)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_Nat_Onat),c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),tc_Nat_Onat),c_COMBC(tc_Nat_Onat,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),tc_Nat_Onat),hAPP(c_COMBB(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),tc_Nat_Onat),c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)))),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat))))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z_2)) # label(fact_minus__int__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 37 (all V_y_2 all V_z_2 all V_f_2 all T_c all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_y_2) -> c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,V_A_2,hAPP(hAPP(V_f_2,V_x_2),V_y_2))))) # label(fact_fold__graph_H_Ointros_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 38 (all V_w all V_v c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 39 (all V_xs all T_a (class_Orderings_Olinorder(T_a) -> ((all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),B_i)),hAPP(hAPP(c_List_Onth(T_a),V_xs),B_j)))))) -> c_List_Olinorder__class_Osorted(T_a,V_xs)))) # label(fact_sorted__nth__monoI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 40 (all V_S_2 (c_Nat__Transfer_Onat__set(V_S_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)))))) # label(fact_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 41 (all V_ys all V_xs all V_x all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_ys) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_append__Cons) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 42 (all V_f_2 all V_a_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) # label(fact_setsum__diff1_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 43 (all V_x all V_ys all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) -> V_xs != hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))) # label(fact_impossible__Cons) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 44 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2)))) # label(fact_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 45 (all V_z_2 all V_w_2 ((exists B_n V_z_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 46 (all V_a_2 all V_f_2 all T_b V_a_2 = hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 47 (all V_a_2 all V_A_2 all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)),V_a_2),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))))) # label(fact_fold1__eq__fold__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 48 (all V_n_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_n_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_n_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_n_2)))) # label(fact_atMost__Int__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 49 (all V_is_2 all V_xs_2 all V_P_2 all T_b hAPP(hAPP(c_List_Omap(tc_prod(T_b,tc_Nat_Onat),T_b),c_Product__Type_Ofst(T_b,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_prod(T_b,tc_Nat_Onat)),V_P_2),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_prod(T_b,tc_Nat_Onat)),c_Nat_OSuc),c_Product__Type_Osnd(T_b,tc_Nat_Onat)))),hAPP(hAPP(c_List_Ozip(T_b,tc_Nat_Onat),V_xs_2),V_is_2))) = hAPP(hAPP(c_List_Omap(tc_prod(T_b,tc_Nat_Onat),T_b),c_Product__Type_Ofst(T_b,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_prod(T_b,tc_Nat_Onat)),V_P_2),c_Product__Type_Osnd(T_b,tc_Nat_Onat))),hAPP(hAPP(c_List_Ozip(T_b,tc_Nat_Onat),V_xs_2),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),V_is_2))))) # label(fact_sublist__shift__lemma__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 50 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 51 (all V_m all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)))) # label(fact_negative__zle) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 52 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__add__distrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 53 (all V_j_2 all V_i_2 all V_P_2 (c_List_Olist__ex(tc_Int_Oint,V_P_2,c_List_Oupto(V_i_2,V_j_2)) <-> -c_List_Oall__interval__int(hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint,c_fNot),V_P_2),V_i_2,V_j_2))) # label(fact_list__ex__iff__not__all__inverval__int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 54 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),V_xs_2)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))) # label(fact_length__insort) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 55 (all V_B_2 all V_a_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 56 (all V_x_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c))),V_r_2)),V_x_2)))) # label(fact_INF__INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 57 (all V_m all V_n hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m)))) # label(fact_div__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 58 (all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))))) # label(fact_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 59 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)))) # label(fact_setprod_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 60 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2))) # label(fact_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 61 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))),V_a) = V_a)) # label(fact_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 62 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_i_2),V_j_2) = hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_j_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_atLeastLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 63 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)))) # label(fact_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 64 (all V_n_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) # label(fact_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 65 (all V_k_2 all V_l_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) -> c_SetInterval_Oord_OgreaterThanAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanAtMost__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 66 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))))) # label(fact_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 67 (all V_q all T_a c_Nitpick_Oinverse__frac(T_a,V_q) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),hAPP(c_Nitpick_Odenom(T_a),V_q)),hAPP(c_Nitpick_Onum(T_a),V_q))) # label(fact_Nitpick_Oinverse__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 68 (all T_b hAPP(c_Set_OPow(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Pow__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 69 (all V_A_2 all T_b c_Relation_OId__on(T_b,V_A_2) = hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),c_fequal))),V_A_2))) # label(fact_Id__on__def_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 70 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> (V_m_2 = V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2)))))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 71 (all V_x_2 all V_xs_2 all T_b (hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)) = hAPP(c_List_Orev(T_b),V_xs_2) <-> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)) = V_xs_2)) # label(fact_rev__singleton__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 72 (all V_f_2 all T_c all T_b all V_S_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) != V_S_2 -> ((all B_A (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_A),V_S_2)) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,B_A))) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_S_2))))) # label(fact_inj__on__Inter) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 73 (all V_w_2 all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))))) # label(fact_less__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 74 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card__psubset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 75 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Big__Operators_Ocomm__monoid__big(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_F_2,V_g_2),V_A_2) = V_z_2))) # label(fact_comm__monoid__big_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 76 (all V_k_2 (c_Int_OBit0(V_k_2) = c_Int_OPls <-> c_Int_OPls = V_k_2)) # label(fact_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.63 77 (all V_B_2 all V_X_2 all V_f_2 all T_c all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,tc_fun(T_c,tc_HOL_Obool),V_f_2) -> c_Orderings_Oorder__class_Omono(T_b,tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_f_2)),V_X_2))),V_B_2))))) # label(fact_coinduct3__mono__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 78 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_x_2 all T_b hAPP(hAPP(hAPP(hAPP(c_New__Random__Sequence_Oneg__single(T_b),V_x_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Oneg__single(T_b,V_x_2)) # label(fact_neg__single__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 79 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> c_Nat__Transfer_Onat__set(c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_x_2,V_y_2)))) # label(fact_SetInterval_Otransfer__nat__int__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 80 (all V_v_2 all V_v_H_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H_2),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H_2),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_v_H_2)))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Int_Onat))))) # label(fact_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 81 (all V_g_2 all V_B_2 all T_c all V_A_2 all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,tc_prod(T_b,T_c),c_Groups_Otimes__class_Otimes(T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_g_2)),c_Groups_Oone__class_Oone(T_d)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_b,c_Groups_Otimes__class_Otimes(T_d)),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(hAPP(c_COMBC(T_b,T_d,tc_fun(tc_fun(T_c,tc_HOL_Obool),T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_d),tc_fun(T_d,tc_fun(tc_fun(T_c,tc_HOL_Obool),T_d)),T_b),c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d))),V_g_2)),c_Groups_Oone__class_Oone(T_d))),V_B_2)),c_Groups_Oone__class_Oone(T_d)),V_A_2))))) # label(fact_fold__image__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 82 (all T_b c_Wellfounded_Owf(T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))) # label(fact_wf__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 83 (all V_a_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_a_2),V_A_2)))) # label(fact_Inf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 84 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 85 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> V_a = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_b)))) # label(fact_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 86 (all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)))) # label(fact_setprod__infinite) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 87 (all V_i_2 all V_yq_2 all V_xq_2 all T_b hAPP(c_New__DSequence_Opos__union(T_b,V_xq_2,V_yq_2),V_i_2) = c_Lazy__Sequence_Oappend(T_b,hAPP(V_xq_2,V_i_2),hAPP(V_yq_2,V_i_2))) # label(fact_New__DSequence_Opos__union__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 88 (all V_C_2 all V_s_2 all T_d all V_B_2 all V_A_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_s_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_B_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_2),V_s_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_b),V_C_2))))))) # label(fact_rel__comp__subset__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 89 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__all(T_b,V_P_2,hAPP(c_List_Orev(T_b),V_xs_2)) <-> c_List_Olist__all(T_b,V_P_2,V_xs_2))) # label(fact_list__all__rev) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 90 (all V_w all T_a (class_Int_Onumber__ring(T_a) & class_Int_Oring__char__0(T_a) -> -c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w))))) # label(fact_iszero__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 91 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_Int_Ointrel)) <-> hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))) = hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_equiv__intrel__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 92 (all T_b c_Relation_Osym(T_b,c_Relation_OId(T_b))) # label(fact_sym__Id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 93 (all V_A_2 all V_F_2 all V_f_2 all T_b (c_Big__Operators_Osemilattice__big(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),V_A_2)))) # label(fact_semilattice__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 94 (all V_w_2 all T_b (class_Int_Onumber__ring(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Int_Oring__1__class_OInts(T_b))))) # label(fact_Ints__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 95 (all T_b all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 96 (all V_R_2 all T_b hAPP(c_Relation_OField(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),V_R_2)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),c_Relation_OField(T_b)),V_R_2))) # label(fact_Field__Union) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 97 (all V_b_H all V_a_H all V_b all V_a all T_b all T_a (hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a_H),V_b_H) -> -(V_a_H = V_a -> V_b != V_b_H))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 98 (all V_xs all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Otake(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_xs)) # label(fact_take__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 99 (all V_r all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_r)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_r),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))))) # label(fact_half__gt__zero) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 100 (all V_f_2 all T_b all T_c ((all B_x hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),B_x)) = B_x) <-> hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_surj__iff__all) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 101 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_b))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 102 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)))) # label(fact_sup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 103 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orotate1(T_a),V_xs)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) # label(fact_length__rotate1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 104 (all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2))) # label(fact_inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 105 (all V_b_2 all V_a_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_c,V_r_2),V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(V_r_2,V_b_2),V_a_2)))) # label(fact_conversepD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 106 (all V_xs_2 all V_x_2 all T_b ((hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Oinsert(T_b,V_x_2,V_xs_2) = V_xs_2) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Oinsert(T_b,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)))) # label(fact_List_Oinsert__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 107 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2))))))) # label(fact_Min_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 108 (all V_N_2 all V_h_2 all T_b (class_Lattices_Olattice(T_b) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice__class_OSup__fin(T_b,V_N_2))))))) # label(fact_Sup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 109 (all V_z all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_w))),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_z))) = hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z)))) # label(fact_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 110 (all V_X_2 all V_a_2 all V_f_2 all T_b (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))))))) # label(fact_coinduct__set) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 111 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_b,T_b,c_Fun_Oid(T_b)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_surj__id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 112 (all V_y_2 all V_x_2 all V_r_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),T_b),hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d))))),V_r_2)),V_x_2),V_y_2)))) # label(fact_INF__INT__eq2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 113 (all V_xs_2 all V_x_2 all V_P_2 all T_b (hBOOL(hAPP(V_P_2,V_x_2)) | c_List_Olist__ex(T_b,V_P_2,V_xs_2) <-> c_List_Olist__ex(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)))) # label(fact_list__ex__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 114 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 115 (all V_xs_2 all V_f_2 all T_c all V_n_2 all T_b hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2)) = hAPP(hAPP(c_List_Odrop(T_b),V_n_2),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_drop__map) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 116 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 117 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),V_u_2)))) # label(fact_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 118 (all V_i all V_j -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_i))) # label(fact_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 119 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 120 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_negative__zless__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 121 (all T_b (class_Power_Opower(T_b) -> c_Power_Opower__class_Opower(T_b) = c_Power_Opower_Opower(T_b,c_Groups_Oone__class_Oone(T_b),c_Groups_Otimes__class_Otimes(T_b)))) # label(fact_power__power__power) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 122 (all V_P_2 all V_n_2 ((all B_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2)) -> hBOOL(hAPP(V_P_2,B_m)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 123 (all V_r_H_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_r_H_2)))) # label(fact_wf__Int1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 124 (all V_D_2 all V_B_2 all T_c all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_B_2,B_x)),hAPP(V_D_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_C_2),V_D_2)))))) # label(fact_Sigma__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 125 (all V_q all T_a c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_a),V_q))),hAPP(c_Nitpick_Odenom(T_a),V_q))) = c_Nitpick_Ouminus__frac(T_a,V_q)) # label(fact_Nitpick_Ouminus__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 126 (all V_d all V_b all V_c all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_c),V_d))))))) # label(fact_sup__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 127 (all V_A_2 all V_f_2 all T_c all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> hAPP(c_Groups_Oabs__class_Oabs(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Oabs__class_Oabs(T_b)),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Oabs__class_Oabs(T_b)),V_f_2)),V_A_2))) # label(fact_abs__setsum__abs) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 128 (all V_X_2 all V_a_2 all V_f_2 all T_b all V_A_2 (V_A_2 = c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_A_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))))) # label(fact_def__coinduct__set) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 129 (all V_h_2 all V_B_2 all V_f_2 all T_c all T_b all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_h_2),V_f_2)),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_h_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))))) # label(fact_setsum__reindex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 130 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(T_b,c_Transitive__Closure_Otrancl(T_b,V_r_2)))) # label(fact_wf__trancl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 131 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n))) # label(fact_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 132 (all V_xs_2 all V_n_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Odrop(T_b),V_n_2),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_in__set__dropD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 133 (all V_f_2 all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Oorder__class_Omono(tc_Nat_Onat,T_b,V_f_2) <-> (all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,B_n)),hAPP(V_f_2,hAPP(c_Nat_OSuc,B_n)))))))) # label(fact_mono__iff__le__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 134 (all V_zs all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Oappend(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_zs))),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_ys),hAPP(hAPP(c_List_Odrop(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_zs))) = hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_zs)) # label(fact_zip__append1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 135 (all V_x_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Wellfounded_Oacc(T_b,V_r_2))) <-> hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_r_2)),V_x_2)))) # label(fact_accp__acc__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 136 (all V_B_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_folding__image__simple_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 137 (all V_x_2 -(all B_y (V_x_2 = hAPP(c_Int_OAbs__Integ,B_y) -> -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B_y),c_Int_OInteg))))) # label(fact_Abs__Integ__cases) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 138 (all V_v all V_w ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))))) # label(fact_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 139 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_x = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)))) # label(fact_min__max_Osup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 140 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2))) # label(fact_INT__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 141 (all V_A_2 all V_x_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2)))))) # label(fact_Sup__upper) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 142 (all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2)) # label(fact_inv__f__f) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 143 (all V_B_2 all V_A_2 all T_b all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_OCollect(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c)),c_fequal)),V_B_2))))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 144 (all V_A_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)) = V_A_2) # label(fact_Collect__mem__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 145 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = V_B_2))) # label(fact_INT__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 146 (all V_P_2 ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,B_x))))) <-> (all B_x1 hBOOL(hAPP(V_P_2,B_x1))))) # label(fact_all__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 147 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_DiffD1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 148 (all V_xss_2 all V_xs_2 all T_b hAPP(c_List_Ofoldl(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Oappend(T_b),V_xs_2),V_xss_2) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),c_List_Oconcat(T_b,V_xss_2))) # label(fact_foldl__conv__concat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 149 (all V_y_2 all V_x_2 all V_r_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),T_b),hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d))))),V_r_2)),V_x_2),V_y_2)))) # label(fact_SUP__UN__eq2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 150 (all V_x_2 all V_A_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_A_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_x))))))))) # label(fact_below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 151 (all V_k_2 all V_C_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),V_C_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_C_2))) -> ((all B_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_c),V_C_2)) -> hAPP(c_Finite__Set_Ocard(T_b),B_c) = V_k_2)) -> ((all B_c1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_c1),V_C_2)) -> (all B_c2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_c2),V_C_2)) -> (B_c2 != B_c1 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),B_c1),B_c2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hAPP(c_Finite__Set_Ocard(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_C_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_fun(T_b,tc_HOL_Obool)),V_C_2))))))) # label(fact_card__partition) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 152 (all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_x))) # label(fact_inf__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 153 (all V_q all V_p c_Divides_Odivmod__int(V_p,V_q) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_p),V_q)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_p,V_q))) # label(fact_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 154 (all V_n all V_b all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))) # label(fact_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 155 (all V_A_2 all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)))))) # label(fact_surj__Compl__image__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 156 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> c_Relation_Otrans(tc_List_Olist(T_b),c_List_Olexord(T_b,V_r_2)))) # label(fact_lexord__transI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 157 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremove1(T_a,V_x,c_List_Oremdups(T_a,V_xs)) = c_List_Oremdups(T_a,c_List_Oremove1(T_a,V_x,V_xs)))) # label(fact_remove1__remdups) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 158 (all V_P_2 all V_f_2 all T_b all T_c c_Random__Sequence_Omap(T_c,T_b,V_f_2,V_P_2) = c_Random__Sequence_Obind(T_c,T_b,V_P_2,hAPP(c_Fun_Ocomp(T_b,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))))),T_c,c_Random__Sequence_Osingle(T_b)),V_f_2))) # label(fact_Random__Sequence_Omap__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 159 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))))) # label(fact_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 160 (all V_xs_2 all V_f_2 all T_c all T_b c_List_Oremdups(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = c_List_Oremdups(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Oremdups(T_c,V_xs_2)))) # label(fact_remdups__map__remdups) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 161 (all V_z_2 all V_y_2 all V_f_2 all V_r_2 all T_c all T_b (c_Equiv__Relations_Ocongruent(T_b,T_c,V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_z_2)),V_r_2)) -> hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_z_2)))) # label(fact_congruentD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 162 (all V_A_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 <-> (exists B_x hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 163 (all V_f_2 all V_A_2 all V_a_2 all T_b ((hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))) # label(fact_setsum__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 164 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_c) & class_Orderings_Olinorder(T_b) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))))))) # label(fact_strict__mono__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 165 (all V_w all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 166 (all V_a_2 all V_b_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)))))) # label(fact_subset__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 167 (all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Obij__betw(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_bij__imp__bij__inv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 168 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 169 (all T_b c_Equiv__Relations_Oequivp(T_b,c_fequal)) # label(fact_identity__equivp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 170 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_x))))))) # label(fact_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 171 (all V_a_2 all V_f_2 all V_A_2 all T_b all T_c (class_Fields_Ofield(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hAPP(V_f_2,V_a_2) != c_Groups_Ozero__class_Ozero(T_c) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2))))))) # label(fact_setprod__diff1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.64 172 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 173 (all V_k_2 all V_R_2 all V_t_2 all T_c all V_s_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_s_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_t_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_t_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_s_2))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),V_R_2),B_x)))) = hAPP(V_k_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_Nat_Onat,T_b),c_Finite__Set_Ocard(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),c_Set_OCollect(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_t_2)))),V_R_2)))),V_s_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_k_2),V_t_2))))) # label(fact_setsum__multicount__gen) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 174 (all V_b_2 all V_c_2 all V_a_2 all T_b (class_Groups_Oordered__ab__semigroup__add__imp__le(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 175 (all V_v (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 176 (all T_b c_Relation_OId__on(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) # label(fact_Id__on__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 177 (all V_A_2 all V_g_2 all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Oplus__class_Oplus(T_b)),V_f_2)),V_g_2)),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_g_2),V_A_2)))) # label(fact_setsum__addf) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 178 (all V_A_2 all V_B_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 179 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 180 (all V_y_2 all V_x_2 all T_b (class_Int_Onumber__ring(T_b) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) <-> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2))))))) # label(fact_eq__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 181 (all V_v_2 all V_c_2 all V_b_2 all V_a_2 all T_b all T_c all T_d (V_c_2 = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_a_2),V_b_2) -> hAPP(V_a_2,hAPP(V_b_2,V_v_2)) = hAPP(V_c_2,V_v_2))) # label(fact_o__eq__dest__lhs) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 182 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = V_n))) # label(fact_less__SucE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 183 (all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OSup__class_OSup(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(T_b))) # label(fact_Sup__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 184 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2)),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))))) # label(fact_gfp__lemma2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 185 (all V_zs_2 all V_xs_2 all V_ys_2 all T_b (hAPP(hAPP(c_List_Oappend(T_b),V_zs_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),V_xs_2) <-> V_zs_2 = V_ys_2)) # label(fact_append__same__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 186 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b)))))) # label(fact_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 187 (all T_b (class_Orderings_Olinorder(T_b) -> c_Orderings_Oord__class_Omin(T_b) = c_Orderings_Oord_Omax(T_b,hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_b))))) # label(fact_dual__max) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 188 (all V_b all V_a ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))) # label(fact_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 189 (all T_b all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 190 (all V_r_2 all T_b all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Relation_ORange(T_c,T_b),V_r_2) <-> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)))) # label(fact_Range__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 191 (all T_c all T_b all V_b_2 all V_a_2 all V_f1_2 (hBOOL(hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))))) # label(fact_prod__caseI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 192 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 193 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))) # label(fact_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 194 (all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_b (hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),c_List_Olist_ONil(T_b))) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b))) <-> V_x_2 = V_y_2 & V_ys_2 = V_xs_2)) # label(fact_append1__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 195 (all V_A_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2)))) -> c_Relation_Orefl__on(T_b,V_A_2,V_r_2)))) # label(fact_refl__onI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 196 (all V_k hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k) = V_k) # label(fact_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 197 (all V_xs_2 all V_P_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_Olist_ONil(T_b) = c_List_OdropWhile(T_b,V_P_2,V_xs_2))) # label(fact_dropWhile__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 198 (all V_x_2 all V_i_2 all V_xs_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_x_2))),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_set__update__subset__insert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 199 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)))))) # label(fact_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 200 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> V_x_2 != V_z_2) -> -(all B_y all B_m (hAPP(c_Nat_OSuc,B_m) = V_n_2 -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_y)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),B_m),V_R_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_z_2)),V_R_2)))))))) # label(fact_rel__pow__E) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 201 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))),c_List_Olistrel(T_b,V_r_2))) -> -(all B_x all B_xs (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),B_x),B_xs) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),V_y_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),B_xs),V_ys_2)),c_List_Olistrel(T_b,V_r_2)))))))) # label(fact_listrel__Cons2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 202 (all V_xs_2 all T_b all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Otake(T_b),V_m_2),V_xs_2))),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2)))))) # label(fact_set__take__subset__set__take) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 203 (all V_z_2 all T_b (class_Int_Oring__char__0(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2) <-> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_z_2))) # label(fact_of__int__0__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 204 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2))))))))) # label(fact_min__max_OSup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 205 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_B_2))))))) # label(fact_Inf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 206 (all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))))) # label(fact_atLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 207 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)))))) # label(fact_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 208 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 209 (all T_c all V_A_2 all V_a_2 all V_f_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_a_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_A_2))))) # label(fact_vimageI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 210 (all V_z all V_y all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 211 (all V_r_1 all V_y all V_b_1 all V_a_1 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_y)),V_r_1) = V_a_1 -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_r_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r_1),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_1),V_b_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_1))) -> (V_b_1 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_1),V_b_1) = V_y)))) # label(fact_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 212 (all V_f_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_Oimage(T_b,T_b,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_b,T_b,V_f_2,V_A_2)))) # label(fact_finite__surj__inj) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 213 (all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> V_x_2 = hAPP(c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),hAPP(V_f_2,V_x_2))))) # label(fact_the__inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 214 (all V_B_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_B_2,V_a_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UN__upper) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 215 (all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_b))) # label(fact_setprod__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 216 (all V_j all V_i all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,hAPP(c_List_Orev(T_a),V_xs)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j)),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)))))))) # label(fact_sorted__rev__nth__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 217 (all V_ys all V_y all V_xs all V_x all T_a c_List_Osplice(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Osplice(T_a,V_xs,V_ys)))) # label(fact_splice_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 218 (all V_P_2 all V_xs_2 all T_b all V_n_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i)))))) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> -hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_n_2)))) -> c_List_OtakeWhile(T_b,V_P_2,V_xs_2) = hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2)))) # label(fact_takeWhile__eq__take__P__nth) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 219 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 220 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 221 (all V_m all V_n all V_k c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) # label(fact_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 222 (all V_n_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)))))) # label(fact_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 223 (all V_b_2 all V_r_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Wellfounded_Oacc(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_a_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Wellfounded_Oacc(T_b,V_r_2)))))) # label(fact_acc__downwards) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 224 (all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> V_A_2 = hAPP(c_Set_Oimage(T_b,T_c,V_Rep_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_type__definition_ORep__range) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 225 (all V_g_2 all T_d all T_e all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) -> (c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_e,T_d,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool))) -> c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_prod(T_c,T_e),tc_prod(T_b,T_d),c_Product__Type_Omap__pair(T_c,T_b,T_e,T_d,V_f_2,V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)))))) # label(fact_map__pair__surj) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 226 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b all T_d hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),hAPP(V_f_2,V_x_2)),V_y_2) = hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_x_2),V_y_2))) # label(fact_apfst__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 227 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)))) # label(fact_min__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 228 (all V_x_2 all V_P_2 all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) -> (hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_P_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),hAPP(c_List_Ofilter(T_c,V_P_2),V_xs_2)))))) # label(fact_filter__insort) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 229 (all V_l (c_List_Onat__list(V_l) -> c_List_Oembed__list(c_List_Oreturn__list(V_l)) = V_l)) # label(fact_transfer__nat__int__list__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 230 (all V_M_2 all V_u_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_u_2),hAPP(V_M_2,B_i))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_u_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),V_M_2)))))) # label(fact_le__INFI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 231 (all V_X_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_C_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),V_X_2))))) # label(fact_InterE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 232 (all V_h_2 all V_g_2 all T_e all V_f_2 all T_b all T_c all T_d hAPP(c_Fun_Ocomp(T_e,T_c,T_b,hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_h_2) = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_2),hAPP(c_Fun_Ocomp(T_e,T_d,T_b,V_g_2),V_h_2))) # label(fact_o__assoc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 233 (all V_xs_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> V_xs_2 = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_a_2),c_List_Oremove1(T_b,V_a_2,V_xs_2)))))) # label(fact_insort__remove1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 234 (all V_y_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))))) # label(fact_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 235 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) & V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 236 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)))) # label(fact_IntD2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 237 (all V_x_2 all V_xs_2 all T_b all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_n_2),V_x_2)))))) # label(fact_set__update__memI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 238 (all V_A_2 all T_b hAPP(c_Set_OCollect(tc_List_Olist(T_b)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),c_List_Oset(T_b))),V_A_2)) = c_List_Olists(T_b,V_A_2)) # label(fact_lists__eq__set) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 239 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_folding__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 240 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Compl__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 241 (all V_P_2 all T_b c_List_OtakeWhile(T_b,V_P_2,c_List_Olist_ONil(T_b)) = c_List_Olist_ONil(T_b)) # label(fact_takeWhile_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 242 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 243 (all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),V_A_2))))))) # label(fact_Sup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 244 (all V_x_2 all V_A_2 all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2)) -> (c_Fun_Oinj__on(T_d,T_b,V_g_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2)))) -> hAPP(hAPP(c_Fun_Ocomp(T_b,T_d,T_c,c_Fun_Othe__inv__into(T_d,T_b,V_A_2,V_g_2)),c_Fun_Othe__inv__into(T_b,T_c,hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2),V_f_2)),V_x_2) = hAPP(c_Fun_Othe__inv__into(T_d,T_c,V_A_2,hAPP(c_Fun_Ocomp(T_b,T_c,T_d,V_f_2),V_g_2)),V_x_2))))) # label(fact_the__inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 245 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),V_a_2))))) # label(fact_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 246 (all T_b c_Relation_Oantisym(T_b,c_Relation_OId(T_b))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 247 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),c_Orderings_Obot__class_Obot(T_b)),V_A_2)))) # label(fact_Sup__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 248 (all V_r_2 all V_s_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Relation_OId(T_b)),V_s_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_s_2)),V_r_2)),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_s_2))))) # label(fact_rtrancl__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 249 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_atLeastAtMost__upto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 250 (all V_zs_2 all V_ys_2 all V_xs_2 all T_b (V_zs_2 = V_ys_2 <-> hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_zs_2) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) # label(fact_same__append__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 251 (all V_a all T_a (class_Power_Opower(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 252 (all V_B_2 all V_A_2 all V_l_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_l_2),c_List_Olists(T_b,V_A_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_l_2),c_List_Olists(T_b,V_B_2))) -> hBOOL(hAPP(c_List_Olistsp(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))),V_l_2))))) # label(fact_lists__IntI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 253 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)))))) # label(fact_psubset__card__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 254 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))) # label(fact_min__max_OInf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 255 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))))) # label(fact_atLeastLessThan__add__Un) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 256 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)))))) # label(fact_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 257 (all V_zs_2 all V_ys_2 all T_c all V_xs_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) -> (hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_b),c_Product__Type_Ofst(T_b,T_c)),V_zs_2) = V_xs_2 & hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_c),c_Product__Type_Osnd(T_b,T_c)),V_zs_2) = V_ys_2 <-> V_zs_2 = hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)))) # label(fact_zip__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 258 (all V_B_2 all V_A_2 all T_d all V_f_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Set_Oimage(T_c,T_b,V_f_2)),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_image__UN) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 259 (all V_f_2 all V_xq_2 all T_b all T_c c_Lazy__Sequence_Oflat(T_b,c_Lazy__Sequence_Omap(T_c,tc_Lazy__Sequence_Olazy__sequence(T_b),V_f_2,V_xq_2)) = hAPP(c_Lazy__Sequence_Obind(T_c,T_b,V_xq_2),V_f_2)) # label(fact__05) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 260 (all V_a_2 all V_b_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)))) # label(fact_atLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 261 (all V_b_2 all V_a_2 all V_c_2 all T_b all T_d all T_c hAPP(hAPP(V_c_2,V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_Ointernal__split(T_c,T_d,T_b),V_c_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 262 (all V_x_2 all V_f_2 all V_A_2 all T_c all T_b hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),V_x_2) = c_Hilbert__Choice_OEps(T_b,hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),V_x_2)))) # label(fact_inv__into__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 263 (all V_g_2 all V_f_2 all T_e all T_c all T_d all T_b hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_prod(T_d,T_e)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,T_e),tc_fun(T_c,tc_prod(T_d,T_e))),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(tc_fun(T_c,T_e),tc_fun(T_c,tc_prod(T_d,T_e))),T_b),c_COMBB(T_e,tc_prod(T_d,T_e),T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_e,tc_prod(T_d,T_e)),T_b),c_Product__Type_OPair(T_d,T_e)),V_f_2))),V_g_2)) = c_Product__Type_Omap__pair(T_b,T_d,T_c,T_e,V_f_2,V_g_2)) # label(fact_map__pair__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 264 (all V_y_2 all V_f_2 all T_c all V_A_2 all V_x_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_fun__upd__image) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 265 (all V_y_2 all V_R_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_Relation_ODomain(T_b,T_b),V_R_2))) -> (V_x_2 = V_y_2 <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_R_2)))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 266 (all V_Y_2 all T_b hAPP(c_Set_Ovimage(T_b,T_b,c_COMBI(T_b)),V_Y_2) = V_Y_2) # label(fact_vimage__ident) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 267 (all V_A_2 all V_B_2 all V_c_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnCI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 268 (all V_xs_2 all V_P_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),c_List_OdropWhile(T_b,V_P_2,V_xs_2)))) -> -hBOOL(hAPP(V_P_2,B_x)))) -> hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2) = c_List_OtakeWhile(T_b,V_P_2,V_xs_2))) # label(fact_takeWhile__eq__filter) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 269 (all V_j_2 all V_i_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(tc_Int_Oint,T_b),V_f_2),c_List_Oupto(V_i_2,V_j_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),V_f_2),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,V_j_2))))) # label(fact_setsum__set__upto__conv__listsum__int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 270 (all V_ys_2 all V_xs_2 all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_b),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)),hAPP(c_List_Ofilter(T_b,V_P_2),V_ys_2))) # label(fact_filter__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 271 (all V_r_2 all T_b (class_Int_Onumber__ring(T_b) & class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_r_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_r_2))))) # label(fact_half__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 272 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (exists B_b hBOOL(hAPP(hAPP(c_member(T_b),B_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))))) # label(fact_psubset__imp__ex__mem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 273 (all V_B_2 all V_f_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Set_Ovimage(T_b,T_c,V_f_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Set_Oinsert(T_c)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2)) # label(fact_vimage__eq__UN) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 274 (all V_xs_2 all V_P_2 all T_b all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_b),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_j_2))) # label(fact_takeWhile__nth) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 275 (all V_x_2 all V_X_2 all V_a_2 all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_X_2)),V_x_2)) -> -(all B_a all B_A (hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_X_2) = hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_A) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,B_a,B_A),V_x_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A))))))) # label(fact_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 276 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> hAPP(c_Set_Oimage(T_c,T_b,c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) = V_A_2)) # label(fact_the__inv__into__onto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 277 (all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)),V_r_2))))) # label(fact_refl__on__comp__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 278 (all V_x_2 all V_A_2 all V_a_2 all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,V_a_2,V_A_2),V_x_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),V_x_2))))) # label(fact_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 279 (all V_ys_2 all V_xs_2 all V_f_2 all T_d all T_c all T_b hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_c),tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_prod(T_b,T_c)),T_d),c_Product__Type_OPair(T_b,T_c)),V_f_2))),hAPP(hAPP(c_List_Ozip(T_d,T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Ozip(T_b,T_c),hAPP(hAPP(c_List_Omap(T_d,T_b),V_f_2),V_xs_2)),V_ys_2)) # label(fact_zip__map1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 280 (all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))))) # label(fact_atLeastatMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 281 (all V_m all V_n (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_m))) -> V_m = V_n))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 282 (all V_X_2 hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_FunDef_Omin__weak))) # label(fact_wmin__emptyI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 283 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 284 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 285 (all V_k all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_k)),V_x))))) # label(fact_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 286 (all T_b c_Fun_Obij__betw(T_b,T_b,c_Fun_Oid(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_bij__id) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 287 (all T_a (class_Int_Onumber__ring(T_a) -> -c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_not__iszero__Numeral1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 288 (all V_P_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_P_2))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),V_P_2)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_finite__Collect__not) # label(axiom) # label(non_clause). [assumption]. 8.62/8.65 289 (all V_A_2 all V_f_2 all T_c all T_b (class_Fields_Olinordered__field(T_b) -> hAPP(c_Groups_Oabs__class_Oabs(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Oabs__class_Oabs(T_b)),V_f_2)),V_A_2))) # label(fact_abs__setprod) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 290 (all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Orderings_Otop__class_Otop(T_a)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_compl__top__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 291 (all V_c_2 all V_R_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_R_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_R_2)))))) # label(fact_r__r__into__trancl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 292 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> V_x_2 = c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)))))) # label(fact_Inf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 293 (all T_d all T_c all V_b_2 all V_a_2 all V_c_2 all V_z_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(hAPP(V_c_2,V_a_2),V_b_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_fun(T_b,tc_HOL_Obool)),V_c_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)))))) # label(fact_mem__splitI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 294 (all V_m_2 all V_n_2 (hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 295 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),V_y_2))))) # label(fact_le__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 296 (all V_A_2 all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2)) = c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))) # label(fact_Sup__insert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 297 (all V_z_2 all V_y_2 all V_x_2 all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(V_R_2,V_y_2),V_z_2)) -> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_z_2)))))) # label(fact_equivp__transp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 298 (all V_A_2 all T_b hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),c_List_Olist_ONil(T_b)))) # label(fact_listsp_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 299 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y)))) # label(fact_divide__add__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 300 (all V_i_2 all V_x_2 all V_A_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_x_2))),V_A_2))))) # label(fact_set__update__subsetI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 301 (all V_y all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_y) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) # label(fact_list__update__code_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 302 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))) # label(fact_bij__image__Compl__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 303 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2))))) # label(fact_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 304 (all V_s_2 all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> (c_Relation_Osym(T_b,V_s_2) -> c_Relation_Osym(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_sym__Int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 305 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)))) # label(fact_min__max_Osup__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 306 (all V_a all T_a (class_Orderings_Olinorder(T_a) -> V_a = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_a))) # label(fact_min__max_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 307 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))))) # label(fact_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 308 (all V_b_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))))) # label(fact_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 309 (all V_d_2 c_Int_Oint__ge__less__than(V_d_2) = hAPP(c_Set_OCollect(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_d_2)))),c_Orderings_Oord__class_Oless(tc_Int_Oint))))) # label(fact_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 310 (all V_y_2 all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) -> (V_y_2 = V_x_2 <-> hAPP(V_Abs_2,V_x_2) = hAPP(V_Abs_2,V_y_2)))))) # label(fact_type__definition_OAbs__inject) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 311 (all V_h_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_COMBS(T_b,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Groups_Otimes__class_Otimes(T_c)),V_g_2)),V_h_2)),c_Groups_Oone__class_Oone(T_c)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_h_2),c_Groups_Oone__class_Oone(T_c)),V_A_2))))) # label(fact_fold__image__distrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 312 (all V_P_2 all T_b all T_c (class_Enum_Oenum(T_b) & class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_Enum_Oenum__class_Oenum__ex(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_HOL_Obool,T_c),V_P_2)),c_Product__Type_OPair(T_b,T_c))))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(tc_prod(T_b,T_c)),V_P_2))))) # label(fact_enum__ex__prod__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 313 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 314 (all V_x all V_j all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> (V_i != V_j -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j)) & (V_j = V_i -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j) = V_x))) # label(fact_nth__list__update) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 315 (all V_B_2 all V_A_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))) # label(fact_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 316 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_a))) # label(fact_min__max_Osup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 317 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_2) & class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_fun(T_2,T_1)))) # label(arity_fun__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 318 (all V_n all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_m))) # label(fact_diff__le__self) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 319 (all V_xs_2 all V_P_2 all T_b (hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2) = c_List_Olist_ONil(T_b) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> -hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_filter__empty__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 320 (all V_r_2 all V_xs_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_xs_2)),c_List_Olistrel1(T_b,V_r_2)))) # label(fact_not__Nil__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 321 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_set__upto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 322 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_B_2)))))) # label(fact_fold__image__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 323 (all V_x_2 all T_c all T_b (class_Orderings_Obot(T_b) -> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,T_b)),V_x_2) = c_Orderings_Obot__class_Obot(T_b))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 324 (all V_A_2 all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Oimage(T_b,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),V_A_2)) = V_A_2)) # label(fact_image__surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 325 (all V_a_2 all T_b all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 326 (all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) <-> c_Relation_Orefl__on(T_b,V_A_2,V_r_2))) # label(fact_refl__on__converse) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 327 (all V_a_2 all V_xs_2 all T_b (class_Groups_Oab__semigroup__add(T_b) -> c_List_Ofoldr(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_xs_2,V_a_2) = hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_xs_2))) # label(fact_foldr__conv__foldl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 328 (all V_C_2 all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 329 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 330 (all V_B_2 all V_A_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(V_B_2,B_x))) <-> c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_INTER__UNIV__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 331 (all V_r_2 all V_b_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_OImage(T_b,T_c,V_r_2),V_A_2)))))) # label(fact_rev__ImageI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 332 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_x_2)),hAPP(V_F_2,V_A_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2))))) # label(fact_folding_Ocommute__comp_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 333 (all V_r_2 all V_b_2 all V_a_2 all T_b all T_c hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),hAPP(c_Relation_ORange(T_c,T_b),V_r_2)) = hAPP(c_Relation_ORange(T_c,T_b),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_a_2),V_b_2)),V_r_2))) # label(fact_Range__insert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 334 (all V_j_2 all V_i_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_List_Oupto__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_i_2),V_j_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),V_j_2)) -> c_List_Olist_ONil(tc_Int_Oint) = c_List_Oupto(V_i_2,V_j_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),V_j_2)) -> hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),V_i_2),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2)) = c_List_Oupto(V_i_2,V_j_2)))) # label(fact_upto_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 335 (all V_p all V_m ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m)) & (V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))))))) # label(fact_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 336 (all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Int_Opred(V_k)) # label(fact_pred__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 337 (all V_v_H ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))))) # label(fact_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 338 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2))))))) # label(fact_Inf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 339 (all V_v_2 all V_w_2 all T_b (class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(hAPP(c_HOL_OLet(T_b,T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_b),c_Groups_Otimes__class_Otimes(T_b)),hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)))),c_COMBI(T_b))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> c_Groups_Oone__class_Oone(T_b) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))))) # label(fact_power__number__of__odd__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 340 (all V_xs_2 all V_f_2 all T_b hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(T_b,tc_Nat_Onat),V_f_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)) = c_List_Olist_Olist__size(T_b,V_f_2,V_xs_2)) # label(fact_list__size__conv__listsum) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 341 (all V_r_2 all V_xs_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),c_List_Olist_ONil(T_b))),c_List_Olex(T_b,V_r_2)))) # label(fact_Nil2__notin__lex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 342 (all V_b_2 all V_a_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(V_r_2,V_b_2),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_c,V_r_2),V_a_2),V_b_2)))) # label(fact_conversep__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 343 (all V_xs_2 all V_P_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))) # label(fact_length__takeWhile__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 344 (all V_n all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n))) # label(fact_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 345 (all V_P_2 all T_c all T_b c_Hilbert__Choice_OEps(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_HOL_Obool,T_c),V_P_2)),c_Product__Type_OPair(T_b,T_c)))) = c_Hilbert__Choice_OEps(tc_prod(T_b,T_c),V_P_2)) # label(fact_split__paired__Eps) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 346 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))))) # label(fact_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 347 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2))))) # label(fact_PowI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 348 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_A_2) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Compl__partition2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 349 (all V_p_2 all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_p_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_p_2)),hAPP(hAPP(c_List_Onth(T_c),V_ys_2),V_p_2)))))) # label(fact_list__all2__nthD2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 350 (all V_r all V_q all V_r_H all V_q_H all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),V_q))))))) # label(fact_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 351 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))))) # label(fact_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 352 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)))))) # label(fact_Sup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 353 (all T_b all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 354 (all V_i_2 all V_j_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_j_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),V_j_2))))) # label(fact_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 355 (all V_l_2 all V_k_2 (-(hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2))) <-> c_SetInterval_Oord_OgreaterThanAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 356 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)))) # label(fact_min__max_Osup__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 357 (all V_n_2 all V_d_2 all V_a_2 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),V_d_2))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_d_2))))) # label(fact_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 358 (all V_z_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)))))) # label(fact_Inf__greatest) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 359 (all V_ys_2 all V_xs_2 all V_P_2 all T_b (c_List_Olist__ex(T_b,V_P_2,V_xs_2) | c_List_Olist__ex(T_b,V_P_2,V_ys_2) <-> c_List_Olist__ex(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)))) # label(fact_list__ex__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 360 (all V_r_2 all V_x_2 all T_b ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_x_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),B_y),c_Wellfounded_Oacc(T_b,V_r_2))))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Wellfounded_Oacc(T_b,V_r_2))))) # label(fact_acc_OaccI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 361 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (V_y != V_x -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 362 (all V_xs_2 all T_b all V_x_2 all V_P_2 (-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_Oremove1(T_b,V_x_2,hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))) # label(fact_remove1__filter__not) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 363 (all V_f_2 all V_n_2 all T_b hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),hAPP(c_Nat_OSuc,V_n_2)),V_f_2) = hAPP(c_Fun_Ocomp(T_b,T_b,T_b,V_f_2),hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_n_2),V_f_2))) # label(fact_funpow_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 364 (all V_l all V_k c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit0(V_k)),V_l)) # label(fact_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 365 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 366 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> V_x_2 = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_OInf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 367 (all V_a3_2 all V_a1_2 all V_s_2 all V_r_2 all T_d all T_c all T_b ((exists B_b (hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_b)) & hBOOL(hAPP(hAPP(V_s_2,B_b),V_a3_2)))) <-> hBOOL(hAPP(hAPP(c_Predicate_Opred__comp(T_b,T_c,T_d,V_r_2,V_s_2),V_a1_2),V_a3_2)))) # label(fact_pred__comp_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 368 (all V_B_2 all V_A_2 all V_a_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 369 (all V_n all V_a all T_a (class_Power_Opower(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 370 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (V_c = V_b -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 371 (all V_Q_2 all V_P_2 (hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2)) | -hBOOL(V_P_2))) # label(help_c__fdisj__1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 372 (all V_ys_2 all V_g_2 all T_d all V_xs_2 all V_f_2 all T_b all T_c (hAPP(hAPP(c_List_Omap(T_d,T_b),V_g_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2))) # label(fact_map__eq__imp__length__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 373 (all V_A_2 all V_f_2 all T_b hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_b),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2)) # label(fact_int__setprod) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 374 (all V_xa_2 all V_x_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_xa_2)),c_List_Olistrel(T_b,V_r_2))) <-> c_List_Olistrelp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_r_2),V_x_2,V_xa_2))) # label(fact_listrelp__listrel__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 375 (all V_ys_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,c_List_Olist_ONil(T_b),V_ys_2) <-> c_List_Olist_ONil(T_c) = V_ys_2)) # label(fact_list__all2__Nil) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 376 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))),V_b))))) # label(fact_gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 377 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) -> V_n = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n)) # label(fact_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 378 (all V_B_2 all V_A_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 379 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2)) -> V_d_2 = V_b_2))))) # label(fact_atLeastLessThan__inj_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 380 (all V_a all V_b all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b))))) # label(fact_nonzero__minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 381 (all V_xs_2 all V_n_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2))),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_set__take__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 382 (all T_b c_Relation_Oantisym(T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))) # label(fact_antisym__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 383 (all V_n all V_i hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_i)),V_n) = c_Int_Onat__aux(V_i,V_n)) # label(fact_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 384 (all T_a (class_Int_Onumber__ring(T_a) -> c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)))) # label(fact_iszero__Numeral0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 385 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> V_x_2 = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) # label(fact_Min_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 386 (all V_N_2 all V_h_2 all T_b (class_Lattices_Olattice(T_b) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,c_Big__Operators_Olattice__class_OInf__fin(T_b,V_N_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2))))))) # label(fact_Inf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 387 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls)),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)))))) # label(fact_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 388 (all V_B_H_2 all V_B_2 all V_A_H_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_A_H_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> (hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2) = V_B_H_2 -> c_Fun_Obij__betw(T_b,T_c,V_f_2,V_B_2,V_B_H_2))))) # label(fact_bij__betw__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 389 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.66 390 (all V_xs all T_a all V_ys ((c_List_Olist_ONil(T_a) != V_ys -> c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = c_List_Olast(T_a,V_ys)) & (c_List_Olist_ONil(T_a) = V_ys -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))))) # label(fact_last__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 391 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Odrop(T_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_xs)),hAPP(hAPP(c_List_Otake(T_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_xs)) = hAPP(c_List_Orotate(T_a,V_n),V_xs)) # label(fact_rotate__drop__take) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 392 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_divide__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 393 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_u_2))))) # label(fact_ivl__disj__un_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 394 (all V_k_2 (c_Int_OMin = V_k_2 <-> c_Int_OMin = c_Int_OBit1(V_k_2))) # label(fact_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 395 (all V_A_2 all V_y_2 all T_c all T_b (class_Rings_Osemiring__1(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))),V_y_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(c_COMBK(T_b,T_c),V_y_2)),V_A_2))) # label(fact_setsum__constant) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 396 (all V_a all T_a (class_Rings_Omult__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 397 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_negative__zle__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 398 (all V_k_2 all V_R_2 all V_T_2 all T_c all V_S_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_T_2)) -> V_k_2 = hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),V_R_2),B_x)))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_Nat_Onat,T_b),c_Finite__Set_Ocard(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),c_Set_OCollect(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_T_2)))),V_R_2)))),V_S_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(T_c),V_T_2)))))) # label(fact_setsum__multicount) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 399 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2) <-> V_m_2 = V_n_2)) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 400 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2 & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2)) # label(fact_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 401 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))) # label(fact_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 402 (all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> V_x_2 = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_min__max_OSup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 403 (all V_n hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_n) = V_n) # label(fact_nat__size) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 404 (all V_k c_Int_OBit1(V_k) != c_Int_OPls) # label(fact_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 405 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Nitpick_Onorm__frac__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_a all B_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Nitpick_Onorm__frac__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b))) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B_a)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B_b)))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 406 (all V_n_2 all V_v_2 hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_OSuc,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Int_Oint),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Int_Onat)),V_n_2))))) # label(fact_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 407 (all V_a all V_b all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)))))) # label(fact_min__max_Ole__supI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 408 (all V_b all V_a all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),V_c) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)))) # label(fact_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 409 (all V_B_2 all V_A_2 all V_C_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_C_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_inj__on__image__set__diff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 410 (all T_c all V_x_2 all V_f_2 all T_b hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_rangeI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 411 (all V_z all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 412 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_a))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)))))) # label(fact_finite__SigmaI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 413 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 414 (all V_c all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b)))))))) # label(fact_divide__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 415 (all V_r_2 all T_b ((all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))) -> c_Wellfounded_Oacyclic(T_b,V_r_2))) # label(fact_acyclicI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 416 (all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) -> (c_Fun_Oinj__on(T_b,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_b,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_finite__UNIV__inj__surj) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 417 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 418 (all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (c_Relation_Osym(T_b,V_r_2) -> (c_Relation_Otrans(T_b,V_r_2) -> c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2))))) # label(fact_equivI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 419 (all V_a_2 all V_w_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_a_2))))) # label(fact_divide__le__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 420 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2))))) # label(fact_image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 421 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_k_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2))))) # label(fact_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 422 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b = V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_a)))))) # label(fact_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 423 (all V_us all V_ys all V_zs all V_xs1 all V_xs all T_a (V_zs = hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_xs1) -> (hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_us) = V_ys -> hAPP(hAPP(c_List_Oappend(T_a),V_zs),V_us) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_append__eq__appendI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 424 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)))) # label(fact_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 425 (all T_b c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Inter__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 426 (all V_z_2 all V_h_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hAPP(V_h_2,B_x) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_h_2),V_z_2),V_A_2))))) # label(fact_fold__image__cong) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 427 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit1(V_l_2))))) # label(fact_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 428 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))) # label(fact_sup__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 429 (all V_P_2 all V_a_2 all V_f_2 all T_b all V_A_2 (c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),V_f_2) = V_A_2 -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_b),V_P_2))))) -> hBOOL(hAPP(V_P_2,B_x)))) -> hBOOL(hAPP(V_P_2,V_a_2))))))) # label(fact_def__lfp__induct__set) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 430 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) # label(fact_folding__image__simple_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 431 (all V_R_2 all T_b (c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_R_2)) <-> c_Wellfounded_Owf(T_b,V_R_2))) # label(fact_wf__comp__self) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 432 (all V_z_H all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_z_H))))) # label(fact_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 433 (all T_a (class_Int_Onumber__ring(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)))) # label(fact_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 434 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_a_2),V_A_2)))) # label(fact_min__max_OSup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 435 (all V_P_2 all T_b (class_Enum_Oenum(T_b) -> ((all B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_b),V_P_2))))) # label(fact_enum__all) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 436 (all V_i_2 all V_h_2 all V_g_2 all V_f_2 all T_c all T_b all T_e all T_d all T_g all T_f hAPP(c_Fun_Ocomp(tc_prod(T_f,T_g),tc_prod(T_d,T_e),tc_prod(T_b,T_c),c_Product__Type_Omap__pair(T_f,T_d,T_g,T_e,V_f_2,V_g_2)),c_Product__Type_Omap__pair(T_b,T_f,T_c,T_g,V_h_2,V_i_2)) = c_Product__Type_Omap__pair(T_b,T_d,T_c,T_e,hAPP(c_Fun_Ocomp(T_f,T_d,T_b,V_f_2),V_h_2),hAPP(c_Fun_Ocomp(T_g,T_e,T_c,V_g_2),V_i_2))) # label(fact_map__pair_Ocomp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 437 (all V_a all T_a (class_Orderings_Olinorder(T_a) -> V_a = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_a))) # label(fact_min__max_Osup_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 438 (all V_l_2 (c_List_Onat__list(V_l_2) <-> c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),V_l_2)))) # label(fact_nat__list__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 439 (all V_R_2 all V_n_2 all T_b (c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2)) -> c_Wellfounded_Owf(T_b,V_R_2))) # label(fact_wf__exp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 440 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_b (class_Lattices_Olattice(T_b) -> hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))) # label(fact_inf__apply) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 441 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Code__Numeral_Onat__of,V_n),hAPP(c_Code__Numeral_Onat__of,V_m))) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_n,V_m)) # label(fact_mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 442 (all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OPls),V_w) = c_Int_OPls) # label(fact_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 443 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))))) # label(fact_length__pos__if__in__set) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 444 (all V_P_2 ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) & hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,B_x))))))) # label(fact_ex__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 445 (all V_k_2 all V_f_2 all V_n_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2))) -> (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)) & V_k_2 = hAPP(V_f_2,B_i))))))) # label(fact_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 446 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),V_A_2)))) # label(fact_finite__UnionD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 447 (all V_A_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_A_2)) # label(fact_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 448 (all V_P_2 all T_b c_HOL_OThe(T_b,V_P_2) = c_Enum_Oenum__the(T_b,V_P_2)) # label(fact_enum__the__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 449 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> V_A_2 = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)))) # label(fact_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 450 (all V_f_2 all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_B_2))))) # label(fact_setsum__diff__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 451 (all V_y_2 all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_x_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2)) -> V_y_2 = V_x_2)))) # label(fact_fun__left__comm_Ofold__graph__determ) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 452 (all T_b (class_Nat_Osemiring__char__0(T_b) -> c_Fun_Oinj__on(tc_Nat_Onat,T_b,c_Nat_Osemiring__1__class_Oof__nat(T_b),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_inj__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 453 (all V_ys_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_ys_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),hAPP(c_List_Oset(T_b),V_ys_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),B_xa)))))) & c_List_Olinorder__class_Osorted(T_b,V_xs_2) <-> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))))) # label(fact_sorted__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 454 (all V_j_2 all V_i_2 all V_xs_2 all T_b (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_b),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2),V_j_2)),V_i_2) = hAPP(hAPP(c_List_Onth(T_b),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),c_List_Otranspose(T_b,V_xs_2)),V_i_2)),V_j_2))))) # label(fact_nth__nth__transpose__sorted) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 455 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)))))) # label(fact_pos__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 456 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_divide__nonneg__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 457 (all V_xs all T_a V_xs = hAPP(hAPP(c_List_Odrop(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_xs)) # label(fact_drop__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 458 (all V_S_2 all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_S_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_S_2),c_COMBI(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Union__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 459 (all V_r_2 all T_c all V_a_2 all T_b ((exists B_y hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_y),V_a_2)),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Relation_ORange(T_c,T_b),V_r_2))))) # label(fact_Range__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 460 (all V_Q_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,T_b,tc_HOL_Obool),V_Q_2),B_x)))))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),V_Q_2)))))))) # label(fact_finite__Collect__bex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 461 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c (class_Lattices_Osemilattice__inf(T_b) & class_Lattices_Osemilattice__inf(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2))))))) # label(fact_mono__inf) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 462 (all V_B_2 all V_A_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2))))) # label(fact_disjoint__eq__subset__Compl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 463 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_k))))) # label(fact_le__trans) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 464 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 465 (all V_r_2 all V_x_2 all T_b all V_y_2 ((exists B_b exists B_z hAPP(hAPP(c_List_Olist_OCons(T_b),B_b),B_z) = V_y_2) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),hAPP(hAPP(c_List_Oappend(T_b),V_x_2),V_y_2))),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__append__rightI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 466 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y))) # label(fact_min__max_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 467 (all V_A_2 all T_b c_Relation_OId__on(T_b,V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),T_b),c_Set_Oinsert(tc_prod(T_b,T_b))),hAPP(hAPP(c_COMBS(T_b,T_b,tc_prod(T_b,T_b)),c_Product__Type_OPair(T_b,T_b)),c_COMBI(T_b)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))))) # label(fact_Id__on__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 468 (all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)) = V_f_2)) # label(fact_inv__inv__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 469 (all V_G_2 all T_b c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) # label(fact_empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 470 (all T_b (class_Rings_Osemiring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Nat_Osemiring__1__class_ONats(T_b))))) # label(fact_Nats__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 471 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)))) # label(fact_ivl__disj__un_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 472 (all V_y_2 all V_x_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2) = c_Hilbert__Choice_OEps(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(c_fequal,V_x_2)))),hAPP(c_fequal,V_y_2))))) # label(fact_Eps__split__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 473 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 474 (all V_k all T_a (class_Rings_Oring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_k)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k)))) # label(fact_of__int__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 475 (all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_finite__PlusD_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 476 (all V_A_2 all V_f_2 all T_c all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_a_2)),V_A_2)))) # label(fact_vimageD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 477 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 478 (all V_a all V_b all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)))))) # label(fact_min__max_Oless__supI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 479 (all V_qr_H all V_qr all V_n all V_m (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr)) -> (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr_H)) -> V_qr = V_qr_H))) # label(fact_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 480 (all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,T_c),tc_HOL_Obool)))) -> (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != hAPP(c_Finite__Set_Ocard(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_finite__fun__UNIVD1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 481 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Ofinite__psubset(T_b))))) # label(fact_in__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 482 (all V_x_2 all T_b (class_Groups_Ozero(T_b) -> (V_x_2 = c_Groups_Ozero__class_Ozero(T_b) <-> c_Groups_Ozero__class_Ozero(T_b) = V_x_2))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 483 (all V_m_2 all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_n_2,hAPP(c_Nat_OSuc,V_m_2))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_n_2,V_m_2)) # label(fact_atLeastAtMost__upt) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 484 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),T_b,T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_b),V_B_2)),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_b),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_b),T_b,T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),V_A_2)),hAPP(c_COMBC(T_c,T_d,T_b),V_f_2))))) # label(fact_SUP__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 485 (all V_C_2 all V_A_2 all T_b ((all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)),V_A_2)),V_C_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_A_2)),V_C_2)))) # label(fact_UN__finite__subset) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 486 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> V_x_2 = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) # label(fact_Sup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 487 (all V_B_2 all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2))) <-> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_psubset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 488 (all V_xs_2 all V_x_2 all T_b ((hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oremove1(T_b,V_x_2,V_xs_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oremove1(T_b,V_x_2,V_xs_2))))) # label(fact_length__remove1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 489 (all V_xs_2 all V_g_2 all V_f_2 all T_c all T_b (class_Groups_Oab__group__add(T_b) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Ominus__class_Ominus(T_b)),V_f_2)),V_g_2)),V_xs_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_g_2),V_xs_2))))) # label(fact_listsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 490 (all V_x all V_n all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Oreplicate(T_a,V_n,V_x)) = c_List_Oreplicate(T_a,hAPP(c_Nat_OSuc,V_n),V_x)) # label(fact_replicate__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 491 (all V_f_2 all V_n_2 all V_m_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2)),V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))))) # label(fact_setsum__head__upt__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.67 492 (all V_S_2 all V_R_2 all T_b c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_R_2)),c_Transitive__Closure_Ortrancl(T_b,V_S_2))) = c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2))) # label(fact_rtrancl__Un__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 493 (all V_f2_2 all V_f1_2 all T_b V_f1_2 = c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_b,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) # label(fact_code__numeral_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 494 (all V_A_2 all V_B_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Un__upper2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 495 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)))))) # label(fact_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 496 (all V_j all V_i c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Oupto(V_i,V_j))) # label(fact_sorted__upto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 497 (all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_finite__insert) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 498 (all V_y_2 all V_x_2 all T_b (class_Int_Oring__char__0(T_b) & class_Int_Onumber__ring(T_b) -> (V_x_2 = V_y_2 <-> hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))) # label(fact_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 499 (all V_y all V_x (c_Nat__Transfer_Ois__nat(V_x) -> (c_Nat__Transfer_Ois__nat(V_y) -> c_Nat__Transfer_Ois__nat(c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y))))) # label(fact_Divides_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 500 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_List_Ofoldl(T_b,T_b,c_Orderings_Oord__class_Omin(T_b),V_x_2),V_xs_2) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_Min__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 501 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n))) # label(fact_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 502 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_y))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 503 (all T_c all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_b,T_c),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)))) # label(fact_Domain__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 504 (all V_ys_2 all V_xs_2 all V_x_2 all T_c all T_b hAPP(hAPP(c_List_Oappend(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Omap(T_c,tc_prod(T_b,T_c)),hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2)),V_ys_2)),c_Enum_Oproduct(T_b,T_c,V_xs_2,V_ys_2)) = c_Enum_Oproduct(T_b,T_c,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2),V_ys_2)) # label(fact_product_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 505 (all V_fs_2 all T_b all V_y_2 all V_x_2 all V_f_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_b,tc_Nat_Onat)),V_f_2),V_fs_2)))))) # label(fact_measures__less) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 506 (all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)))) # label(fact_fun__left__comm__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 507 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n))))) # label(fact_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 508 (all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_x = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_x))) # label(fact_Min_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 509 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 510 (all V_P_2 all T_b (c_FunDef_Oreduction__pair(T_b,V_P_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2)),hAPP(c_Product__Type_Osnd(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2))),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2))) & c_Wellfounded_Owf(T_b,hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2)))) # label(fact_reduction__pair__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 511 (all V_x_2 all T_b all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 -> hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_x_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 512 (all V_z all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)),V_x)))))) # label(fact_min__max_Osup__least) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 513 (all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),c_List_Oupt(V_i,V_j))) # label(fact_length__upt) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 514 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 515 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))))) # label(fact_UnE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 516 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) <-> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),V_xs_2)))))) # label(fact_sorted__insort__key) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 517 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2))))))) # label(fact_min__max_OInf__le__Sup) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 518 (all V_g_2 all V_f_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(V_g_2,V_a_2) = hAPP(c_Fun_Ooverride__on(T_b,T_c,V_f_2,V_g_2,V_A_2),V_a_2))) # label(fact_override__on__apply__in) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 519 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) | V_m = V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 520 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_j),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_k))))) # label(fact_zle__trans) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 521 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_x)))) # label(fact_inf__sup__ord_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 522 (all V_x_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c))),V_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_r_2))))) # label(fact_SUP__UN__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 523 (all V_y all V_x all V_i all V_xs all T_a hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_y) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i),V_y)) # label(fact_list__update__overwrite) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 524 (all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_y_2),c_Int_OInteg)) -> -(all B_x V_y_2 != hAPP(c_Int_ORep__Integ,B_x)))) # label(fact_Rep__Integ__cases) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 525 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Fun_Ocomp(T_c,T_b,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),V_f_2) = c_Fun_Oid(T_b))) # label(fact_inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 526 (all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> -(c_Predicate_Oreflp(T_b,V_R_2) -> (c_Predicate_Osymp(T_b,V_R_2) -> -c_Predicate_Otransp(T_b,V_R_2))))) # label(fact_equivpE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 527 (all T_b hAPP(c_Fun_Ocomp(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,T_b,c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint)),c_Nitpick_ORep__Frac(T_b)) = c_Nitpick_Onum(T_b)) # label(fact_Nitpick_Onum__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 528 (all V_ys_2 all V_xs_2 all V_A_2 all T_b (hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_ys_2)) & hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_xs_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))))) # label(fact_append__in__listsp__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 529 (all V_m all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_le__add2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 530 (all V_u_2 all V_l_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),hAPP(c_Nat_OSuc,V_l_2)) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2))) # label(fact_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 531 (all T_d all T_c all T_b c_Product__Type_Ointernal__split(T_b,T_c,T_d) = c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d)) # label(fact_internal__split__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 532 (all T_b c_Fun_Oid(tc_List_Olist(T_b)) = c_List_Orotate(T_b,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_rotate0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 533 (all V_P_2 (--hBOOL(V_P_2) | hBOOL(hAPP(c_fNot,V_P_2)))) # label(help_c__fNot__2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 534 (all V_v all V_b all V_a all T_b (class_Rings_Oring(T_b) & class_Int_Onumber(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a),V_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))))) # label(fact_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 535 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k))) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 536 (all V_s_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_b)),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Product__Type_Ofst(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Oselect__weight(T_b,V_xs_2),V_s_2))),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),T_b),c_Product__Type_Osnd(tc_Code__Numeral_Ocode__numeral,T_b)),V_xs_2)))))) # label(fact_select__weight__member) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 537 (all V_f_2 all V_B_2 all T_c all V_A_2 all T_b all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(T_b,T_c),T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_d),tc_fun(tc_fun(T_c,tc_HOL_Obool),T_d),T_b),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d)),V_f_2)),V_B_2)),V_A_2))))) # label(fact_setsum__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 538 (all V_B_2 all V_a_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 539 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))))) # label(fact_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 540 (all V_Q_2 all T_b all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_P_2),V_Q_2)) -> hBOOL(hAPP(V_Q_2,V_x_2))))) # label(fact_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 541 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel1(T_b,V_r_2)),c_List_Olistrel1(T_b,V_s_2))))) # label(fact_listrel1__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 542 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 543 (all V_b_2 all V_A_2 all T_b ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) -> c_Partial__Function_Oflat__lub(T_b,V_b_2,V_A_2) = V_b_2) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) -> c_HOL_OThe(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = c_Partial__Function_Oflat__lub(T_b,V_b_2,V_A_2)))) # label(fact_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 544 (all V_Q_2 all V_P_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2)),V_Q_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),V_P_2)),hAPP(c_Set_OCollect(T_b),V_Q_2))) # label(fact_Collect__conj__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 545 (all T_c all T_b all V_q_2 all V_Q_2 all V_p_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_p_2)) -> (hBOOL(hAPP(V_Q_2,V_q_2)) -> hBOOL(hAPP(V_Q_2,hAPP(c_Product__Type_Osnd(T_b,T_c),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_p_2),V_q_2)))) & hBOOL(hAPP(V_P_2,hAPP(c_Product__Type_Ofst(T_b,T_c),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_p_2),V_q_2))))))) # label(fact_conjI__realizer) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 546 (all V_xs_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_List_Odistinct(T_b),c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2))))) # label(fact_distinct__sort) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 547 (all V_l c_Int_OPls != c_Int_OBit1(V_l)) # label(fact_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 548 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 549 (all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(c_Set_OCollect(T_b),hAPP(c_fequal,V_a_2))) # label(fact_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 550 (all V_f_2 all V_n_2 all V_m_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),V_n_2)))))) # label(fact_setsum__head__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 551 (all V_xs_2 all V_P_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) & hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_Olist__ex(T_b,V_P_2,V_xs_2))) # label(fact_list__ex__iff__raw) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 552 (all V_a2_2 all V_a1_2 all V_R_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a1_2)) & V_a2_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)) & hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_a2_2)))))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a2_2)) <-> c_Wellfounded_Omax__extp(T_b,V_R_2,V_a1_2,V_a2_2))) # label(fact_max__extp_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 553 (all T_1 (class_Lazy__Sequence_Osmall__lazy(T_1) -> class_Lazy__Sequence_Osmall__lazy(tc_List_Olist(T_1)))) # label(arity_List__Olist__Lazy__Sequence_Osmall__lazy) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 554 (all V_x_2 all V_P_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) -> hBOOL(hAPP(V_P_2,hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),B_y))))) -> hBOOL(hAPP(V_P_2,V_x_2))))) # label(fact_inj__transfer) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 555 (all V_ys all V_xs all T_a all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) -> ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_k)) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),B_i) = hAPP(hAPP(c_List_Onth(T_a),V_ys),B_i))) -> hAPP(hAPP(c_List_Otake(T_a),V_k),V_xs) = hAPP(hAPP(c_List_Otake(T_a),V_k),V_ys))))) # label(fact_nth__take__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 556 (all V_n hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 557 (all V_yq all V_xq all V_x all T_a c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,c_Lazy__Sequence_Oappend(T_a,V_xq,V_yq)) = c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,V_xq),V_yq)) # label(fact__01) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 558 (all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_Abs_2),V_A_2))) # label(fact_type__definition_OAbs__image) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 559 (all T_a c_List_Oconcat(T_a,c_List_Olist_ONil(tc_List_Olist(T_a))) = c_List_Olist_ONil(T_a)) # label(fact_concat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 560 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_c all T_b (c_Relation_Osingle__valued(T_b,T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_z_2)),V_r_2)) -> V_y_2 = V_z_2)))) # label(fact_single__valuedD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 561 (all V_xs_2 all V_x_2 all V_P_2 all T_b (hBOOL(hAPP(V_P_2,V_x_2)) & c_List_Olist__all(T_b,V_P_2,V_xs_2) <-> c_List_Olist__all(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)))) # label(fact_list__all__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 562 (all V_b all V_a all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)))))) # label(fact_min__max_Oless__supI1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 563 (all V_m all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)))) # label(fact_le__add1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 564 (all V_g_2 all V_x_2 all T_d all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_A_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_A_2)),V_g_2))))) # label(fact_folding_Ocommute__left__comp_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 565 (all V_nat_2 all V_f2_2 all V_f1_2 all T_b c_Nat_Onat_Onat__case(T_b,V_f1_2,V_f2_2,hAPP(c_Nat_OSuc,V_nat_2)) = hAPP(V_f2_2,V_nat_2)) # label(fact_nat__case__Suc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 566 (all V_M_2 all V_A_2 all T_c all V_u_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_u_2),hAPP(V_M_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_u_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),V_A_2),V_M_2)))))) # label(fact_le__INF__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 567 (all V_n_2 all V_v_2 all V_f_2 all V_a_2 all T_b hAPP(c_Nat_Onat_Onat__rec(T_b,V_a_2,V_f_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),V_n_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_b),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),tc_Int_Oint),c_If(T_b)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_Onat_Onat__rec(T_b,V_a_2,V_f_2),V_n_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_b,T_b),tc_Int_Oint),V_f_2),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Int_Onat)),V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),c_Nat_Onat_Onat__rec(T_b,V_a_2,V_f_2)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Int_Onat)),V_n_2)))))) # label(fact_nat__rec__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 568 (all V_A_2 all V_b_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)))))) # label(fact_le__Inf__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 569 (all V_R2_2 all T_c all V_R1_2 all T_b (c_Relation_Otrans(T_b,V_R1_2) -> (c_Relation_Otrans(T_c,V_R2_2) -> c_Relation_Otrans(tc_prod(T_b,T_c),c_Wellfounded_Olex__prod(T_b,T_c,V_R1_2,V_R2_2))))) # label(fact_trans__lex__prod) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 570 (all V_P_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> c_List_Olinorder__class_Osorted(T_b,c_List_OdropWhile(T_b,V_P_2,V_xs_2))))) # label(fact_sorted__dropWhile) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 571 (all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 572 (all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> V_x_2 = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_Min_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 573 (all V_b_2 all V_a_2 all T_b (class_Rings_Oring__no__zero__divisors(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = V_a_2 | V_b_2 = c_Groups_Ozero__class_Ozero(T_b) <-> c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)))) # label(fact_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 574 (all V_h_2 all T_c all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> (c_Fun_Oinj__on(T_c,T_b,V_h_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Ovimage(T_c,T_b,V_h_2),V_F_2)))))) # label(fact_finite__vimageI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 575 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 576 (all V_h_2 all V_g_2 all T_g all V_f_2 all T_d all T_e all T_f all V_F_2 all V_fa_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_fa_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_g,T_e,T_d,hAPP(c_Fun_Ocomp(T_f,T_e,T_g,V_f_2),V_g_2)),V_h_2) = hAPP(c_Fun_Ocomp(T_f,T_e,T_d,V_f_2),hAPP(c_Fun_Ocomp(T_g,T_f,T_d,V_g_2),V_h_2)))) # label(fact_folding_Ocommute__comps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 577 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) = hAPP(c_List_Ofoldl(T_b,T_b,c_Orderings_Oord__class_Omin(T_b),V_x_2),V_xs_2))) # label(fact_min__max_OInf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 578 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))))) # label(fact_card__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 579 (all V_a_2 all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) -> (-hBOOL(hAPP(hAPP(hAPP(V_B_2,V_a_2),V_b_2),V_c_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))) # label(fact_INF2__E) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 580 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b)) = V_b)) # label(fact_add__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 581 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)))) # label(fact_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 582 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))) # label(fact_sup__assoc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 583 (all T_2 all T_1 (class_Orderings_Opreorder(T_1) -> class_Orderings_Opreorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Opreorder) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 584 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))) # label(fact_ab__diff__minus) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 585 (all V_l_2 all V_P_2 all T_b (class_Rings_Osemiring__0(T_b) & class_Rings_Odvd(T_b) -> ((exists B_x hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_l_2),B_x)))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_l_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),B_x),c_Groups_Ozero__class_Ozero(T_b)))) & hBOOL(hAPP(V_P_2,B_x))))))) # label(fact_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 586 (all V_xs_2 all V_f_2 all T_c all T_b c_List_Otranspose(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_b)),hAPP(c_List_Omap(T_c,T_b),V_f_2)),V_xs_2)) = hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_b)),hAPP(c_List_Omap(T_c,T_b),V_f_2)),c_List_Otranspose(T_c,V_xs_2))) # label(fact_transpose__map__map) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 587 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),V_A_2))))))) # label(fact_min__max_OSup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 588 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_b all T_c hAPP(hAPP(c_Fun_Ocomp(T_c,T_b,T_d,V_f_2),V_g_2),V_x_2) = hAPP(V_f_2,hAPP(V_g_2,V_x_2))) # label(fact_o__apply) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 589 (all V_S_2 all V_R_2 all T_b (hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2) = hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_R_2) <-> V_R_2 = V_S_2)) # label(fact_pred__equals__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 590 (all V_y all V_m all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_y),V_m)) # label(fact_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 591 (all V_a_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,V_f_2),V_a_2),V_A_2)))) # label(fact_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 592 (all V_i all V_d ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_Oempty(tc_Int_Oint) = c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i) = c_Lazy__Sequence_Oappend(tc_Int_Oint,hAPP(c_Lazy__Sequence_Osingle(tc_Int_Oint),V_i),c_Lazy__Sequence_Osmall__lazy_H(V_d,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),c_Groups_Oone__class_Oone(tc_Int_Oint))))))) # label(fact_small__lazy_H_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 593 (all V_P_2 all V_f_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2) = V_A_2 -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_A_2),V_P_2))),V_P_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_A_2),V_P_2))))))) # label(fact_def__lfp__induct) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 594 (all V_x_2 all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_ys_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))))) # label(fact_rtrancl__listrel1__ConsI1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 595 (all V_P_2 all T_b ((all B_x -hBOOL(hAPP(V_P_2,B_x))) <-> hAPP(c_Set_OCollect(T_b),V_P_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 596 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> (c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) -> (c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) -> V_y = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))))) # label(fact_compl__unique) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 597 (all V_R_2 all V_a2_2 all V_a1_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a1_2)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_xa)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_a2_2)))))) & V_a2_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a2_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_a1_2),V_a2_2)),c_Wellfounded_Omax__ext(T_b,V_R_2))))) # label(fact_max__ext_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 598 (all V_y_2 all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_y_2 = hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),V_y_2)))) # label(fact_surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 599 (all V_x_2 all V_g_2 all T_e all V_f_2 all T_c all T_b all T_d hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_x_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_d,T_e),V_x_2))) = hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_f_2),hAPP(c_Product__Type_Oapsnd(T_e,T_c,T_d,V_g_2),V_x_2))) # label(fact_apfst__apsnd) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 600 (all V_n_2 all V_m_2 hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) # label(fact_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 601 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (V_c_2 = V_b_2 & V_a_2 = V_d_2 | V_c_2 = V_a_2 & V_b_2 = V_d_2 <-> hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_d_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 602 (all V_k_2 all V_A_2 all T_b (hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(c_Nat_OSuc,V_k_2) <-> (exists B_b exists B_B (hAPP(hAPP(c_Set_Oinsert(T_b),B_b),B_B) = V_A_2 & -hBOOL(hAPP(hAPP(c_member(T_b),B_b),B_B)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = B_B) & hAPP(c_Finite__Set_Ocard(T_b),B_B) = V_k_2)))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 603 (all V_C_2 all V_J_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_J_2),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2)) # label(fact_Sigma__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 604 (all V_n_2 all V_k_2 all V_m_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 605 (all V_k all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_k),c_Groups_Osgn__class_Osgn(T_a,V_k)))) # label(fact_abs__sgn) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 606 (all V_g_2 all V_f_2 all T_c all V_A_2 all V_a_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(c_Fun_Ooverride__on(T_b,T_c,V_f_2,V_g_2,V_A_2),V_a_2))) # label(fact_override__on__apply__notin) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 607 (all V_x_2 all V_A_2 all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,V_A_2),V_x_2)) -> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 608 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_compl__inf) # label(axiom) # label(non_clause). [assumption]. 8.62/8.68 609 (all V_r_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2)))))) # label(fact_finite__quotient) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 610 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_a)))) # label(fact_div__self) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 611 (all V_r_2 all T_b (c_Relation_Osym(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))) <-> c_Predicate_Osymp(T_b,V_r_2))) # label(fact_symp__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 612 (all V_b all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) # label(fact_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 613 (all V_P_2 all V_a_2 all V_n_2 all T_b (V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & hBOOL(hAPP(V_P_2,V_a_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_a_2)))) & hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_Bex__set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 614 (all V_xs_2 all T_b c_List_Olist_ONil(T_b) = c_List_Osublist(T_b,V_xs_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_sublist__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 615 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 616 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = V_A_2) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 617 (all V_xs_2 all V_n_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(c_List_Orotate(T_b,V_n_2),V_xs_2))))) # label(fact_distinct__rotate) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 618 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Inf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 619 (all V_g_2 all T_d all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) -> (hAPP(c_Set_Oimage(T_b,T_d,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_Fun_Ocomp(T_b,T_d,T_c,V_g_2),V_f_2)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_comp__surj) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 620 (all V_R_2 all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_R_2)),c_Transitive__Closure_Ortrancl(T_b,V_R_2)) = c_Transitive__Closure_Ortrancl(T_b,V_R_2)) # label(fact_rtrancl__idemp__self__comp) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 621 (all V_f_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(c_Wellfounded_Omeasure(T_b),V_f_2))))) # label(fact_in__measure) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 622 (all V_z all V_x all V_y all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)))))) # label(fact_mult__imp__div__pos__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 623 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_inf__assoc) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 624 (all V_C_2 all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2)))) # label(fact_Times__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 625 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))))) # label(fact_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 626 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(c_List_Orev(T_b),V_xs_2))))) # label(fact_distinct__rev) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 627 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_image__Int) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 628 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 629 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 630 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_b,tc_HOL_Obool),T_d),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),V_B_2)) # label(fact_UN__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 631 (all V_A_2 all T_b ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,T_b,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_COMBK(tc_Nat_Onat,T_b),c_Groups_Oone__class_Oone(tc_Nat_Onat))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_card__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 632 (all V_A_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 633 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_drop__tl) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 634 (all V_A_2 all V_r_2 all T_b (hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)),V_r_2) = V_r_2 -> (hAPP(c_Relation_ODomain(T_b,T_b),V_r_2) = V_A_2 -> c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2)))) # label(fact_comp__equivI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 635 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = c_Groups_Osgn__class_Osgn(T_b,V_a_2) <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_sgn__0__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 636 (all V_xs_2 all V_x_2 all T_b ((hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Oremdups(T_b,V_xs_2) = c_List_Oremdups(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Oremdups(T_b,V_xs_2)) = c_List_Oremdups(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_remdups_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 637 (all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist_ONil(T_b) = V_xs_2 <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,c_List_Olist_ONil(T_c)))) # label(fact_list__all2__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 638 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 639 (all V_s all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Orange(V_k),V_s))),V_k)))) # label(fact_range) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 640 (all V_z_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))))) # label(fact_of__int__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 641 (all V_y_2 all V_x_2 all T_c all T_b -hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_x_2),V_y_2))) # label(fact_bot2E) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 642 (all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_S_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_b,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_b,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)))))),c_Wellfounded_Omin__ext(T_b,V_R_2))))) # label(fact_min__ext__compat) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 643 (all V_v_2 all V_f_2 all V_a_2 all T_b c_Nat_Onat_Onat__case(T_b,V_a_2,V_f_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_b),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),tc_Int_Oint),c_If(T_b)),c_Nat__Numeral_Oneg)),V_a_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),V_f_2),c_Int_Onat)))) # label(fact_nat__case__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 644 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) & hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = V_B_2 <-> c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2))) # label(fact_bij__betw__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 645 (all V_u_2 all V_M_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_M_2,B_i)),V_u_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),V_M_2)),V_u_2))))) # label(fact_SUP__leI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 646 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> c_Divides_Odivmod__nat(V_m,V_n) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n))))) # label(fact_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 647 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))))) # label(fact_diff__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 648 (all V_x_2 all T_b (class_Int_Onumber__ring(T_b) & class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 649 (all V_l_2 all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Divides_Odivmod__int(V_k_2,V_l_2)) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (V_l_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2) = c_Divides_Odivmod__int(V_k_2,V_l_2)) & (V_l_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(c_Product__Type_Oapsnd(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint,hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),hAPP(hAPP(hAPP(c_If(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_fdisj,hAPP(hAPP(c_fconj,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l_2)),hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))),hAPP(hAPP(c_fconj,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_l_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),c_Divides_Opdivmod(V_k_2,V_l_2)),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_fun(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),hAPP(c_COMBC(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_If(tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l_2))))),c_Divides_Opdivmod(V_k_2,V_l_2)))) = c_Divides_Odivmod__int(V_k_2,V_l_2))))) # label(fact_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 650 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Nat_OSuc,V_m_2))))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 651 (all V_a_2 all V_r_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_a_2),V_b_2)),V_r_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 652 (all V_m all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_x),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_m)) # label(fact_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 653 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onat,V_z_2))))) # label(fact_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 654 (all V_x_2 all T_b all V_n_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 -> hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_x_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_x_2))))) # label(fact_set__replicate__conv__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 655 (all V_A_2 V_A_2 = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) # label(fact_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 656 (all V_a2_2 all V_a1_2 all V_A_2 all T_b (hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_a2_2)) & hBOOL(hAPP(V_A_2,V_a1_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_a1_2),V_a2_2))))) # label(fact_listsp_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 657 (all V_B_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 658 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 659 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_b),hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_map_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 660 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__ex(T_b,V_P_2,V_xs_2) <-> (exists B_n (hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_n))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))))))) # label(fact_list__ex__length) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 661 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) <-> c_Relation_Osym(T_b,V_r_2) & c_Relation_Otrans(T_b,V_r_2) & c_Relation_Orefl__on(T_b,V_A_2,V_r_2))) # label(fact_equiv__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 662 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 663 (all V_j all V_i ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_i),c_List_Oupt(hAPP(c_Nat_OSuc,V_i),V_j))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i,V_j)))) # label(fact_upt__rec) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 664 (all V_r_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2))))) # label(fact_finite__Domain) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 665 (all V_f_2 all T_d all V_g_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_Oimage(T_c,T_b,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_Oimage(T_c,T_d,hAPP(hAPP(c_COMBB(T_b,T_d,T_c),V_f_2),V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__range__imageI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 666 (all V_z_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olexord(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_y_2),V_z_2)),c_List_Olexord(T_b,V_r_2))) -> (c_Relation_Otrans(T_b,V_r_2) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_z_2)),c_List_Olexord(T_b,V_r_2))))))) # label(fact_lexord__trans) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 667 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_B_2))))))) # label(fact_setsum_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 668 (all V_ys all T_b all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))))) # label(fact_distinct__zipI1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 669 (all V_z_2 all V_z_H_2 ((hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z_H_2)) -> hAPP(c_Int_Onat,V_z_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z_2)),hAPP(c_Int_Onat,V_z_H_2))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z_H_2)) -> hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z_2),V_z_H_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Int_Onat)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z_2)),hAPP(c_Int_Onat,V_z_H_2))))) # label(fact_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 670 (all V_a_2 all V_b_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))))) # label(fact_neg__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 671 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oordered__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2)))))) # label(fact_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 672 (all V_A_H_H_2 all V_f_H_2 all T_d all V_A_H_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_H_2,V_A_H_2,V_A_H_H_2) <-> c_Fun_Obij__betw(T_b,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_f_H_2),V_f_2),V_A_2,V_A_H_H_2)))) # label(fact_bij__betw__comp__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 673 (all V_xs all V_x all T_a all V_v ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_drop__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 674 (all V_m_2 all V_n_2 ((-(c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))) -> c_Divides_Odivmod__nat(V_m_2,V_n_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat)),c_Nat_OSuc)),c_Divides_Odivmod__nat(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2),V_n_2))) & (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2) = c_Divides_Odivmod__nat(V_m_2,V_n_2)))) # label(fact_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 675 (all V_n all V_m all V_u all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i)),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)))) # label(fact_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 676 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) <-> hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_r_2) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_r_2)))))) # label(fact_eq__equiv__class__iff2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 677 (all V_m_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))))) # label(fact_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 678 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all T_b (V_y_2 = V_x_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olex(T_b,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))),c_List_Olex(T_b,V_r_2))))) # label(fact_Cons__in__lex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 679 (all T_1 class_Nat_Osize(tc_Option_Ooption(T_1))) # label(arity_Option__Ooption__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 680 (all V_I_2 all V_xs_2 all T_b hAPP(c_List_Oset(T_b),c_List_Osublist(T_b,V_xs_2,V_I_2)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_b),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_Nat_Onat,T_b),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_b),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_b),V_xs_2))))),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_I_2)))))) # label(fact_set__sublist) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 681 (all V_B_2 all V_C_2 all V_A_2 all T_b (hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) -> (hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_B_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)))) # label(fact_union__comp__emptyL) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 682 (all T_b (class_Groups_Oab__semigroup__mult(T_b) -> c_Finite__Set_Ofun__left__comm(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)))) # label(fact_fun__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 683 (all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_divide__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 684 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnI1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 685 (all V_b all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_c)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))))) # label(fact_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 686 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))))) # label(fact_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 687 (all V_xs_2 all V_x_2 all V_ys_2 all V_P_2 all T_b ((exists B_us exists B_vs (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_b),B_us),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_vs)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),B_us))) -> -hBOOL(hAPP(V_P_2,B_x)))) & hBOOL(hAPP(V_P_2,V_x_2)) & V_xs_2 = hAPP(c_List_Ofilter(T_b,V_P_2),B_vs))) <-> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2) = hAPP(c_List_Ofilter(T_b,V_P_2),V_ys_2))) # label(fact_filter__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 688 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 689 (all V_xs_2 all V_P_2 all T_b (c_List_OdropWhile(T_b,V_P_2,V_xs_2) != c_List_Olist_ONil(T_b) -> -hBOOL(hAPP(V_P_2,hAPP(c_List_Ohd(T_b),c_List_OdropWhile(T_b,V_P_2,V_xs_2)))))) # label(fact_hd__dropWhile) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 690 (all V_x all V_z all V_y all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_z),V_x)))) # label(fact_min__max_Oinf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 691 (all V_b_2 all V_a_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))),V_S_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))),V_S_2) = hAPP(V_b_2,V_a_2))))) # label(fact_setsum__delta_H) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 692 (all V_A_2 all V_f_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Set_Ovimage(T_b,T_c,V_f_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2))) # label(fact_vimage__Union) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 693 (all V_b all V_a_H all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),V_a_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_H),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))))) # label(fact_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 694 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2))) # label(fact_Int__lower2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 695 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))))))) # label(fact_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 696 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) = V_b))) # label(fact_minus__unique) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 697 (all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_A_2) <-> c_Equiv__Relations_Oequivp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_A_2)))) # label(fact_equivp__equiv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 698 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z))) # label(fact_mult__2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 699 (all V_xq_2 all V_g_2 all V_f_2 all T_c all T_b c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_b,T_c,V_f_2,V_g_2,V_xq_2) = c_Option_Ooption_Ooption__case(T_b,tc_prod(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c)),V_f_2,hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c),T_b),V_g_2),hAPP(c_Lazy__Sequence_Oyield(T_c),V_xq_2))) # label(fact_seq__case) # label(axiom) # label(non_clause). [assumption]. 8.62/8.69 700 (all V_S_2 all V_R_2 all V_P_2 all T_b (c_FunDef_Oreduction__pair(T_b,V_P_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_S_2),hAPP(c_Product__Type_Osnd(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2))) -> (c_Wellfounded_Owf(T_b,V_S_2) -> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2))))))) # label(fact_reduction__pair__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 701 (all V_i all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs))))) # label(fact_distinct__take) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 702 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))) # label(fact_card__Diff__singleton) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 703 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i)),hAPP(hAPP(c_List_Onth(T_c),V_ys_2),B_i))))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2))) # label(fact_list__all2__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 704 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) <-> c_Relation_Osym(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) # label(fact_sym__converse) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 705 (all V_a all V_r all V_q all V_c all V_b (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__nat__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_r),V_c))))))) # label(fact_divmod__nat__rel__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 706 (all V_P_2 all V_f_2 all T_c all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_b),V_P_2),V_f_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(c_Set_OCollect(T_c),V_P_2))) # label(fact_vimage__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 707 (all V_A_2 all T_b c_Relation_Otrans(T_b,c_Relation_OId__on(T_b,V_A_2))) # label(fact_trans__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 708 (all V_X_2 all V_a_2 all V_f_2 all T_b (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)))))))) # label(fact_gfp__fun__UnI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 709 (all V_k -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 710 (all V_h_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Groups_Oplus__class_Oplus(T_c)),V_g_2)),V_h_2)),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_h_2),V_A_2))))) # label(fact_setsum_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 711 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Groups_Oab__group__add(T_b) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2) -> (V_b_2 = V_a_2 <-> V_c_2 = V_d_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 712 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x))) # label(fact_compl__sup__top) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 713 (all V_i ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_i -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_i -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i)) -> c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i)) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i))))) # label(fact_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 714 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 715 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_xs) = hAPP(hAPP(c_List_Onth(T_a),V_xs),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_hd__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 716 (all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)) # label(fact_atMost__upto) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 717 (all V_XS_2 all V_x_2 all V_YS_2 all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__leq)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omax__weak)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2)),V_YS_2)),c_FunDef_Omax__weak)))))) # label(fact_wmax__insertI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 718 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_div__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 719 (all V_x all V_i all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) # label(fact_length__list__update) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 720 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> -hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)))))) # label(fact_SUP2__E) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 721 (all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),V_i_2)) <-> hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != V_i_2)) # label(fact_code__numeral__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 722 (all V_n all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Olist_ONil(T_a))) # label(fact_drop__Nil) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 723 (all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_List_Oset(T_b),c_List_Olist_ONil(T_b))) # label(fact_List_Oset_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 724 (all V_ys_2 all V_xs_2 all V_P_2 all T_b (c_List_Olist__all(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) <-> c_List_Olist__all(T_b,V_P_2,V_ys_2) & c_List_Olist__all(T_b,V_P_2,V_xs_2))) # label(fact_list__all__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 725 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) & V_m_2 != V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 726 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 727 (all V_A_2 all V_g_2 all V_f_2 all T_c all T_b (class_Groups_Oab__group__add(T_b) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Ominus__class_Ominus(T_b)),V_f_2)),V_g_2)),V_A_2))) # label(fact_setsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 728 (all V_r_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2)) -> (B_xa != B_x -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_xa)),V_r_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_xa),B_x)),V_r_2))))))) <-> c_Relation_Ototal__on(T_b,V_A_2,V_r_2))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 729 (all V_f_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> c_Groups_Oone__class_Oone(T_c) = hAPP(V_f_2,B_x))) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_f_2),c_Groups_Oone__class_Oone(T_c)),V_S_2))))) # label(fact_fold__image__1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 730 (all V_y all V_xs all T_a all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(c_List_Orev(T_a),V_xs)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_k)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_y) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_y)))) # label(fact_rev__update) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 731 (all V_b all V_a_H all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),V_a_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_H),V_b)))))) # label(fact_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 732 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),T_d),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_UN__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 733 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_OInf__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 734 (all V_x_2 all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,T_b),V_A_2),V_x_2) = c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_fun(T_c,T_b))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_c,T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,T_b))),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_fun(T_c,T_b))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),T_c,T_b),c_COMBI(tc_fun(T_c,T_b))),V_x_2)))))))) # label(fact_Inf__apply) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 735 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_abs__power2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 736 (all V_vs all V_ys all V_us all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_us) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_vs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys) -> hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),hAPP(hAPP(c_List_Oappend(T_b),V_us),V_vs)) = hAPP(hAPP(c_List_Oappend(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_us)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_ys),V_vs))))) # label(fact_zip__append) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 737 (all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_c_2),c_Relation_OId__on(T_b,V_A_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x) != V_c_2)))) # label(fact_Id__onE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 738 (all V_A_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 739 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 740 (all V_m_2 all V_Q_2 all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_n_2)) -> (hBOOL(hAPP(V_Q_2,V_m_2)) -> (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> ((all B_k (hBOOL(hAPP(V_P_2,hAPP(c_Nat_OSuc,B_k))) <-> hBOOL(hAPP(V_Q_2,B_k)))) -> hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_Q_2)) = c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_P_2)))))) # label(fact_Least__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 741 (all V_r_2 all V_A_2 all T_b hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),T_b),c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_b),c_Relation_OImage(T_b,T_b,V_r_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool))))) # label(fact_quotient__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 742 (all V_t_2 all V_s_2 all V_b_2 all V_a_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_s_2),V_t_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_t_2))),c_FunDef_Opair__leq))))) # label(fact_pair__leqI2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 743 (all V_b all V_a all T_a (class_Rings_Oordered__ring__abs(T_a) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))) # label(fact_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 744 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 745 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_card__Suc__Diff1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 746 (all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) & hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2)))) # label(fact_list__all2__Cons) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 747 (all V_Q_2 all V_P_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2),V_Q_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_P_2))) -> (all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_x)),V_Q_2)) -> B_y = B_x)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_P_2)))))) # label(fact_rtrancl__Un__separator__converseE) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 748 (all V_r_2 all V_ys_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_ys_2)),c_List_Olex(T_b,V_r_2)))) # label(fact_Nil__notin__lex) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 749 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),V_n_2)),c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))))) # label(fact_less__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 750 (all V_t all V_s (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_s),V_t)) -> V_s != V_t)) # label(fact_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 751 (all V_v_2 all V_m_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) <-> V_m_2 = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2))))) # label(fact_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 752 (all V_r_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2)),V_r_2)) = c_Transitive__Closure_Otrancl(T_b,V_r_2)) # label(fact_trancl__unfold) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 753 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)))) # label(fact_Int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 754 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 755 (all V_y all V_x (c_Nat__Transfer_Ois__nat(V_x) -> (c_Nat__Transfer_Ois__nat(V_y) -> c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y))))) # label(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 756 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),V_q))))))))) # label(fact_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 757 (all V_b all V_b_H all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b_H)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))))))) # label(fact_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 758 (all V_x_2 all V_r_2 all V_v_2 all V_u_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_u_2),V_v_2)),c_List_Olexord(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_x_2),V_u_2)),hAPP(hAPP(c_List_Oappend(T_b),V_x_2),V_v_2))),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__append__leftI) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 759 (all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)))) # label(fact_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 760 (all V_r_H_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_r_H_2)) -> (c_Relation_Orefl__on(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_r_H_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel1(T_b,V_r_2)),c_List_Olistrel(T_b,V_r_H_2)))))) # label(fact_listrel1__subset__listrel) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 761 (all V_lazy__sequence_H all V_a_H all T_a c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a) != c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a_H,V_lazy__sequence_H)) # label(fact_lazy__sequence_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 762 (all V_xs_2 all V_f_2 all T_c all T_b (c_List_Olist_ONil(T_c) = V_xs_2 <-> c_List_Olist_ONil(T_b) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_Nil__is__map__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 763 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Olinorder__class_Oinsort__insert__key(T_b,T_b,c_COMBI(T_b),V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),V_xs_2)))) # label(fact_insort__insert__insort) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 764 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (V_y_2 != V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))))) # label(fact_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 765 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)))) # label(fact_inf2D1) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 766 (all V_y_2 all V_x_2 all V_R_2 all T_b (-hBOOL(hAPP(hAPP(V_R_2,V_y_2),V_x_2)) & hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_y_2)) <-> c_Partial__Function_Omk__less(T_b,V_R_2,V_x_2,V_y_2))) # label(fact_mk__less__def) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 767 (all V_w_2 hAPP(hAPP(c_HOL_OLet(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2)),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_COMBI(tc_Nat_Onat))) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) # label(fact_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 768 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 769 (all V_A_2 all T_b V_A_2 = hAPP(c_Relation_ORange(T_b,T_b),c_Relation_OId__on(T_b,V_A_2))) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 770 (all V_P_2 all V_Q_2 all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> ((all B_x (hBOOL(hAPP(V_Q_2,B_x)) -> hBOOL(hAPP(V_P_2,B_x)))) -> ((exists B_x1 hBOOL(hAPP(V_Q_2,B_x1))) -> (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Quotient_ORespects(T_b,V_R_2))))))))) # label(fact_bex__reg__right) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 771 (all V_l all V_k (c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k),V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 772 (all V_l_2 all V_k_2 all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),V_l_2))))) # label(fact_greaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 773 (all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 774 (all T_b all V_a_2 all V_P_2 ((hBOOL(hAPP(V_P_2,V_a_2)) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(c_fequal,V_a_2))),V_P_2))) & (-hBOOL(hAPP(V_P_2,V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(c_fequal,V_a_2))),V_P_2))))) # label(fact_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 775 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 776 (all V_f_2 all V_xs_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(c_List_Ofoldl(T_b,T_c,hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_f_2)),c_Orderings_Otop__class_Otop(T_b)),V_xs_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),hAPP(c_List_Oset(T_c),V_xs_2)),V_f_2))) # label(fact_INFI__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 777 (all V_k_2 all V_P_2 all V_d_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2)) -> ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_d_2))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))))))))) # label(fact_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 778 (all V_m_2 all V_n_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))) <-> V_n_2 = V_m_2))) # label(fact_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 779 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 780 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__commute) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 781 (all V_xs_2 all V_x_2 all V_j_2 all V_i_2 (hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_x_2),V_xs_2) = c_List_Oupt(V_i_2,V_j_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_j_2)) & V_x_2 = V_i_2 & V_xs_2 = c_List_Oupt(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_j_2))) # label(fact_upt__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 782 (all V_j_2 all V_i_2 all V_k_2 hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))) # label(fact_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.62/8.70 783 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 784 (all V_B_2 all V_J_2 all T_d all V_A_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2))),V_B_2)))) # label(fact_Un__INT__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 785 (all T_b all V_R_2 all T_c all V_P_2 ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> c_Wellfounded_Owf(T_c,hAPP(V_R_2,B_x)))) -> c_Wellfounded_Owf(tc_prod(T_b,T_c),c_Recdef_Osame__fst(T_b,T_c,V_P_2,V_R_2)))) # label(fact_wf__same__fst) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 786 (all V_l all V_k c_Int_OBit1(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l))) # label(fact_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 787 (all V_P_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(c_Set_OCollect(T_b),V_P_2)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2)))),V_P_2))) # label(fact_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 788 (all V_a_2 all V_f_2 all T_b all V_g_2 (V_g_2 = c_Finite__Set_Ofold1(T_b,V_f_2) -> hAPP(V_g_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_a_2)) # label(fact_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 789 (all T_b c_Relation_OId(T_b) = hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_prod(T_b,T_b)),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(tc_prod(T_b,T_b),tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_HOL_Obool)),tc_prod(T_b,T_b)),c_COMBB(tc_prod(T_b,T_b),tc_HOL_Obool,T_b)),c_fequal)),hAPP(hAPP(c_COMBS(T_b,T_b,tc_prod(T_b,T_b)),c_Product__Type_OPair(T_b,T_b)),c_COMBI(T_b)))))) # label(fact_Id__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 790 (all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),T_c),c_Set_OPow(T_b)),V_B_2)) = hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Pow__INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 791 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OMin),V_k) = c_Int_Opred(V_k)) # label(fact_add__Min) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 792 (all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 793 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2)) # label(fact_INT__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 794 (all V_n hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_min__0L) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 795 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2)))) # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 796 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_n))) # label(fact_le__refl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 797 (all V_A_2 all V_B_2 all T_c all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Union__image__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 798 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 799 (all V_a4_2 all V_a2_2 all V_a3_2 all V_a1_2 all V_r_2 all T_b (c_List_Olistrelp(T_b,V_r_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_a1_2),V_a3_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_a2_2),V_a4_2)) <-> hBOOL(hAPP(hAPP(V_r_2,V_a1_2),V_a2_2)) & c_List_Olistrelp(T_b,V_r_2,V_a3_2,V_a4_2))) # label(fact_listrelp_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 800 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> c_Nat__Transfer_Onat__set(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.70 801 (all V_a_2 all V_c_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2))))) # label(fact_divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 802 (all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2))))))) # label(fact_Inf__le__Sup) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 803 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 804 (all V_xss_2 all T_b c_List_Oconcat(T_b,V_xss_2) = hAPP(c_List_Ofoldl(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Oappend(T_b),c_List_Olist_ONil(T_b)),V_xss_2)) # label(fact_concat__conv__foldl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 805 (all V_B_2 all V_b_2 all V_A_2 all V_a_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2))) = hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2)))),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2))))) # label(fact_insert__times__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 806 (all V_ys all V_y all V_xs all V_x all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y),V_ys)) = hAPP(hAPP(c_List_Olist_OCons(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))) # label(fact_zip__Cons__Cons) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 807 (all V_B_2 all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2)))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))),V_f_2)),c_Groups_Ozero__class_Ozero(T_c))),V_A_2)))) # label(fact_setsum__restrict__set_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 808 (all V_B_2 all V_A_2 all T_c all T_b (hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) | c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_B_2)) # label(fact_Times__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 809 (all V_B_2 all V_r_2 all T_b all T_c hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Relation_OImage(T_c,T_b,V_r_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Set_Oinsert(T_c)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 810 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_subset__trans) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 811 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2))))) # label(fact_atLeastLessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 812 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_odd__power__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 813 (all V_z_H all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z))),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z_H))) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_z_H)))) # label(fact_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 814 (all V_x_2 all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)))) # label(fact_equivp__reflp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 815 (all V_n_2 all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),hAPP(c_List_Olexn(T_b,V_r_2),V_n_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = V_n_2 & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = V_n_2)) # label(fact_lexn__length) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 816 (all V_b_2 all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),V_b_2))))) # label(fact_abs__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 817 (all V_B_2 all V_A_2 all T_c all V_b_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_INT__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 818 (all V_B_H_2 all V_B_2 all V_g_2 all T_d all T_e all V_A_H_2 all V_A_2 all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = V_A_H_2 -> (V_B_H_2 = hAPP(c_Set_Oimage(T_e,T_d,V_g_2),V_B_2) -> hAPP(hAPP(c_Product__Type_OSigma(T_b,T_d),V_A_H_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_b),V_B_H_2)) = hAPP(c_Set_Oimage(tc_prod(T_c,T_e),tc_prod(T_b,T_d),c_Product__Type_Omap__pair(T_c,T_b,T_e,T_d,V_f_2,V_g_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_e),V_A_2),hAPP(c_COMBK(tc_fun(T_e,tc_HOL_Obool),T_c),V_B_2)))))) # label(fact_map__pair__surj__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 819 (all V_a_2 all V_b_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_atLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 820 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 821 (all V_list all V_a all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_a),V_list)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_list)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_list_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 822 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x = c_List_Olast(T_a,c_List_Oreplicate(T_a,V_n,V_x)))) # label(fact_last__replicate) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 823 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> hAPP(c_Finite__Set_Ocard(T_c),V_B_2) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) # label(fact_bij__betw__same__card) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 824 (all V_b all V_a hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_b))) # label(fact_zmod__int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 825 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 826 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Osgn__class_Osgn(T_b,V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_sgn__less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 827 (all V_b c_Divides_OposDivAlg(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 828 (all V_x all V_xs all T_a all V_k ((hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) != V_k -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = hAPP(hAPP(c_List_Olist__update(T_a,c_List_Obutlast(T_a,V_xs)),V_k),V_x)) & (V_k = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = c_List_Obutlast(T_a,V_xs)))) # label(fact_butlast__list__update) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 829 (all V_A_2 all T_b c_List_Olist_ONil(T_b) = c_List_Osublist(T_b,c_List_Olist_ONil(T_b),V_A_2)) # label(fact_sublist__nil) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 830 (all V_A_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) # label(fact_Compl__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 831 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))))) # label(fact_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 832 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))))) # label(fact_Sup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 833 (all V_ys_2 all V_xs_2 all T_c all T_b hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)) = hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_prod(T_b,T_c)),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_prod(T_b,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_prod(T_b,T_c)),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_prod(T_b,T_c)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(tc_prod(T_b,T_c),tc_fun(tc_Nat_Onat,tc_prod(T_b,T_c)),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_prod(T_b,T_c)),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_prod(T_b,T_c)),c_COMBB(tc_prod(T_b,T_c),tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_c,tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_prod(T_b,T_c)),tc_Nat_Onat),c_Product__Type_OPair(T_b,T_c)),hAPP(c_List_Onth(T_b),V_xs_2))),hAPP(c_List_Onth(T_c),V_ys_2)))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))))))) # label(fact_set__zip) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 834 (all V_xs_2 all V_P_2 all T_b hAPP(c_List_Orev(T_b),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(c_List_Orev(T_b),V_xs_2))) # label(fact_rev__filter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 835 (all T_b all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_2)) -> hBOOL(hAPP(V_P_2,c_Hilbert__Choice_OEps(T_b,V_P_2))))) # label(fact_someI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 836 (all V_d_2 all V_m_2 ((exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_d_2),B_q)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,V_d_2))) # label(fact_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 837 (all V_xs_2 all V_a_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_a_2),V_xs_2) != c_List_Olist_ONil(T_b))) # label(fact_insort__not__Nil) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 838 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 839 (all V_b_2 all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (V_a_2 = V_b_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 840 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) # label(fact_diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 841 (all V_y_2 all V_x_2 (V_y_2 = V_x_2 | -hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)))) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 842 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a))) # label(fact_minus__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 843 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_z))) # label(fact_mult__minus1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 844 (all V_n all V_m all V_u all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_i)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_u)),V_m)),V_n))) # label(fact_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 845 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 846 (all T_a (class_Int_Onumber__ring(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)))) # label(fact_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 847 (all V_xs_2 all T_b (c_List_Olist_ONil(T_b) = V_xs_2 <-> c_List_Olist_ONil(T_b) = hAPP(c_List_Orev(T_b),V_xs_2))) # label(fact_rev__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 848 (all T_b c_Predicate_Oconversep(T_b,T_b,c_fequal) = c_fequal) # label(fact_conversep__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 849 (all V_r_2 all T_b hAPP(c_Relation_ORange(T_b,T_b),V_r_2) = hAPP(c_Relation_ORange(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2))) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 850 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2))))))) # label(fact_finite__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 851 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 852 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))))) # label(fact_refl__onD1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 853 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 854 (all V_ys_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> (c_List_Olinorder__class_Osorted(T_b,V_ys_2) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_ys_2)) -> (hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(c_List_Oset(T_b),V_ys_2) -> V_xs_2 = V_ys_2))))))) # label(fact_sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 855 (all V_D_2 all V_C_2 all V_R_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Omax__ext(T_b,V_R_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)),c_Wellfounded_Omax__ext(T_b,V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2))),c_Wellfounded_Omax__ext(T_b,V_R_2)))))) # label(fact_max__ext__additive) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 856 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x))) # label(fact_sup__commute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 857 (all V_ys_2 all V_xs_2 all T_b (c_List_Olist_ONil(T_b) = V_ys_2 <-> hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) = V_xs_2)) # label(fact_append__self__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 858 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2)),V_a_2)),c_Groups_Ozero__class_Ozero(T_b))))))) # label(fact_Ints__odd__less__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 859 (all V_B_2 all T_c all T_b all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(c_Set_Oimage(tc_prod(T_b,T_c),T_c,c_Product__Type_Osnd(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2))) = V_B_2) & (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(tc_prod(T_b,T_c),T_c,c_Product__Type_Osnd(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_snd__image__times) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 860 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),V_A_2))))))) # label(fact_Max_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 861 (all V_x_2 all T_b (class_Int_Onumber__ring(T_b) & class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Oone__class_Oone(T_b)))))) # label(fact_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 862 (all V_I_2 all V_xs_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),c_List_Osublist(T_b,V_xs_2,V_I_2))),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_set__sublist__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 863 (all V_w_2 all V_v_2 all T_b (class_Int_Onumber__ring(T_b) -> hAPP(hAPP(c_HOL_OLet(T_b,T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),c_Groups_Otimes__class_Otimes(T_b)),c_COMBI(T_b))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))))) # label(fact_power__number__of__even__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 864 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)))) # label(fact_min__max_Oinf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 865 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)))) # label(fact_INT__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 866 (all V_k V_k = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OPls)) # label(fact_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 867 (all V_xs_2 all V_x_2 all V_P_2 all T_b c_List_Opartition(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),tc_fun(tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),tc_List_Olist(T_b)),c_COMBS(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(tc_List_Olist(T_b),tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_List_Olist(T_b)),hAPP(c_If(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(V_P_2,V_x_2)))),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_List_Olist(T_b)),c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(c_List_Olist_OCons(T_b),V_x_2))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)))),tc_List_Olist(T_b)),c_COMBB(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_List_Olist(T_b))),c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)))),hAPP(c_List_Olist_OCons(T_b),V_x_2)))),c_List_Opartition(T_b,V_P_2,V_xs_2))) # label(fact_partition_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 868 (all V_a2_2 all V_a1_2 all V_f_2 all T_d all V_r2_2 all V_A2_2 all T_c all V_r1_2 all V_A1_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_c,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_b,T_c,tc_fun(T_d,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a1_2),V_A1_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a2_2),V_A2_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r1_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a1_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a2_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))),V_f_2)) = hAPP(hAPP(V_f_2,V_a1_2),V_a2_2))))))) # label(fact_UN__equiv__class2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 869 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (V_B_2 = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2)) -> hAPP(V_g_2,B_a) = hAPP(V_h_2,hAPP(V_f_2,B_a)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_h_2),V_B_2)))))) # label(fact_setsum__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 870 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 871 (all V_B_2 all V_C_2 all T_c all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(V_B_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_INT__greatest) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 872 (all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))) # label(fact_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 873 (all V_Q_2 all V_P_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_Q_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_b),V_P_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Set_OCollect(T_b),V_Q_2))))))) # label(fact_Int__Collect__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 874 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> (exists B_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k) = V_n_2))) # label(fact_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 875 (all V_y_2 all V_x_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_x_2),V_y_2)))) # label(fact_bot__empty__eq2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 876 (all V_y_2 all V_x_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2) = c_HOL_OThe(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(c_fequal,V_x_2)))),hAPP(c_fequal,V_y_2))))) # label(fact_The__split__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 877 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) # label(fact_power__odd__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 878 (all V_x all V_k all V_i all T_a c_List_Oreplicate(T_a,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_i),V_k),V_x) = hAPP(hAPP(c_List_Otake(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x))) # label(fact_take__replicate) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 879 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 880 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)))) # label(fact_max__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 881 (all V_b all V_a all T_b all T_a hAPP(c_Nat_Osize__class_Osize(tc_Nitpick_Opair__box(T_a,T_b)),c_Nitpick_Opair__box_OPairBox(T_a,T_b,V_a,V_b)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_pair__box_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 882 (all V_f_2 all V_a_2 all V_F_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_F_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_F_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_F_2)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 883 (all V_B_2 all V_A_2 all V_a_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_B_2,V_a_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),V_B_2)) # label(fact_UN__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 884 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2) = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_min__max_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 885 (all V_B_2 all T_b V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_B_2)) # label(fact_Int__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 886 (all V_b_2 all V_a_2 all T_b (class_Rings_Oidom(T_b) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_b_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_a_2) <-> V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2) | V_b_2 = V_a_2))) # label(fact_square__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 887 (all V_w all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))))) # label(fact_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 888 (all V_y_2 all T_b (class_Int_Onumber__ring(T_b) & class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Oone__class_Oone(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))))) # label(fact_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 889 (all V_T_2 all V_S_2 all V_R_2 all T_c all T_d all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_R_2),V_S_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_R_2),V_T_2))) # label(fact_rel__comp__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 890 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_inf__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 891 (all V_Q_2 all T_c all V_P_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_P_2))) -> ((all B_y (hBOOL(hAPP(V_P_2,B_y)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,T_b,tc_HOL_Obool),V_Q_2),B_y)))))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2))),V_Q_2)))))))) # label(fact_finite__Collect__bounded__ex) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 892 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 893 (all V_B_2 all V_A_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_c,T_b),V_A_2)),hAPP(c_Relation_ORange(T_c,T_b),V_B_2))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 894 (all T_a c_List_Oremdups(T_a,c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_remdups_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 895 (all V_c_2 all T_c all T_b all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_c_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) & (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> V_c_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_c_2))))) # label(fact_INT__constant) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 896 (all V_g_2 all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) & class_Groups_Oordered__cancel__ab__semigroup__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)))))))) # label(fact_setsum__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 897 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_n))) # label(fact_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 898 (all V_q all V_r all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)) = V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),c_Groups_Oone__class_Oone(tc_Int_Oint))))))) # label(fact_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 899 (all T_a c_Nitpick_Ozero__frac(T_a) = c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_Nitpick_Ozero__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 900 (all V_r_2 all V_xs_2 all V_ys_2 all V_y_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2)),V_xs_2)),c_List_Olistrel(T_b,V_r_2))) -> -(all B_y all B_ys (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),B_y),B_ys) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),B_y)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_ys_2),B_ys)),c_List_Olistrel(T_b,V_r_2)))))))) # label(fact_listrel__Cons1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 901 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k))) # label(fact_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.71 902 (all V_ys_2 all V_xs_2 all V_g_2 all T_e all V_f_2 all T_b all T_d all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_e,T_d),T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_d,T_b),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,T_b),T_e),hAPP(c_COMBB(tc_prod(T_c,T_d),T_b,T_d),V_f_2)),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_prod(T_c,T_d)),T_e),c_Product__Type_OPair(T_c,T_d)),V_g_2)))),hAPP(hAPP(c_List_Ozip(T_e,T_d),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_b),V_f_2),hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Omap(T_e,T_c),V_g_2),V_xs_2)),V_ys_2))) # label(fact_map__zip__map) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 903 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_minus__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 904 (all V_f_2 all V_a_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))))) # label(fact_setprod__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 905 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> V_a = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_power__one__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 906 (all V_xs_2 all T_b hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_finite__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 907 (all V_r_2 all T_b all T_c c_Predicate_Oconversep(T_c,T_b,c_Predicate_Oconversep(T_b,T_c,V_r_2)) = V_r_2) # label(fact_conversep__conversep) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 908 (all T_b all V_ys_2 all V_xs_2 (V_ys_2 = V_xs_2 <-> (all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i) = hAPP(hAPP(c_List_Onth(T_b),V_ys_2),B_i))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) # label(fact_list__eq__iff__nth__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 909 (all T_b c_Nat_Ocompow(T_b,T_b) = c_Nat_Ofunpow(T_b)) # label(fact_funpow__code__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 910 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_rtrancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 911 (all V_P_2 all V_f_2 all T_b all T_c c_New__Random__Sequence_Opos__map(T_c,T_b,V_f_2,V_P_2) = c_New__Random__Sequence_Opos__bind(T_c,T_b,V_P_2,hAPP(c_Fun_Ocomp(T_b,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b))))),T_c,c_New__Random__Sequence_Opos__single(T_b)),V_f_2))) # label(fact_pos__map__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 912 (all T_b all V_A_2 all V_x_2 all V_B_2 ((-hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(V_A_2,V_x_2))) -> hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1CI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 913 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2))))) # label(fact_SigmaD2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 914 (all V_S_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) & class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_b,V_S_2)))))) # label(fact_less__Sup__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 915 (all V_y all T_a -hBOOL(hAPP(hAPP(c_List_Omember(T_a),c_List_Olist_ONil(T_a)),V_y))) # label(fact_member__rec_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 916 (all V_A_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_c) & class_Orderings_Olinorder(T_b) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2)))) # label(fact_strict__mono__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 917 (all V_x_2 all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_fun(T_c,T_b))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_c,T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,T_b))),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_fun(T_c,T_b))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),T_c,T_b),c_COMBI(tc_fun(T_c,T_b))),V_x_2)))))) = hAPP(c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,T_b),V_A_2),V_x_2))) # label(fact_Sup__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 918 (all V_z all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w))),V_z))) # label(fact_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 919 (all V_S_2 all T_b c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_S_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_S_2),c_COMBI(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Inter__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 920 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))))) # label(fact_distrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 921 (all V_xs all V_x all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_xs))) # label(fact_replicate__app__Cons__same) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 922 (all V_G_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_G_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_G_2)))))) # label(fact_finite__UnI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 923 (all V_ys_2 all V_xs_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_ys_2))) # label(fact_map__append) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 924 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Int__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 925 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_i)),V_k))))) # label(fact_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 926 (all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_vimage__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 927 (all V_i_2 all V_P_2 ((all B_n (V_i_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) -> hBOOL(hAPP(V_P_2,B_n)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) <-> hBOOL(hAPP(V_P_2,hAPP(c_Int_Onat,V_i_2))))) # label(fact_split__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 928 (all V_b_2 all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (V_b_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) <-> V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))) # label(fact_equation__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 929 (all V_k_2 all V_f_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_k_2)))) # label(fact_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 930 (all V_a all V_b all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)))))) # label(fact_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 931 (all V_g_2 all V_xs_2 all V_f_2 all T_b all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x))) <-> hAPP(hAPP(c_List_Omap(T_c,T_b),V_g_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_map__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 932 (all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) <-> c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) & hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_bij__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 933 (all V_P_2 all V_d_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2)) -> ((all B_x all B_k (hBOOL(hAPP(V_P_2,B_x)) <-> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B_k),V_d_2)))))) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_d_2))) & hBOOL(hAPP(V_P_2,B_x)))))))) # label(fact_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 934 (all V_n_2 all V_m_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))))) # label(fact_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 935 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) -> (-hBOOL(hAPP(V_A_2,V_x_2)) -> hBOOL(hAPP(V_B_2,V_x_2))))) # label(fact_sup1E) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 936 (all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Option_Ooption(T_b)),c_Orderings_Otop__class_Otop(tc_fun(tc_Option_Ooption(T_b),tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_finite__option__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 937 (all V_n_2 all V_a_2 all V_times_2 all V_one_2 all T_b hAPP(hAPP(V_times_2,V_a_2),hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),V_n_2)) = hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_power_Opower_Opower__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 938 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a0_2),V_a1_2))) -> ((all B_x all B_y (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B_x),B_y))) -> ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B_y -> hBOOL(hAPP(hAPP(V_P_2,B_y),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B_x,B_y)))) -> hBOOL(hAPP(hAPP(V_P_2,B_x),B_y))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 939 (all V_b_2 all V_a_2 all V_r_2 all T_b (c_Relation_Oantisym(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_a_2)),V_r_2)) -> V_a_2 = V_b_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 940 (all V_b_2 all V_a_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))),V_S_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(V_b_2,V_a_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))),V_S_2))))) # label(fact_setsum__delta) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 941 (all V_r_2 all T_b hAPP(c_Relation_OField(T_b),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) = hAPP(c_Relation_OField(T_b),V_r_2)) # label(fact_Field__converse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 942 (all V_j_2 all V_i_2 (c_List_Oupt(V_i_2,V_j_2) = c_List_Olist_ONil(tc_Nat_Onat) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_j_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)))) # label(fact_upt__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 943 (all V_ys all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) # label(fact_length__append) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 944 (all V_n all V_i ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_i -> c_Code__Numeral_Onat__of__aux(V_i,V_n) = c_Code__Numeral_Onat__of__aux(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_OSuc,V_n))) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_i -> V_n = c_Code__Numeral_Onat__of__aux(V_i,V_n)))) # label(fact_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 945 (all V_r_2 all V_A_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_quotient__is__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 946 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l)))))) # label(fact_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 947 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Oab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_setsum__Un) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 948 (all V_x_2 all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_fun(T_c,T_b))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_c,T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,T_b))),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_fun(T_c,T_b))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),T_c,T_b),c_COMBI(tc_fun(T_c,T_b))),V_x_2)))))) = hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,T_b),V_A_2),V_x_2))) # label(fact_Inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 949 (all V_B_2 all V_A_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_c,T_b),V_A_2)),hAPP(c_Relation_ORange(T_c,T_b),V_B_2))),hAPP(c_Relation_ORange(T_c,T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Range__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 950 (all V_n_2 all V_P_2 all V_Q_2 ((all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_Q_2,B_x)),hAPP(V_P_2,B_x)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_P_2)),V_Q_2)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_P_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_Q_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))))) # label(fact_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 951 (all V_r_2 all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_r_2) = c_Transitive__Closure_Otrancl(T_b,V_r_2)) # label(fact_trancl__unfold__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 952 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_x_2 all T_b hAPP(hAPP(hAPP(hAPP(c_New__Random__Sequence_Opos__single(T_b),V_x_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Opos__single(T_b,V_x_2)) # label(fact_pos__single__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 953 (all V_P_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_OCollect(T_b),V_P_2))) -> hBOOL(hAPP(V_P_2,V_a_2)))) # label(fact_CollectE) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 954 (all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_of__nat__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 955 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_PowD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 956 (all V_k_2 all T_b (class_Rings_Oring__1(T_b) -> (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_k_2) = c_Groups_Ozero__class_Ozero(T_b)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_k_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_k_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,T_b),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(T_b,T_b)),tc_fun(tc_Int_Oint,T_b)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_b),T_b),tc_fun(tc_fun(tc_Int_Oint,tc_fun(T_b,T_b)),tc_fun(tc_Int_Oint,T_b)),tc_Int_Oint),c_COMBB(tc_fun(T_b,T_b),T_b,tc_Int_Oint)),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_fun(T_b,T_b),T_b),tc_Int_Oint),c_HOL_OLet(T_b,T_b)),c_Int_Oring__1__class_Oof__int(T_b)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(T_b,T_b),tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,T_b)),tc_Int_Oint),c_COMBS(T_b,T_b,T_b)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(T_b,T_b),tc_fun(T_b,tc_fun(T_b,T_b))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_fun(T_b,T_b))),tc_Int_Oint),c_COMBB(T_b,tc_fun(T_b,T_b),T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),tc_Int_Oint),c_If(T_b)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),c_Groups_Oplus__class_Oplus(T_b)),c_COMBI(T_b))))),hAPP(hAPP(c_COMBC(T_b,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_b),c_Groups_Oplus__class_Oplus(T_b)),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),c_Groups_Oplus__class_Oplus(T_b)),c_COMBI(T_b)))),c_Groups_Oone__class_Oone(T_b))))),c_Divides_Odivmod__int(V_k_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))))) # label(fact_of__int__num) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 957 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I20_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 958 (all V_C_2 all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2)))) # label(fact_Times__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 959 (all V_r_2 all V_S_2 all T_d all T_b all T_c hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_S_2),V_r_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_S_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),T_d),c_Relation_Oconverse(T_c,T_b)),V_r_2))) # label(fact_converse__INTER) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 960 (all V_n_2 all V_m_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = c_Nat_Onat_Onat__case(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),c_COMBI(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 961 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)))) # label(fact_inf__top__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 962 (all V_ys_2 all V_xs_2 all T_b (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) <-> V_xs_2 = c_List_Olist_ONil(T_b))) # label(fact_append__self__conv2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 963 (all V_r_2 all T_b hAPP(c_Relation_OImage(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Olistrel(T_b,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)))) # label(fact_listrel__Nil) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 964 (all V_a all V_b all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)))))) # label(fact_less__supI2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 965 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (V_t_2 != B_x -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2) != V_t_2)))))) # label(fact_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 966 (all V_s_2 all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> (c_Relation_Otrans(T_b,V_s_2) -> c_Relation_Otrans(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_trans__Int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 967 (all V_ys all V_x all V_xs all T_a hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) = V_x) # label(fact_nth__append__length) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 968 (all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__divide) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 969 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 970 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b)),V_c)) # label(fact_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 971 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_trancl__converseD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 972 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 973 (all V_bs_2 all V_as_2 all V_f_2 all T_d all V_P_2 all T_c all T_b (c_List_Olist__all2(T_d,T_c,hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_HOL_Obool),T_d),V_P_2),V_f_2),V_as_2,V_bs_2) <-> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Omap(T_d,T_b),V_f_2),V_as_2),V_bs_2))) # label(fact_list__all2__map1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 974 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))) # label(fact_le0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 975 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_xs) = c_List_Olast(T_a,hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_last__rev) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 976 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 977 (all V_B_2 all V_A_2 all V_R_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_B_2)) = hAPP(c_Relation_OImage(T_c,T_b,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Image__Un) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 978 (all V_r_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),c_Transitive__Closure_Otrancl(T_b,V_r_2))))) # label(fact_finite__trancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 979 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Min_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 980 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)),V_z) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 981 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 982 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2))))) # label(fact_DomainI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 983 (all V_xs_2 all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,V_P_2),c_List_Oremdups(T_b,V_xs_2)) = c_List_Oremdups(T_b,hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))) # label(fact_remdups__filter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 984 (all T_b all V_B_2 all V_x_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_x_2)) -> (hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))))) # label(fact_inf1I) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 985 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i))) # label(fact_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 986 (all V_d c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_Int_Oint,V_d) = c_Lazy__Sequence_Osmall__lazy_H(V_d,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_d))) # label(fact_small__lazy__int__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 987 (all V_l_2 all V_k_2 ((V_k_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l_2 -> hAPP(c_Product__Type_Oapsnd(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint,hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),hAPP(hAPP(hAPP(c_If(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_fequal,c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_k_2)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),c_Divides_Opdivmod(V_k_2,V_l_2)),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_fun(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),hAPP(c_COMBC(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_If(tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l_2))))),c_Divides_Opdivmod(V_k_2,V_l_2)))) = c_Divides_Odivmod__int(V_k_2,V_l_2)) & (V_l_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2) = c_Divides_Odivmod__int(V_k_2,V_l_2))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Divides_Odivmod__int(V_k_2,V_l_2)))) # label(fact_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 988 (all V_A_2 all T_c all T_b c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Sigma__empty2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 989 (all V_xs_2 all V_x_2 all T_b (class_Lattices_Olattice(T_b) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) = hAPP(c_List_Ofoldl(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_xs_2))) # label(fact_Sup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 990 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(V_f_2,V_x_2) = V_y_2 -> V_x_2 = hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),V_y_2))))) # label(fact_inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 991 (all V_k_2 all V_f_2 all V_n_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2))) -> (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)) & V_k_2 = hAPP(V_f_2,B_i))))))) # label(fact_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 992 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 993 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2))) # label(fact_atMost__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 994 (all V_Q all V_P all T_b all T_a V_P = hAPP(hAPP(c_COMBK(T_a,T_b),V_P),V_Q)) # label(help_c__COMBK__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 995 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))) # label(fact_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 996 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_On__lists(T_a,V_n,V_xs)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) # label(fact_length__n__lists) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 997 (all V_x_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_List_Omember(T_b),V_xs_2),V_x_2)))) # label(fact_List_Omember__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 998 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_b (class_Lattices_Olattice(T_b) -> hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 8.70/8.72 999 (all V_n all V_ns all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),V_ns))) -> hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Olist__update(tc_Nat_Onat,V_ns),V_k),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),V_ns)),V_n)),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),V_ns),V_k)))) # label(fact_listsum__update__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1000 (all V_f_2 all V_xs_2 all V_x_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),c_List_Oremove1(T_b,V_x_2,V_xs_2))))))) # label(fact_listsum__map__remove1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1001 (all V_q all T_a all T_b (class_Rings_Oring__1(T_b) & class_Rings_Oinverse(T_b) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Nitpick_Onum(T_a),V_q))),hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Nitpick_Odenom(T_a),V_q))) = c_Nitpick_Oof__frac(T_a,T_b,V_q))) # label(fact_Nitpick_Oof__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1002 (all V_A_2 all V_f_2 all T_c all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Oabs__class_Oabs(T_b)),V_f_2)),V_A_2))))) # label(fact_setsum__abs) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1003 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)))))) # label(fact_distrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1004 (all V_xs_2 all V_n_2 all T_b c_Enum_On__lists(T_b,hAPP(c_Nat_OSuc,V_n_2),V_xs_2) = c_List_Oconcat(tc_List_Olist(T_b),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),tc_List_Olist(T_b)),c_List_Omap(T_b,tc_List_Olist(T_b))),hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)))),V_xs_2)),c_Enum_On__lists(T_b,V_n_2,V_xs_2)))) # label(fact_n__lists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1005 (all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),c_List_Olist_ONil(T_a))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs))) # label(fact_take__Suc__conv__app__nth) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1006 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1007 (all V_ys_2 all V_P_2 all V_xs_2 all V_x_2 all T_b ((exists B_us exists B_vs (V_xs_2 = hAPP(c_List_Ofilter(T_b,V_P_2),B_vs) & hBOOL(hAPP(V_P_2,V_x_2)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),B_us))) -> -hBOOL(hAPP(V_P_2,B_x)))) & hAPP(hAPP(c_List_Oappend(T_b),B_us),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_vs)) = V_ys_2)) <-> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2) = hAPP(c_List_Ofilter(T_b,V_P_2),V_ys_2))) # label(fact_Cons__eq__filter__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1008 (all V_n_2 all V_R_2 all T_b (c_Relation_Osingle__valued(T_b,T_b,V_R_2) -> c_Relation_Osingle__valued(T_b,T_b,hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2)))) # label(fact_single__valued__rel__pow) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1009 (all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_A_2) = c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)))) # label(fact_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1010 (all V_xs_2 all T_b c_List_Ofoldr(tc_List_Olist(T_b),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_b)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2))) # label(fact_length__transpose) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1011 (all V_b all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_m)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1012 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> c_List_Olast(T_a,V_xs) = hAPP(c_List_Ohd(T_a),hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_hd__rev) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1013 (all V_lazy__sequence_H_2 all V_a_H_2 all V_lazy__sequence_2 all V_a_2 all T_b (c_Lazy__Sequence_Olazy__sequence_OInsert(T_b,V_a_H_2,V_lazy__sequence_H_2) = c_Lazy__Sequence_Olazy__sequence_OInsert(T_b,V_a_2,V_lazy__sequence_2) <-> V_lazy__sequence_H_2 = V_lazy__sequence_2 & V_a_H_2 = V_a_2)) # label(fact_lazy__sequence_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1014 (all T_a c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = c_Nitpick_Oone__frac(T_a)) # label(fact_Nitpick_Oone__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1015 (all V_f_2 all T_d all T_c all T_b hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,T_d),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),T_d,T_c),V_f_2)),c_Product__Type_OPair(T_b,T_c))) = V_f_2) # label(fact_split__eta) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1016 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),c_List_Oupt(V_j,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))) = c_List_Oupt(V_i,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k)))) # label(fact_upt__add__eq__append) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1017 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> V_y_2 = V_x_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1018 (all V_f_2 all V_A_2 all T_b all T_c (class_Rings_Ocomm__semiring__1(T_c) & class_Rings_Ono__zero__divisors(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & c_Groups_Ozero__class_Ozero(T_c) = hAPP(V_f_2,B_x))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c))))) # label(fact_setprod__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1019 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2)))))) # label(fact_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1020 (all V_r_2 all V_q_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_b),c_Nitpick_Oplus__frac(T_b,V_q_2,c_Nitpick_Ouminus__frac(T_b,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> c_Nitpick_Oless__frac(T_b,V_q_2,V_r_2))) # label(fact_Nitpick_Oless__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1021 (all V_f_2 all V_A_2 all T_c all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) & class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),V_A_2),V_f_2))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(V_f_2,B_x)))))))) # label(fact_less__SUP__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1022 (all V_f2_2 all V_f1_2 all T_c all T_b V_f1_2 = hAPP(c_List_Olist_Olist__case(T_b,T_c,V_f1_2,V_f2_2),c_List_Olist_ONil(T_c))) # label(fact_list_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1023 (all V_xs_2 all V_g_2 all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olinorder__class_Osorted(T_b,hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),hAPP(V_g_2,V_xs_2))),V_xs_2)))) # label(fact_sorted__same) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1024 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y))) # label(fact_inf__commute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1025 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w)) # label(fact_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1026 (all T_b all V_as_2 (V_as_2 != c_List_Olist_ONil(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),c_List_Olast(T_b,V_as_2)),hAPP(c_List_Oset(T_b),V_as_2))))) # label(fact_last__in__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1027 (all V_k_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2))))) # label(fact_setsum__shift__lb__Suc0__0__upt) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1028 (all T_c all V_B_2 all T_b all V_b_2 all V_a_2 all V_f_2 (V_b_2 = hAPP(V_f_2,V_a_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_B_2)))))) # label(fact_vimageI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1029 (all V_f_2 all T_b all T_c c_List_Olist_ONil(T_b) = c_List_Omaps(T_c,T_b,V_f_2,c_List_Olist_ONil(T_c))) # label(fact_maps__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1030 (all V_f_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (V_A_2 = c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hAPP(V_f_2,V_A_2) = V_A_2)))) # label(fact_def__gfp__unfold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1031 (all V_xs_2 all T_b all V_l_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_l_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hAPP(c_Random_Opick(T_b,hAPP(hAPP(c_List_Omap(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,T_b)),hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),V_xs_2)),hAPP(c_Code__Numeral_Oof__nat,V_l_2)) = hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_l_2))) # label(fact_pick__same) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1032 (all V_w_2 all V_z_2 all T_b (class_Int_Onumber__ring(T_b) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) = hAPP(hAPP(c_HOL_OLet(T_b,T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),c_Groups_Otimes__class_Otimes(T_b)),c_COMBI(T_b))))) # label(fact_power__number__of__even) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1033 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1034 (all V_code__numeral_H_1 c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_1)) # label(fact_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1035 (all V_k all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_k),V_n)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_k),V_m)))))) # label(fact_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1036 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1037 (all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)) -> hBOOL(hAPP(V_A_2,B_x)))) <-> hBOOL(hAPP(c_Predicate_OPowp(T_b,V_A_2),V_B_2)))) # label(fact_Powp__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1038 (all V_A_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),tc_Nat_Onat),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool))),c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),V_A_2))) # label(fact_UN__UN__finite__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1039 (all V_x_2 all V_A_2 all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2)) -> (c_Fun_Oinj__on(T_d,T_b,V_g_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2)))) -> hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,hAPP(c_Fun_Ocomp(T_b,T_c,T_d,V_f_2),V_g_2)),V_x_2) = hAPP(hAPP(c_Fun_Ocomp(T_b,T_d,T_c,c_Hilbert__Choice_Oinv__into(T_d,T_b,V_A_2,V_g_2)),c_Hilbert__Choice_Oinv__into(T_b,T_c,hAPP(c_Set_Oimage(T_d,T_b,V_g_2),V_A_2),V_f_2)),V_x_2))))) # label(fact_inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1040 (all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Onat,V_z) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1041 (all V_R_2 all T_c all T_d all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool))),V_R_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))) # label(fact_rel__comp__empty1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1042 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))))) # label(fact_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1043 (all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_B_2)) -> B_x != B_xa)))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1044 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2)))) # label(fact_Un__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1045 (all V_xs_2 all V_P_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OdropWhile(T_b,V_P_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))) # label(fact_length__dropWhile__le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1046 (all V_m hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) # label(fact_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1047 (all V_r_2 all T_b c_List_Olistrel1(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) = hAPP(c_Relation_Oconverse(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olistrel1(T_b,V_r_2))) # label(fact_listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1048 (all V_f_2 all V_R_2 all T_b (c_Wellfounded_Owf(T_b,V_R_2) -> c_Wellfounded_Owf(T_b,c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2)))) # label(fact_wf__mlex) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1049 (all V_A_2 all T_b hAPP(c_Set_OPow(T_b),V_A_2) = hAPP(c_Set_OCollect(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),V_A_2))) # label(fact_Pow__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1050 (all V_A_2 all T_c all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Compl__Times__UNIV2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1051 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1052 (all V_xs_2 all T_b c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Orev(T_b)),hAPP(c_List_Orev(tc_List_Olist(T_b)),V_xs_2))) = hAPP(c_List_Orev(T_b),c_List_Oconcat(T_b,V_xs_2))) # label(fact_rev__concat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1053 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_OposDivAlg(V_a,V_b)))))) # label(fact_posDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1054 (all V_n_2 all V_m_2 all V_k_2 (V_n_2 = V_m_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2))) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1055 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_x)))) & c_List_Olinorder__class_Osorted(T_b,V_xs_2) <-> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_sorted__Cons) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1056 (all V_x_2 all V_y_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2)) -> (V_x_2 = V_y_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1057 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Lattices_Osemilattice__sup(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2))),hAPP(V_f_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_A_2),V_B_2))))))) # label(fact_min__max_Omono__sup) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1058 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)))) # label(fact_abs__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1059 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) <-> hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) & hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2))))) # label(fact_equiv__class__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1060 (all V_P_2 all V_B_2 all V_A_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),hAPP(V_B_2,B_x))) -> hBOOL(hAPP(V_P_2,B_xa)))))))) # label(fact_ball__UN) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1061 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (V_y != V_x -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1062 (all V_r_2 all V_y_2 all V_b_2 all V_x_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_2),V_x_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_b_2),V_y_2))),c_List_Olexord(T_b,V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olexord(T_b,V_r_2))) & V_b_2 = V_a_2 | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)))) # label(fact_lexord__cons__cons) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1063 (all V_xs all V_x all T_a V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nth__Cons__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1064 (all V_c_2 all V_b_2 all V_w_2 all T_b (class_Int_Onumber(T_b) & class_Fields_Ofield__inverse__zero(T_b) -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) <-> (V_c_2 != c_Groups_Ozero__class_Ozero(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2) = V_b_2) & (c_Groups_Ozero__class_Ozero(T_b) = V_c_2 -> hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b))))) # label(fact_eq__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1065 (all V_k_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2)))) # label(fact_finite__atMost) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1066 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_card__Diff__singleton__if) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1067 (all V_z hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_z)))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1068 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),V_c)) # label(fact_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1069 (all V_A_2 all V_f_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2)) -> (hAPP(V_f_2,B_xa) = hAPP(V_f_2,B_x) -> B_x = B_xa))))) <-> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1070 (all V_A_2 all V_x_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)),V_x_2))))) # label(fact_Inf__lower) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1071 (all V_r_H all V_q_H all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> V_r = V_r_H)))) # label(fact_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1072 (all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m))))) # label(fact_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1073 (all V_y_2 all V_x_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2))) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_y_2 -> V_x_2 = c_Nitpick_Onat__gcd(V_x_2,V_y_2)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_y_2 -> c_Nitpick_Onat__gcd(V_y_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x_2,V_y_2)) = c_Nitpick_Onat__gcd(V_x_2,V_y_2)))) # label(fact_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1074 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),c_List_Oremove1(T_a,V_x,V_xs))))) # label(fact_distinct__remove1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1075 (all V_P_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))))) # label(fact_distinct__filter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1076 (all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_x_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)))) # label(fact_image__uminus__greaterThan) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1077 (all V_r_2 all T_b (c_Relation_Otrans(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) <-> c_Relation_Otrans(T_b,V_r_2))) # label(fact_trans__converse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1078 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_min__max_OSup__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1079 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_y != V_x))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1080 (all V_b_2 all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),V_A_2))))))) # label(fact_min__max_Osup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1081 (all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_fequal,V_y_2))) <-> V_y_2 = V_x_2)) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1082 (all V_B_2 all V_A_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hAPP(V_B_2,B_x) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_UNION__empty__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1083 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_psubset__subset__trans) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1084 (all V_l all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_n)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_m))))) # label(fact_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1085 (all V_Q_2 all V_P_2 (-hBOOL(V_P_2) | hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2)) | -hBOOL(V_Q_2))) # label(help_c__fconj__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1086 (all V_a_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> V_a_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_a_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)))))) # label(fact_inf__Sup__absorb) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1087 (all V_A_2 all V_f_2 all T_c all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_vimage__Compl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1088 (all V_xs_2 all V_g_2 all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_g_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Oplus__class_Oplus(T_b)),V_f_2)),V_g_2)),V_xs_2)))) # label(fact_listsum__addf) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1089 (all V_X_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_X_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_X_2)),c_FunDef_Omax__weak)))) # label(fact_wmax__emptyI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1090 (all V_x all V_y ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Nat__Transfer_Otsub(V_x,V_y)))) # label(fact_tsub__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1091 (all V_r_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2)))) <-> c_Relation_Orefl__on(T_b,V_A_2,V_r_2))) # label(fact_refl__on__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1092 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))))))) # label(fact_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1093 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1094 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_y_2)))))) # label(fact_greaterThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1095 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_d_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_d_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_a_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2)))))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1096 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1097 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c hAPP(c_Fun_Ofun__upd(T_c,T_b,V_f_2,V_x_2,V_y_2),V_x_2) = V_y_2) # label(fact_fun__upd__same) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1098 (all V_xs_2 all V_y_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_y_2),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_b,hAPP(c_COMBC(T_b,T_c,T_c),V_f_2),V_y_2),V_xs_2))) # label(fact_fun__left__comm__idem_Ofold__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.73 1099 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_inj__on__Int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1100 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1101 (all V_R_2 all T_b hAPP(c_Relation_ODomain(T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_R_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1102 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Orotate1(T_a),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Otl(T_a),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),V_xs)),c_List_Olist_ONil(T_a))))) # label(fact_rotate1__hd__tl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1103 (all V_k hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k))) # label(fact_dvd__1__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1104 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1105 (all V_x_2 all T_b hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_x_2))) # label(fact_top1I) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1106 (all V_x_2 all V_g_2 all V_f_2 all T_e all T_d all T_c all T_b hAPP(c_Product__Type_Ofst(T_b,T_c),hAPP(c_Product__Type_Omap__pair(T_d,T_b,T_e,T_c,V_f_2,V_g_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_x_2))) # label(fact_fst__map__pair) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1107 (all V_y_2 all V_x_2 all V_Y_2 all V_X_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_Y_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> V_Y_2 = V_X_2))))))) # label(fact_quotient__eqI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1108 (all V_P_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_OCollect(T_b),V_P_2))) -> hBOOL(hAPP(V_P_2,V_a_2)))) # label(fact_CollectD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1109 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)),V_n))) # label(fact_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1110 (all V_lst_2 all V_k_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(c_COMBK(T_b,T_c),V_k_2)),V_lst_2) = c_List_Oreplicate(T_b,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_lst_2),V_k_2)) # label(fact_map__replicate__const) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1111 (all V_xs_2 all V_f_2 all T_c all V_n_2 all T_b hAPP(hAPP(c_List_Otake(T_b),V_n_2),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2))) # label(fact_take__map) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1112 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))))))) # label(fact_le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1113 (all V_f_2 all V_T_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_c))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_f_2),c_Groups_Oone__class_Oone(T_c)),V_S_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_f_2),c_Groups_Oone__class_Oone(T_c)),V_T_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_f_2),c_Groups_Oone__class_Oone(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))))))) # label(fact_fold__image__Un__one) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1114 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1115 (all V_r_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_xs_2)),c_List_Olistrel(T_b,V_r_2))) -> V_xs_2 = c_List_Olist_ONil(T_b))) # label(fact_listrel__Nil1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1116 (all V_R_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Ortrancl(T_b,V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2)))))) # label(fact_rtrancl__imp__UN__rel__pow) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1117 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (((exists B_x hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)))))) # label(fact_card_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1118 (all V_xs all V_x all T_a all V_v ((hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs))) & (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olist_ONil(T_a)))) # label(fact_take__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1119 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_x)))) # label(fact_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1120 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k))) # label(fact_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1121 (all V_x_2 all T_b (class_Int_Onumber__ring(T_b) -> (c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) = c_Groups_Oone__class_Oone(T_b)))) # label(fact_eq__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1122 (all V_b_2 all V_a_2 all V_g_2 all V_f_2 all T_c all T_e all T_b all T_d hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2)) = hAPP(c_Product__Type_Omap__pair(T_d,T_b,T_e,T_c,V_f_2,V_g_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2))) # label(fact_map__pair__simp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1123 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1124 (all V_j_2 all V_i_2 all V_P_2 (c_List_Olist__all(tc_Nat_Onat,V_P_2,c_List_Oupt(V_i_2,V_j_2)) <-> c_List_Oall__interval__nat(V_P_2,V_i_2,V_j_2))) # label(fact_list__all__iff__all__interval__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1125 (all V_t_2 all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(c_List_Orev(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) -> c_List_OtakeWhile(T_c,hAPP(hAPP(c_COMBB(T_b,tc_HOL_Obool,T_c),hAPP(c_Orderings_Oord__class_Oless(T_b),V_t_2)),V_f_2),V_xs_2) = hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(T_b,tc_HOL_Obool,T_c),hAPP(c_Orderings_Oord__class_Oless(T_b),V_t_2)),V_f_2)),V_xs_2)))) # label(fact_filter__equals__takeWhile__sorted__rev) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1126 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),V_k_2)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_atLeast__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1127 (all V_w_2 all V_b_2 all V_a_2 all T_b (class_Fields_Ofield__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = V_a_2) & (hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) != c_Groups_Ozero__class_Ozero(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) = V_b_2) <-> V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))) # label(fact_eq__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1128 (all V_X_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> V_X_2 = hAPP(c_Set_Oimage(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_X_2)))) # label(fact_inv__image__comp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1129 (all V_r_2 all T_c all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_b,T_c),V_r_2) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)) = V_r_2)) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1130 (all V_r_2 all T_b c_Relation_Orefl__on(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) # label(fact_refl__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1131 (all V_B_2 all T_b all T_c c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_B_2)) # label(fact_INT__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1132 (all V_A_2 all V_b_2 all V_a_2 all V_r_2 all T_b (hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)))))) # label(fact_eq__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1133 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))) # label(fact_tl__take) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1134 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1135 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_divide__pos__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1136 (all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> c_Fun_Oid(T_c) = hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_folding_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1137 (all V_A_2 all V_f_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_A_2))),V_A_2))) # label(fact_image__vimage__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1138 (all V_v_2 all V_f_2 all V_a_2 all T_b hAPP(c_Nat_Onat_Onat__rec(T_b,V_a_2,V_f_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_b),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),tc_Int_Oint),c_If(T_b)),c_Nat__Numeral_Oneg)),V_a_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_b,T_b),tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),c_Nat_Onat_Onat__rec(T_b,V_a_2,V_f_2)),c_Int_Onat))))) # label(fact_nat__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1139 (all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)) = c_Divides_OnegDivAlg(V_k,V_l)))) # label(fact_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1140 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k))) # label(fact_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1141 (all V_xs_2 all T_b c_Random_Oselect__weight(T_b,V_xs_2) = c_Random_Oselect__weight(T_b,hAPP(c_List_Ofilter(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Code__Numeral_Ocode__numeral,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_COMBK(tc_HOL_Obool,T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))))),V_xs_2))) # label(fact_select__weigth__drop__zero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1142 (all V_w all V_v hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))) # label(fact_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1143 (all V_y_2 all V_x_2 (hAPP(c_Code__Numeral_Onat__of,V_y_2) = hAPP(c_Code__Numeral_Onat__of,V_x_2) <-> V_x_2 = V_y_2)) # label(fact_code__numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1144 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1145 (all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_Odivmod__int(V_a,V_b))) # label(fact_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1146 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),T_d),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_UN__UN__flatten) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1147 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1148 (all V_j all V_i -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_i))) # label(fact_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1149 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_A_2) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_bij__betw__empty1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1150 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_r_2),V_r_2)),V_r_2)))) # label(fact_trans__O__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1151 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = V_A_2) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_UN__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1152 (all V_y_2 all T_b (class_Int_Onumber__ring(T_b) & class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_y_2))))) # label(fact_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1153 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m) = c_Divides_Odivmod__nat(V_m,V_n))) # label(fact_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1154 (all V_y_2 all V_x_2 all V_u_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_u_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_2),V_x_2))),hAPP(hAPP(c_List_Oappend(T_b),V_u_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_b_2),V_y_2)))),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__append__left__rightI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1155 (all V_B_2 all V_A_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),T_c),c_Set_OPow(T_b)),V_B_2))),hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UN__Pow__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1156 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) -> -(all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_r_2)) -> (all B_us all B_vs (hAPP(hAPP(c_List_Oappend(T_b),B_us),hAPP(hAPP(c_List_Olist_OCons(T_b),B_x),B_vs)) = V_xs_2 -> V_ys_2 != hAPP(hAPP(c_List_Oappend(T_b),B_us),hAPP(hAPP(c_List_Olist_OCons(T_b),B_y),B_vs)))))))) # label(fact_listrel1E) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1157 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))))) # label(fact_div__geq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1158 (all T_b c_List_Olists(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))) # label(fact_lists__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1159 (all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_xs_2))),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_listrel1I1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1160 (all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_compl__bot__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1161 (all V_ys_2 all V_P_2 all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))) -> c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),c_List_OtakeWhile(T_b,V_P_2,V_ys_2)))) # label(fact_takeWhile__append2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1162 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)) = V_a)) # label(fact_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1163 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))) # label(fact_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1164 (all V_y_2 all V_x_2 all T_b (V_y_2 != V_x_2 <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_b,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot)),c_fequal)),V_x_2),V_y_2)))) # label(fact_conversep__noteq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1165 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_c))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_d))))))) # label(fact_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1166 (all V_l c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1167 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1168 (all V_xs_2 all V_n_2 all V_ys_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_ys_2),hAPP(c_List_Oset(tc_List_Olist(T_b)),c_Enum_On__lists(T_b,V_n_2,V_xs_2)))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = V_n_2)) # label(fact_length__n__lists__elem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1169 (all V_ys all T_b all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_ys)) -> hBOOL(hAPP(c_List_Odistinct(tc_prod(T_a,T_b)),c_Enum_Oproduct(T_a,T_b,V_xs,V_ys)))))) # label(fact_distinct__product) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1170 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1171 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))))))) # label(fact_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1172 (all V_a_2 all V_A_2 all V_r_2 all T_b (c_Fun_Oinj__on(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),T_b),c_Equiv__Relations_Oquotient(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_r_2),V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_r_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_r_2))))) # label(fact_quotient__diff1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1173 (all V_b all V_a ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))))) # label(fact_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1174 (all V_zs_2 all V_ys_2 all V_xs_2 all T_b (V_zs_2 = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) <-> V_xs_2 = hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),V_zs_2) & V_ys_2 = hAPP(hAPP(c_List_Odrop(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),V_zs_2))) # label(fact_append__eq__conv__conj) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1175 (all V_y all V_ys all V_x all V_i all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),hAPP(hAPP(c_List_Olist__update(T_b,V_ys),V_i),V_y)) = hAPP(hAPP(c_List_Olist__update(tc_prod(T_a,T_b),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y))) # label(fact_zip__update) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1176 (all V_l all V_k c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l))) # label(fact_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1177 (all V_r_2 all T_c all V_a_2 all T_b ((exists B_y hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),B_y)),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2))))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1178 (all V_A_2 all V_I_2 all T_c all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(V_A_2,B_x))))))) # label(fact_INT__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1179 (all V_xs all V_x all T_a V_xs = hAPP(hAPP(c_List_Odrop(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_drop__1__Cons) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1180 (all V_z all V_w all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_w),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))))))))) # label(fact_frac__less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1181 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_x_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_y_2))) # label(fact_rtrancl__listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1182 (all V_c all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_q,V_c))),V_r)))))))) # label(fact_divmod__nat__rel__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1183 (all V_B_2 all V_A_2 all T_b (V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) & c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 <-> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1184 (all V_B_2 all V_g_2 all T_e all T_d all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_d,T_e,V_g_2,V_B_2) -> c_Fun_Oinj__on(tc_prod(T_b,T_d),tc_prod(T_c,T_e),c_Product__Type_Omap__pair(T_b,T_c,T_d,T_e,V_f_2,V_g_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_b),V_B_2)))))) # label(fact_map__pair__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1185 (all V_y_2 all V_x_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2) = hAPP(c_Int_Onat,hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))))) # label(fact_nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1186 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1187 (all V_b_2 all V_a_2 all T_b (class_Rings_Osemiring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Nat_Osemiring__1__class_ONats(T_b))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Nat_Osemiring__1__class_ONats(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_b))))))) # label(fact_Nats__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1188 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1189 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1190 (all V_r_2 all T_b c_Transitive__Closure_Ortrancl(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)) = c_Transitive__Closure_Ortrancl(T_b,V_r_2)) # label(fact_rtrancl__idemp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1191 (all V_a_2 all V_b_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) -> (-hBOOL(hAPP(hAPP(V_B_2,V_a_2),V_b_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))) # label(fact_INF1__E) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1192 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_a))) # label(fact_sup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1193 (all V_n all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__one__over) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1194 (all V_r all V_q all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_a),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_a -> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_q))) # label(fact_self__quotient) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1195 (all V_xs_2 all V_p_2 all T_b hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_b,tc_HOL_Obool,tc_Nat_Onat),V_p_2),hAPP(c_List_Onth(T_b),V_xs_2))))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_p_2),V_xs_2))) # label(fact_length__filter__conv__card) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1196 (all V_xs all V_x all T_a c_List_Oconcat(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),V_x),V_xs)) = hAPP(hAPP(c_List_Oappend(T_a),V_x),c_List_Oconcat(T_a,V_xs))) # label(fact_concat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1197 (all V_A_2 all V_f_2 all T_c all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_b,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_b,V_A_2,V_f_2),V_B_2))) # label(fact_inj__on__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1198 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_y = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)))) # label(fact_min__max_Oinf__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1199 (all V_n_2 all V_l_2 all T_b c_List_Osublist(T_b,V_l_2,hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)) = hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_l_2)) # label(fact_sublist__upt__eq__take) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1200 (all V_x_2 all T_b c_Hilbert__Choice_OEps(T_b,hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_x_2)) = V_x_2) # label(fact_some__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1201 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)),hAPP(c_Nat_OSuc,V_k))) # label(fact_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1202 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1203 (all V_a all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> c_List_Olinorder__class_Osorted(T_a,c_List_Oremove1(T_a,V_a,V_xs))))) # label(fact_sorted__remove1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1204 (all V_k c_Int_Osucc(c_Int_OBit1(V_k)) = c_Int_OBit0(c_Int_Osucc(V_k))) # label(fact_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1205 (all V_r_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_b)),V_r_2)))) # label(fact_finite__converse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1206 (all V_r_2 all T_b c_Transitive__Closure_Otrancl(T_b,V_r_2) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_r_2),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) # label(fact_trancl__unfold__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1207 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_abs__of__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1208 (all V_P_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),hAPP(c_Set_OCollect(tc_prod(T_c,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_b,tc_HOL_Obool),V_P_2))) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_c)),hAPP(c_COMBC(T_c,T_b,tc_HOL_Obool),V_P_2)))) # label(fact_Range__Collect__split) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1209 (all V_k_2 hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_k_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_k_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.74 1210 (all V_n all T_a (class_Rings_Oring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))) # label(fact_power__minus1__even) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1211 (all V_f_2 all T_c all V_P_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_P_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,T_c),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_c),tc_fun(T_b,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_b)),c_fequal)),V_f_2)))),V_P_2))))))) # label(fact_finite__image__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1212 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit1(V_l_2))))) # label(fact_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1213 (all V_xs_2 all T_b (c_List_Olist_ONil(T_b) = V_xs_2 <-> c_List_Olist_ONil(T_b) = hAPP(c_List_Orotate1(T_b),V_xs_2))) # label(fact_rotate1__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1214 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) & -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)))) # label(fact_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1215 (all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))) # label(fact_trancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1216 (all V_xs_2 all V_r_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Wellfounded_Oacc(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),c_Wellfounded_Oacc(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2))))))) # label(fact_Cons__acc__listrel1I) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1217 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b))))) # label(fact_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1218 (all V_xss_2 all V_xs_2 all V_x_2 all T_b hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_Olist__case(tc_List_Olist(T_b),T_b,c_List_Olist_ONil(T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_b),c_COMBK(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b))))),V_xss_2)))),c_List_Otranspose(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),V_xs_2),c_List_Oconcat(tc_List_Olist(T_b),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_b)),T_b,c_List_Olist_ONil(tc_List_Olist(T_b)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_OCons(tc_List_Olist(T_b))),c_List_Olist_ONil(tc_List_Olist(T_b)))))),V_xss_2))))) = c_List_Otranspose(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_xss_2))) # label(fact_transpose_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1219 (all V_f_2 all V_A_2 all T_b all T_c (class_Rings_Olinordered__semidom(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)))))) # label(fact_setprod__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1220 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> V_c = V_b))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1221 (all V_A_2 all V_f_2 all T_b hAPP(c_Int_Onat,hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_b),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_A_2)) # label(fact_transfer__nat__int__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1222 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1223 (all V_r_2 all T_b all T_c hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_b)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_c,T_b),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_c,T_b)))),hAPP(c_COMBC(T_c,T_b,tc_prod(T_c,T_b)),c_Product__Type_OPair(T_c,T_b))))),V_r_2))) = hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)) # label(fact_converse__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1224 (all V_a_2 all V_f_2 all T_b all T_c V_a_2 = c_List_Ofoldr(T_c,T_b,V_f_2,c_List_Olist_ONil(T_c),V_a_2)) # label(fact_foldr_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1225 (all V_g_2 all V_f_2 all T_c all V_A_2 all V_a_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(c_Fun_Ooverride__on(T_b,T_c,V_f_2,V_g_2,V_A_2),V_a_2) = hAPP(V_f_2,V_a_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(c_Fun_Ooverride__on(T_b,T_c,V_f_2,V_g_2,V_A_2),V_a_2) = hAPP(V_g_2,V_a_2)))) # label(fact_override__on__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1226 (all V_l_2 all V_k_2 (c_Int_OBit0(V_l_2) = c_Int_OBit0(V_k_2) <-> V_k_2 = V_l_2)) # label(fact_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1227 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1228 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1229 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),V_n)) # label(fact_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1230 (all V_c all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_c)))) # label(fact_min__max_Osup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1231 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)))))) # label(fact_finite__Plus) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1232 (all V_ys_2 all V_P_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (-hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(hAPP(c_List_Oappend(T_b),c_List_OdropWhile(T_b,V_P_2,V_xs_2)),V_ys_2) = c_List_OdropWhile(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))))) # label(fact_dropWhile__append1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1233 (all V_A_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> c_Fun_Oinj__on(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b),V_A_2))) # label(fact_inj__uminus) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1234 (all V_A_2 all V_f_2 all T_d all V_A_H_H_2 all V_A_H_2 all V_f_H_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_H_2,V_A_H_2,V_A_H_H_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_b,V_f_2),V_A_2)),V_A_H_2)) -> (c_Fun_Obij__betw(T_d,T_c,hAPP(c_Fun_Ocomp(T_b,T_c,T_d,V_f_H_2),V_f_2),V_A_2,V_A_H_H_2) <-> c_Fun_Obij__betw(T_d,T_b,V_f_2,V_A_2,V_A_H_2))))) # label(fact_bij__betw__comp__iff2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1235 (all V_xs all V_m all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Odrop(T_a),V_m),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_m),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_xs))) # label(fact_take__drop) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1236 (all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1237 (all T_b all V_ys_2 all V_xs_2 (V_xs_2 = V_ys_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),B_x),hAPP(c_List_Oset(tc_prod(T_b,T_b)),hAPP(hAPP(c_List_Ozip(T_b,T_b),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),c_fequal),B_x)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) # label(fact_list__eq__iff__zip__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1238 (all V_d all V_b all V_c all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_c),V_d))))))) # label(fact_min__max_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1239 (all V_A_2 all T_b all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Relation_OId__on(T_b,V_A_2)))))) # label(fact_Id__on__eqI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1240 (all V_x all V_k all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_k -> V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x))) & (V_k != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x))))) # label(fact_last__list__update) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1241 (all V_l all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_Osucc(V_l)))) # label(fact_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1242 (all V_xs_2 all T_b c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2))))) # label(fact_sorted__transpose) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1243 (all V_s_2 all T_b all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Product__Type_Ofst(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Oselect(T_b,V_xs_2),V_s_2))),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_select) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1244 (all V_y_2 all V_x_2 all V_R_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_b,T_c,V_R_2),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),V_R_2)))) # label(fact_in__rel__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1245 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_leD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1246 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) # label(fact_of__int__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1247 (all V_xs_2 all T_b hAPP(hAPP(c_Product__Type_Oscomp(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_Code__Numeral_Ocode__numeral,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random_Orange(hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_b)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_Code__Numeral_Ocode__numeral),c_Product__Type_OPair(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random_Opick(T_b,V_xs_2))) = c_Random_Oselect__weight(T_b,V_xs_2)) # label(fact_select__weight__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1248 (all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),V_B_2)) -> (c_Fun_Oinj__on(T_c,T_b,V_g_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_g_2),V_B_2)),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(c_Finite__Set_Ocard(T_c),V_B_2) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))))))) # label(fact_card__bij__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1249 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_u_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_l_2)))) # label(fact_ivl__disj__un_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1250 (all V_f2_2 all V_f1_2 all T_b c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_b,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = V_f1_2) # label(fact_code__numeral_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1251 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__comm__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1252 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Oab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_setsum__diff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1253 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))))) # label(fact_less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1254 (all V_X_2 all V_a_2 all V_f_2 all T_b (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_f_2)),V_X_2))),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)))))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))))))) # label(fact_coinduct3) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1255 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_ivl__disj__un_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1256 (all V_X_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_C_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),V_X_2))))) # label(fact_InterD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1257 (all T_b hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_finite_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1258 (all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) = hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_x_2)))) # label(fact_image__uminus__atMost) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1259 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n)) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1260 (all V_v_2 all V_n_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_OSuc,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n_2)),c_Int_Onat)))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1261 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(V_F_2,V_A_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> hAPP(V_F_2,V_A_2) = V_x_2))))) # label(fact_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1262 (all T_b c_Wellfounded_OwfP(T_b,hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBK(tc_HOL_Obool,T_b),c_fFalse)))) # label(fact_wfP__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1263 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_finite__Diff2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1264 (all V_x all T_a (class_Orderings_Obot(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x)))) # label(fact_bot__least) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1265 (all T_b c_List_Olistset(T_b,c_List_Olist_ONil(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)))) # label(fact_listset_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1266 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_trancl__trans) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1267 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1268 (all V_b all V_a all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c) = V_a)))) # label(fact_eq__divide__imp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1269 (all V_c all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_c))) # label(fact_min__max_Oinf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1270 (all V_p_2 all V_f_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2),V_p_2) = hAPP(hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_c,T_d),V_p_2)),hAPP(c_Product__Type_Osnd(T_c,T_d),V_p_2))) # label(fact_prod__case__beta) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1271 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b)))))) # label(fact_abs__leI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1272 (all V_j_2 all V_B_2 all V_A_2 all T_d all V_C_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_C_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_B_2,B_x)),V_C_2)))) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_j_2),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Set_Oimage(T_b,T_c,V_f_2)),V_B_2)))))) # label(fact_image__INT) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1273 (all V_b_2 all V_a_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)))))) # label(fact_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1274 (all V_k_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_k_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_k_2))) # label(fact_Compl__greaterThan) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1275 (all V_a_2 all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(hAPP(V_B_2,V_a_2),V_b_2),V_c_2))))) # label(fact_INF2__D) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1276 (all V_x_2 all T_b (class_Rings_Oring__1__no__zero__divisors(T_b) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2) = c_Groups_Oone__class_Oone(T_b) <-> V_x_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b)) | V_x_2 = c_Groups_Oone__class_Oone(T_b)))) # label(fact_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1277 (all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)))))) # label(fact_zero__less__divide__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1278 (all V_n hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) # label(fact_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1279 (all V_x all T_a all V_xs ((c_List_Olist_ONil(T_a) != V_xs -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Obutlast(T_a,V_xs)) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) & (c_List_Olist_ONil(T_a) = V_xs -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olist_ONil(T_a)))) # label(fact_butlast_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1280 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Nitpick_Onorm__frac__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = c_Nitpick_Onorm__frac(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (-(V_a_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Nitpick_Oint__gcd,V_a_2),V_b_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2))),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b_2))) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2 | V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))))) # label(fact_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1281 (all V_r_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_b,V_r_2) -> c_Wellfounded_Owf(T_b,V_r_2)))) # label(fact_finite__acyclic__wf) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1282 (all V_z_2 all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_y_2),V_z_2)) = hAPP(hAPP(V_f_2,V_y_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)))) # label(fact_fun__left__comm_Ofun__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1283 (all V_X_2 all V_r_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_X_2)))))) # label(fact_finite__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1284 (all V_n all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1285 (all V_d all V_b all V_c all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_c),V_d))))))) # label(fact_inf__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1286 (all V_r all V_q all V_r_H all V_q_H all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q_H)),V_r_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r_H)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H))))))) # label(fact_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1287 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_divide__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1288 (all V_p_2 all V_f_2 all V_n_2 all V_m_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_p_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_p_2))))))) # label(fact_setsum__ub__add__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1289 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2))))) # label(fact_Union__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1290 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_B_2))) # label(fact_Sigma__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1291 (all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_w_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1292 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_A_2))))))) # label(fact_Min_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1293 (all V_y_2 all V_n_2 all V_x_2 all V_m_2 all T_b (V_m_2 = V_n_2 & (V_m_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x_2 = V_y_2) <-> c_List_Oreplicate(T_b,V_m_2,V_x_2) = c_List_Oreplicate(T_b,V_n_2,V_y_2))) # label(fact_replicate__eq__replicate) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1294 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1295 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__increasing) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1296 (all V_b_2 all V_a_2 all V_f1_2 all T_b all T_d all T_c c_Nitpick_Opair__box_Opair__box__rec(T_c,T_d,T_b,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_pair__box_Orecs) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1297 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_B_2))) # label(fact_Un__Inter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1298 (all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) # label(fact_sup_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1299 (all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2)) # label(fact_Max_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1300 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))))) # label(fact_transfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1301 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_divide__nonpos__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1302 (all V_y_2 all T_b (class_Int_Onumber__ring(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2) <-> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2))))))) # label(fact_eq__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1303 (all V_x_2 all V_g_2 all V_f_2 all T_e all T_d all T_b all T_c hAPP(c_Product__Type_Osnd(T_c,T_b),hAPP(c_Product__Type_Omap__pair(T_d,T_c,T_e,T_b,V_f_2,V_g_2),V_x_2)) = hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_d,T_e),V_x_2))) # label(fact_snd__prod__fun) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1304 (all V_b all V_a all T_a all T_b hAPP(c_Product__Type_Osnd(T_b,T_a),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_a),V_b)) = V_b) # label(fact_snd__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1305 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1306 (all V_w_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1307 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)))) # label(fact_Diff__subset__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1308 (all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_list_Osize_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1309 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_B_2)))))) # label(fact_setprod_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1310 (all T_b all V_ys_2 all V_xs_2 (c_List_Olist__all2(T_b,T_b,c_fequal,V_xs_2,V_ys_2) <-> V_xs_2 = V_ys_2)) # label(fact_list__all2__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1311 (all V_m all V_n all V_k c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m)),V_n)) # label(fact_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1312 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)))))) # label(fact_min__max_OSup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1313 (all V_b_2 all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_a_2),V_b_2) = c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_Inf__binary) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1314 (all V_y_2 all V_x_2 all V_Y_2 all V_X_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_Y_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) <-> V_Y_2 = V_X_2))))))) # label(fact_quotient__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.75 1315 (all V_A_2 all V_B_2 all V_x_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2)))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1316 (all T_a all V_ys all V_xs (V_ys = V_xs -> V_xs = hAPP(hAPP(c_List_Oappend(T_a),c_List_Olist_ONil(T_a)),V_ys))) # label(fact_eq__Nil__appendI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1317 (all V_xs_2 all T_b c_Random_Oselect(T_b,V_xs_2) = hAPP(hAPP(c_Product__Type_Oscomp(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_Code__Numeral_Ocode__numeral,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random_Orange(hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_Code__Numeral_Ocode__numeral),c_Product__Type_OPair(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Onth(T_b),V_xs_2)),c_Code__Numeral_Onat__of)))) # label(fact_select__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1318 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)))) # label(fact_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1319 (all V_B_2 all V_A_2 all V_R_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_B_2))))) # label(fact_Image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1320 (all V_t_2 all V_s_2 all V_b_2 all V_a_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_s_2),V_t_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_t_2))),c_FunDef_Opair__less))))) # label(fact_pair__lessI2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1321 (all V_a_2 all V_w_2 all V_b_2 all T_b (class_Int_Onumber(T_b) & class_Fields_Ofield__inverse__zero(T_b) -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) <-> (hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = c_Groups_Ozero__class_Ozero(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = V_a_2) & (hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) != c_Groups_Ozero__class_Ozero(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) = V_b_2)))) # label(fact_divide__eq__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1322 (all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)) # label(fact_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1323 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) -> c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_bijI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1324 (all V_b all V_a all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)))) # label(fact_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1325 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_times__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1326 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_one__le__power) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1327 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)) -> (-hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2))))) # label(fact_sup2E) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1328 (all V_B_2 all V_a_2 all T_c all V_A_2 all V_u_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_u_2),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_Oinsert(T_c),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_INT__insert__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1329 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_psubset__trans) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1330 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> c_Groups_Oone__class_Oone(T_c) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_B_2))))))) # label(fact_setprod_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1331 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Obutlast(T_b,V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_in__set__butlastD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1332 (all V_A_2 all T_c all T_b c_Fun_Oinj__on(tc_prod(T_b,T_c),tc_prod(T_c,T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_prod(T_c,T_b)),hAPP(c_COMBC(T_c,T_b,tc_prod(T_c,T_b)),c_Product__Type_OPair(T_c,T_b))),V_A_2)) # label(fact_swap__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1333 (all V_j all V_k all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k) = hAPP(hAPP(c_List_Onth(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),V_k))) # label(fact_nth__upt) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1334 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_length__rotate) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1335 (all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_i)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))))) # label(fact_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1336 (all V_z all V_z_H all V_w all V_w_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_H),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_H),V_z_H)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)))))) # label(fact_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1337 (all V_ys all V_xs all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs)),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_ys)))) # label(fact_listsum__append) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1338 (all V_zs all V_xs all V_ys all V_xs1 all V_x all T_a (hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs1) = V_ys -> (V_xs = hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_zs) -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),V_ys),V_zs)))) # label(fact_Cons__eq__appendI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1339 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_y),V_x_2)))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = V_x_2))))) # label(fact_Max__eqI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1340 (all V_B_2 all V_A_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_c),V_A_2)),hAPP(c_Relation_ODomain(T_b,T_c),V_B_2))))) # label(fact_Domain__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1341 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_min__max_OInf__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1342 (all V_y_2 all V_x_2 all T_b (class_Lattices_Obounded__lattice__bot(T_b) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2) = c_Orderings_Obot__class_Obot(T_b) <-> c_Orderings_Obot__class_Obot(T_b) = V_y_2 & V_x_2 = c_Orderings_Obot__class_Obot(T_b)))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1343 (all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(V_f_2,B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_F_2,V_A_2)),V_A_2))))))) # label(fact_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1344 (all V_i_2 all V_B_2 all V_f_2 all V_s_2 all T_b all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_s_2)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_i))))) -> (hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_s_2) = V_B_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_i_2)),V_B_2)))))))) # label(fact_setsum__nonneg__leq__bound) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1345 (all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_b,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_surj__imp__inj__inv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1346 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1347 (all V_S_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_S_2 -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),V_S_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_S_2)))))) # label(fact_minus__Max__eq__Min) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1348 (all V_n_2 all V_m_2 hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),c_List_Oupt(V_m_2,V_n_2)) = c_List_Oupt(hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_map__Suc__upt) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1349 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 | V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1350 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oordered__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2)),V_d_2))))) # label(fact_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1351 (all V_A_2 all T_b c_Relation_Oantisym(T_b,c_Relation_OId__on(T_b,V_A_2))) # label(fact_antisym__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1352 (all V_xs_2 all T_b (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2))) -> (V_xs_2 != c_List_Olist_ONil(tc_List_Olist(T_b)) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (V_xs_2 = c_List_Olist_ONil(tc_List_Olist(T_b)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2))))) # label(fact_length__transpose__sorted) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1353 (all V_xs all V_x all T_b all T_a (hAPP(c_List_Orotate1(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),V_xs)) = hAPP(hAPP(c_List_Oappend(T_b),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),c_List_Olist_ONil(T_b))) & hAPP(c_List_Orotate1(T_a),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a))) # label(fact_rotate__simps) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1354 (all V_A_2 all V_f_2 all T_c all T_b (class_Rings_Osemiring__1(T_b) & class_Groups_Ocomm__monoid__mult(T_b) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,tc_Nat_Onat),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,T_c),c_Nat_Osemiring__1__class_Oof__nat(T_b)),V_f_2)),V_A_2))) # label(fact_of__nat__setprod) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1355 (all V_a_2 all V_times_2 all V_one_2 all T_b hAPP(hAPP(c_Power_Opower_Opower(T_b,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_one_2) # label(fact_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1356 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1357 (all V_f_2 all V_T_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))) -> c_Groups_Oone__class_Oone(T_c) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_S_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_T_2))))))) # label(fact_setprod__Un__one) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1358 (all V_R_2 all T_b (c_Equiv__Relations_Opart__equivp(T_b,V_R_2) <-> c_Predicate_Otransp(T_b,V_R_2) & c_Predicate_Osymp(T_b,V_R_2) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))))) # label(fact_part__equivp__refl__symp__transp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1359 (all V_k_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_k_2)))) # label(fact_Compl__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1360 (all V_B_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),hAPP(c_Set_OPow(T_b),V_B_2)))) # label(fact_Pow__bottom) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1361 (all V_b_2 all V_c_2 all V_a_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1362 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2)))) # label(fact_finite__greaterThanAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1363 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__idem_Oin__comp__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1364 (all V_r_2 all V_b_2 all V_a_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OField(T_b),V_r_2)) = hAPP(c_Relation_OField(T_b),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2))) # label(fact_Field__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1365 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))) # label(fact_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1366 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))))) # label(fact_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1367 (all V_a_2 all V_b_2 all T_b (class_Rings_Odivision__ring(T_b) -> (V_b_2 != c_Groups_Ozero__class_Ozero(T_b) -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2) = c_Groups_Oone__class_Oone(T_b) <-> V_b_2 = V_a_2)))) # label(fact_right__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1368 (all V_xs_2 all V_x_2 all V_P_2 all T_b (c_List_Olist__ex1(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) <-> (-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_Olist__ex1(T_b,V_P_2,V_xs_2)) & (hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_Olist__all(T_b,hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),V_P_2))),hAPP(c_fequal,V_x_2)),V_xs_2)))) # label(fact_list__ex1__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1369 (all V_i_2 all V_d_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Lazy__Sequence_Osmall__lazy_H__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_d_2),V_i_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d_2),V_i_2)) -> c_Lazy__Sequence_Oempty(tc_Int_Oint) = c_Lazy__Sequence_Osmall__lazy_H(V_d_2,V_i_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d_2),V_i_2)) -> c_Lazy__Sequence_Oappend(tc_Int_Oint,hAPP(c_Lazy__Sequence_Osingle(tc_Int_Oint),V_i_2),c_Lazy__Sequence_Osmall__lazy_H(V_d_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = c_Lazy__Sequence_Osmall__lazy_H(V_d_2,V_i_2)))) # label(fact_small__lazy_H_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1370 (all V_f_2 all T_c all V_P_2 all T_b (c_FunDef_Oreduction__pair(T_b,V_P_2) -> c_FunDef_Oreduction__pair(T_c,hAPP(hAPP(c_FunDef_Orp__inv__image(T_b,T_c),V_P_2),V_f_2)))) # label(fact_rp__inv__image__rp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1371 (all V_P_2 all V_xs_2 all T_b ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i))))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_all__set__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1372 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_le__SucI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1373 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2) = hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))) # label(fact_gfp__unfold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1374 (all V_A_2 all T_b c_Fun_Oinj__on(T_b,T_b,c_Fun_Oid(T_b),V_A_2)) # label(fact_inj__on__id) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1375 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,V_xs_2)))) # label(fact_sort__key__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1376 (all V_f_2 all V_h_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (V_h_2 = c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hAPP(V_f_2,V_h_2) = V_h_2)))) # label(fact_def__lfp__unfold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1377 (all T_c all V_b_2 all V_B_2 all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(V_B_2,V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2))))) # label(fact_SUP1__I) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1378 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1379 (all V_x all V_xs all T_a V_xs = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))))) # label(fact_butlast__snoc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1380 (all V_no_2 all V_yes_2 all V_xs_2 all V_P_2 all T_b (c_List_Opartition(T_b,V_P_2,V_xs_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_yes_2),V_no_2) -> hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_yes_2)),hAPP(c_List_Oset(T_b),V_no_2)))) # label(fact_partition__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1381 (all V_r_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),V_r_2) = hAPP(c_Relation_ODomain(T_b,T_c),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2))) # label(fact_Range__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1382 (all V_B_2 all V_A_2 all V_g_2 all V_f_2 all T_c all T_b all T_d (class_Rings_Osemiring__0(T_d) -> (c_Fun_Oinj__on(tc_prod(T_b,T_c),T_d,hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,T_d),tc_fun(T_c,T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_d),tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,T_d)),T_b),c_COMBB(T_d,T_d,T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_b),c_Groups_Otimes__class_Otimes(T_d)),V_f_2))),V_g_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_d),c_Fun_Oid(T_d)),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_d),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),T_d),c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_d),hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_d),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj))),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_b,tc_fun(T_c,T_d)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_c,T_d)),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),T_d),c_COMBB(tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool)),T_d),c_COMBB(T_d,tc_HOL_Obool,T_c)),c_fequal))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,T_d),tc_fun(T_c,T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_d),tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,T_d)),T_b),c_COMBB(T_d,T_d,T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_b),c_Groups_Otimes__class_Otimes(T_d)),V_f_2))),V_g_2)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2)))))))))) # label(fact_setsum__mult__setsum__if__inj) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1383 (all V_a_2 all V_c_2 all V_b_2 all T_b (class_Fields_Ofield__inverse__zero(T_b) -> ((V_c_2 = c_Groups_Ozero__class_Ozero(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = V_a_2) & (c_Groups_Ozero__class_Ozero(T_b) != V_c_2 -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)) <-> V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))) # label(fact_divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1384 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1385 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> c_Relation_Osym(tc_List_Olist(T_b),c_List_Olistrel(T_b,V_r_2)))) # label(fact_listrel__sym) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1386 (all V_S_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_S_2),c_Relation_ORange(T_c,T_b))) # label(fact_Range__Union) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1387 (all V_n all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_add__leD1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1388 (all V_F_2 all V_E_2 all V_B_2 all V_A_2 all T_c all T_b all T_e all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_prod(T_d,T_e),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_e),V_A_2),hAPP(c_COMBK(tc_fun(T_e,tc_HOL_Obool),T_d),V_B_2))),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),tc_fun(T_e,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_e,tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),tc_fun(T_e,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),T_d),c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),T_d),c_Product__Type_OSigma(T_b,T_c)),V_E_2))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),T_e),c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b)),V_F_2)))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_E_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_F_2)))) # label(fact_UN__Times__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1389 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1390 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))) # label(fact_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1391 (all V_y all V_x all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)))) # label(fact_minus__max__eq__min) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1392 (all V_B_2 all T_c all V_A_2 all T_b ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))) & (-(hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))) -> hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_card__Plus__conv__if) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1393 (all V_w hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1394 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)))) # label(fact_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1395 (all V_f_2 all V_S_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_S_2)))))) # label(fact_setprod__pos__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1396 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omax(T_b)),V_A_2)))) # label(fact_min__max_OSup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1397 (all V_y all V_x all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> V_y = V_x))))) # label(fact_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1398 (all V_f_2 all T_d all T_c all T_b V_f_2 = hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),c_Product__Type_Ocurry(T_b,T_c,T_d,V_f_2))) # label(fact_split__curry) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1399 (all V_n_2 all T_b all V_xs_2 ((c_List_Olist_ONil(tc_List_Olist(T_b)) = V_xs_2 -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2) -> ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2),B_i)) = V_n_2)) -> c_List_Otranspose(T_b,V_xs_2) = hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_List_Olist(T_b)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(tc_Nat_Onat),tc_List_Olist(T_b)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,T_b),tc_fun(tc_List_Olist(tc_Nat_Onat),tc_List_Olist(T_b)),tc_Nat_Onat),c_List_Omap(tc_Nat_Onat,T_b)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_Nat_Onat,T_b),tc_Nat_Onat),c_List_Onth(T_b)),hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2))))),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2)))),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))))) # label(fact_transpose__rectangle) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1400 (all V_P_2 all V_n_2 all V_m_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((V_m_2 = V_n_2 -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2))) -> hBOOL(hAPP(hAPP(V_P_2,V_n_2),V_m_2)))))) # label(fact_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1401 (all V_B_2 all V_A_2 all T_d all V_f_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Set_Ovimage(T_b,T_c,V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__INT) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1402 (all V_g_2 all V_x_2 all T_d all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),V_g_2))) # label(fact_folding__idem_Oidem__left__comp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1403 (all V_x all V_y all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_y))) # label(fact_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1404 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1405 (all V_r_2 all T_b (c_Wellfounded_OwfP(T_b,V_r_2) <-> c_Wellfounded_Owf(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))))) # label(fact_wfP__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1406 (all V_n_2 all V_m_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1407 (all V_G_2 all V_tsa_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)),V_tsa_2),V_G_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_tsa_2))) # label(fact_asm) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1408 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1409 (all V_xqq all V_xq all T_a c_Lazy__Sequence_Oappend(T_a,V_xq,c_Lazy__Sequence_Oflat(T_a,V_xqq)) = c_Lazy__Sequence_Oflat(T_a,c_Lazy__Sequence_Olazy__sequence_OInsert(tc_Lazy__Sequence_Olazy__sequence(T_a),V_xq,V_xqq))) # label(fact__03) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1410 (all V_ys_2 all V_P_2 all V_xs_2 all T_c all T_b hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),c_List_OtakeWhile(T_c,V_P_2,V_ys_2)) = c_List_OtakeWhile(tc_prod(T_b,T_c),hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,tc_prod(T_b,T_c),V_P_2),c_Product__Type_Osnd(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2))) # label(fact_zip__takeWhile__snd) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1411 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,V_r_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) # label(fact_listrel__eq__len) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1412 (all V_i_2 all V_M_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_M_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2))))) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))))) # label(fact_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1413 (all V_r_2 all V_g_2 all V_f_2 all T_d all T_b all T_c hAPP(c_Set_Oimage(T_d,T_b,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,V_g_2),V_r_2)) = hAPP(c_Set_Oimage(T_c,T_b,hAPP(c_Fun_Ocomp(T_d,T_b,T_c,V_f_2),V_g_2)),V_r_2)) # label(fact_image__compose) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1414 (all V_ys all V_xs all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Orev(T_a),V_ys)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__append) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1415 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2)) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2) <-> V_d_2 = V_b_2 & V_c_2 = V_a_2))))) # label(fact_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1416 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_b_2),V_a_2)),V_r_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2))))) # label(fact_converse__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1417 (all V_R2_2 all V_R1_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),V_R1_2),V_R2_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Wellfounded_Oaccp(T_b,V_R2_2)),c_Wellfounded_Oaccp(T_b,V_R1_2))))) # label(fact_accp__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1418 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Nitpick_Onat__gcd(hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_y)))) = hAPP(hAPP(c_Nitpick_Oint__gcd,V_x),V_y)) # label(fact_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1419 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))))))) # label(fact_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1420 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_INT__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1421 (all V_r_2 all T_b c_Relation_Otrans(T_b,c_Transitive__Closure_Otrancl(T_b,V_r_2))) # label(fact_trans__trancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1422 (all T_b c_Fun_Oid(tc_List_Olist(T_b)) = hAPP(c_List_Omap(T_b,T_b),c_COMBI(T_b))) # label(fact_List_Omap_Oidentity) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1423 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)))))) # label(fact_compl__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1424 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> V_A_2 = V_B_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.76 1425 (all V_a_2 all V_A_H_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_A_H_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_b,V_A_H_2,c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2)),V_a_2) = hAPP(V_f_2,V_a_2)))) # label(fact_inv__into__inv__into__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1426 (all V_n_2 hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)) # label(fact_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1427 (all V_x all T_a all V_xs (c_List_Olist_ONil(T_a) = V_xs -> V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_last__ConsL) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1428 (all V_r_2 all T_b ((all B_x hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2))) <-> c_Nitpick_Orefl_H(T_b,V_r_2))) # label(fact_Nitpick_Orefl_H__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1429 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_u_2))))) # label(fact_ivl__disj__un_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1430 (all V_a_2 all T_b (class_Int_Oring__char__0(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_b) <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2)))) # label(fact_Ints__double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1431 (all V_p_2 all T_c all T_b hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_b),c_COMBK(T_b,T_c)),V_p_2) = hAPP(c_Product__Type_Ofst(T_b,T_c),V_p_2)) # label(fact_fst__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1432 (all V_N_2 all V_k_2 c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2),V_N_2)) # label(fact_inj__on__add__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1433 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) # label(fact_Inf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1434 (all V_f_2 all T_b all T_d all T_c c_Product__Type_Oapsnd(T_c,T_d,T_b,V_f_2) = c_Product__Type_Omap__pair(T_b,T_b,T_c,T_d,c_Fun_Oid(T_b),V_f_2)) # label(fact_apsnd__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1435 (all V_A_2 all V_g_2 all T_d all V_f_2 all T_b all T_c hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Oimage(T_d,T_c,V_g_2),V_A_2)) = hAPP(c_Set_Oimage(T_d,T_b,hAPP(hAPP(c_COMBB(T_c,T_b,T_d),V_f_2),V_g_2)),V_A_2)) # label(fact_image__image) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1436 (all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) # label(fact_Compl__eq__Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1437 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2)) # label(fact_bij__betw__empty2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1438 (all V_G_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_G_2)) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_G_2))))) # label(fact_finite__Int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1439 (all T_b c_Finite__Set_Ofun__left__comm__idem(T_b,tc_fun(T_b,tc_HOL_Obool),c_Set_Oinsert(T_b))) # label(fact_fun__left__comm__idem__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1440 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1441 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1442 (all V_k V_k = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),V_k)) # label(fact_add__Pls) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1443 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1444 (all V_m_2 all V_n_2 hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2) = c_Nat_Onat_Onat__case(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n_2)),V_m_2)) # label(fact_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1445 (all V_xs_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) # label(fact_length__sort) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1446 (all V_c all V_a all V_b all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1447 (all V_x_2 all V_f_2 all V_g_2 all T_d all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_d,V_f_2),hAPP(c_Set_Ovimage(T_d,T_c,V_g_2),V_x_2)) = hAPP(c_Set_Ovimage(T_b,T_c,hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_g_2),V_f_2)),V_x_2)) # label(fact_vimage__compose) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1448 (all V_xs_2 all T_b hAPP(c_List_Oset(T_b),hAPP(c_List_Orev(T_b),V_xs_2)) = hAPP(c_List_Oset(T_b),V_xs_2)) # label(fact_set__rev) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1449 (all V_P_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2))) -> (hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)) -> ((all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2))) -> (all B_aa all B_ba (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_aa),B_ba))),V_r_2)) -> (hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)) -> hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba))))))) -> hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)))))) # label(fact_rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1450 (all V_r_2 all T_c all T_b hAPP(c_Relation_ODomain(T_b,T_c),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)) = hAPP(c_Relation_ORange(T_c,T_b),V_r_2)) # label(fact_Domain__converse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1451 (all V_f_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_setsum__Un__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1452 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2))))) # label(fact_fun__left__comm_Ofold__rec) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1453 (all V_n V_n = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) # label(fact_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1454 (all T_a (class_Int_Onumber__ring(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls))) # label(fact_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1455 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = V_m)) # label(fact_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1456 (all V_z_2 all V_f_2 all T_c all T_b hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_z_2))) # label(fact_fold__graph_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1457 (all V_B_2 all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))),V_f_2)),c_Groups_Ozero__class_Ozero(T_c))),V_A_2)))) # label(fact_setsum__restrict__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1458 (all V_w all V_v hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w))) # label(fact_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1459 (all V_r_2 all V_a_2 all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_a_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Wellfounded_Oacc(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Wellfounded_Oacc(T_b,V_r_2)))))) # label(fact_acc__downwards__aux) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1460 (all V_n_2 all V_d_2 all V_a_2 all T_b (class_Rings_Ocomm__semiring__1(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),V_d_2)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),c_Groups_Oone__class_Oone(T_b))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(T_b)),c_Nat_Osemiring__1__class_Oof__nat(T_b))),V_d_2))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))))) # label(fact_arith__series__general) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1461 (all V_P_2 ((all B_x hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1462 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_Transitive__Closure_Otrancl__into__trancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1463 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (V_x_2 = V_y_2 <-> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_x_2) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_y_2)))) # label(fact_greaterThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1464 (all V_r_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),c_Relation_OId(T_b))) # label(fact_reflcl__trancl) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1465 (all V_A_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)),V_A_2))))))) # label(fact_fold1__in) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1466 (all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Nitpick_Osetsum_H(T_b,T_c,V_f_2,V_A_2) = c_Groups_Ozero__class_Ozero(T_c)) & (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Nitpick_Osetsum_H(T_b,T_c,V_f_2,V_A_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),c_Hilbert__Choice_OEps(tc_List_Olist(T_b),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_List_Oset(T_b))),V_A_2))),c_List_Odistinct(T_b)))))))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1467 (all V_z_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_of__int__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1468 (all V_y_2 all V_x_2 hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_y_2),V_x_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_minus) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1469 (all V_xs_2 all T_b all V_b_2 all V_a_2 (V_a_2 != V_b_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_List_Oset(T_b),c_List_Oremove1(T_b,V_b_2,V_xs_2))))))) # label(fact_in__set__remove1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1470 (all V_x_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_setsum_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1471 (all V_n_2 all V_m_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_m_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_n_2)))))) # label(fact_lessThan__strict__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1472 (all V_xs all V_n all T_a c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs))) # label(fact_drop__butlast) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1473 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__ex(T_b,V_P_2,V_xs_2) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) & hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_list__ex__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1474 (all V_A_2 all V_f_2 all T_c all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_b,V_A_2,V_f_2),V_x_2)),V_A_2)))) # label(fact_inv__into__into) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1475 (all V_b_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2)),V_b_2)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_b_2))))))) # label(fact_Sup__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1476 (all V_b_2 all V_c_2 all V_a_2 all T_b (class_Rings_Oidom(T_b) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_a_2),V_b_2)) | c_Groups_Ozero__class_Ozero(T_b) = V_c_2))) # label(fact_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1477 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) <-> hAPP(c_Fun_Ocomp(T_c,T_b,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),V_f_2) = c_Fun_Oid(T_b))) # label(fact_inj__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1478 (all V_b all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1479 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> c_Finite__Set_Ofun__left__comm(T_d,T_c,hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,T_c),T_d),V_f_2),V_g_2)))) # label(fact_fun__left__comm_Ofun__left__comm__apply) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1480 (all V_f_2 all V_S_2 all V_T_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_S_2)))))) # label(fact_setsum__mono__zero__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1481 (all V_B_2 all V_A_2 all T_b all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(V_B_2,B_x))))) # label(fact_UNION__empty__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1482 (all V_y_2 all V_x_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_c,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_r_2)),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_b,T_c),V_r_2))))) # label(fact_conversep__converse__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1483 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_abs__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1484 (all V_xs_2 all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,T_b,c_fNot),V_P_2)),V_xs_2) = hAPP(c_Product__Type_Osnd(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Opartition(T_b,V_P_2,V_xs_2))) # label(fact_partition__filter2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1485 (all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> V_z_2 = hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_folding__image__simple_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1486 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,V_n),V_xs) = hAPP(c_List_Orotate(T_a,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_xs)) # label(fact_rotate__conv__mod) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1487 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n)) # label(fact_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1488 (all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)))) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))))) # label(fact_finite__cartesian__productD2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1489 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1490 (all V_z2 all V_z1 all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2))) # label(fact_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1491 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> ((V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_z_2 != V_x_2) -> -(all B_y all B_m (V_n_2 = hAPP(c_Nat_OSuc,B_m) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_y)),V_R_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),B_m),V_R_2))))))))) # label(fact_rel__pow__E2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1492 (all V_B_2 all V_A_2 all V_C_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_Int__greatest) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1493 (all V_f_2 all T_c all V_A_2 all T_b (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & -hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),hAPP(V_f_2,B_x))))))))))) # label(fact_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1494 (all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2)) # label(fact_subset__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1495 (all V_g_2 all V_x_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_setprod_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1496 (all V_x_2 all V_xs_2 all T_b all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (V_x_2 = hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_i_2) <-> V_xs_2 = hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_x_2)))) # label(fact_list__update__same__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1497 (all V_f_2 all T_c all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))))) # label(fact_imageI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1498 (all V_r_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_List_Olists(T_b,c_Wellfounded_Oacc(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))))) # label(fact_lists__accD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1499 (all V_uu all T_b all T_a c_Enum_Oproduct(T_a,T_b,c_List_Olist_ONil(T_a),V_uu) = c_List_Olist_ONil(tc_prod(T_a,T_b))) # label(fact_product_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1500 (all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hAPP(c_Int_Onat,V_i_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1501 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)))) # label(fact_min__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1502 (all V_m all V_n V_m = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_n)) # label(fact_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1503 (all V_n -hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_not__neg__int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1504 (all V_P_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(c_Set_OCollect(T_b),V_P_2)) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(T_b,tc_HOL_Obool,T_c),V_P_2),c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2))))) # label(fact_bij__image__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1505 (all V_x_2 all V_y_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_A_2)))) # label(fact_inj__on__fun__updI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1506 (all V_f_2 all V_xs_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2))))) # label(fact_listsum__distinct__conv__setsum__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1507 (all T_b c_List_Omember(T_b) = c_List_Oset(T_b)) # label(fact_member__set) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1508 (all V_r all V_q all V_b all V_a (V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r))) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)))))) # label(fact_divmod__int__relI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1509 (all T_b (class_Orderings_Olinorder(T_b) -> c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)))) # label(fact_min__max_Ofun__left__comm__idem__sup) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1510 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1511 (all V_A_2 all T_b -(exists B_f hAPP(c_Set_OPow(T_b),V_A_2) = hAPP(c_Set_Oimage(T_b,tc_fun(T_b,tc_HOL_Obool),B_f),V_A_2))) # label(fact_Cantors__paradox) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1512 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (V_a != V_b -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1513 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),V_l_2)) # label(fact_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1514 (all V_n_2 all V_m_2 all V_k_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 | V_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) # label(fact_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1515 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_e all T_d all T_c hAPP(hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_d,T_e),hAPP(V_f_2,V_x_2))),hAPP(c_Product__Type_Osnd(T_d,T_e),hAPP(V_f_2,V_x_2))) = hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_e,T_b),V_f_2),V_g_2),V_x_2)) # label(fact_scomp__unfold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1516 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (V_b != V_a -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1517 (all V_b all V_x all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x))))) # label(fact_less__infI1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1518 (all V_k all V_l ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l -> c_Divides_OposDivAlg(hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l)) = c_Divides_Opdivmod(V_k,V_l)) & (V_l = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Opdivmod(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k))))) # label(fact_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1519 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n)))) # label(fact_abs__power__minus) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1520 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_z) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y)))) # label(fact_divide__diff__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1521 (all V_xs_2 all V_x_2 all T_b hAPP(c_List_Oset(T_b),c_List_Oinsert(T_b,V_x_2,V_xs_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) # label(fact_List_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1522 (all V_x_2 all V_f_2 all V_A_2 all T_c all T_b hAPP(c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),V_x_2) = c_HOL_OThe(T_b,hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),V_x_2)))) # label(fact_the__inv__into__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1523 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)))) # label(fact_arith__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1524 (all V_f_2 all V_A_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) & class_Groups_Ocomm__monoid__mult(T_b) -> (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_b,tc_Nat_Onat),V_f_2),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))))) # label(fact_transfer__int__nat__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1525 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))))) # label(fact_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1526 (all V_q all V_r all V_b all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) # label(fact_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1527 (all V_YS_2 all V_y_2 all V_XS_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__less)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omin__strict)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2))),c_FunDef_Omin__strict)))))) # label(fact_smin__insertI) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1528 (all V_n all V_m hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))))) # label(fact_divmod__nat__rel) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1529 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1530 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1531 (all V_A_2 all V_f_2 all V_f_H_2 all T_d all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_H_2),V_f_2),V_A_2) -> c_Fun_Oinj__on(T_b,T_d,V_f_2,V_A_2))) # label(fact_inj__on__imageI2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1532 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_x_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),B_x),V_x_2)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))))))))) # label(fact_fold1__strict__below__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1533 (all V_n all V_ys all V_xs all T_a hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) = hAPP(hAPP(c_List_Onth(T_a),V_ys),V_n)) # label(fact_nth__append__length__plus) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1534 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(c_List_Orev(T_b),V_xs_2),V_ys_2) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,hAPP(c_List_Orev(T_c),V_ys_2)))) # label(fact_list__all2__rev1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1535 (all V_xs_2 all V_x_2 all T_b (class_Lattices_Olattice(T_b) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) = hAPP(c_List_Ofoldl(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),V_xs_2))) # label(fact_Inf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1536 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_y)))) # label(fact_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1537 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2),V_u_2) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastSucAtMost__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 8.70/8.77 1538 (all V_ys_2 all V_xs_2 all V_f_2 all T_b all T_c (hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_ys_2) -> (c_Fun_Oinj__on(T_c,T_b,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_ys_2))) -> V_ys_2 = V_xs_2))) # label(fact_map__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1539 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> ((exists B_f c_Fun_Obij__betw(T_b,T_c,B_f,V_A_2,V_B_2)) <-> hAPP(c_Finite__Set_Ocard(T_c),V_B_2) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_BIJ) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1540 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__image__simple_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1541 (all V_ys_2 all T_c all V_xs_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) -> hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_c),c_Product__Type_Osnd(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)) = V_ys_2)) # label(fact_map__snd__zip) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1542 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))))) # label(fact_bij__betw__finite) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1543 (all V_n all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n)))) # label(fact_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1544 (all V_x all T_a (class_Groups_Osgn__if(T_a) -> (V_x = c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Osgn__class_Osgn(T_a,V_x)) & (V_x != c_Groups_Ozero__class_Ozero(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Oone__class_Oone(T_a))))) # label(fact_sgn__if) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1545 (all V_Q_2 all V_P_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fimplies),V_P_2)),V_Q_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),V_P_2))),hAPP(c_Set_OCollect(T_b),V_Q_2))) # label(fact_Collect__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1546 (all V_z all T_a (class_Rings_Oring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_z))))) # label(fact_of__nat__nat) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1547 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1548 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),c_List_Obutlast(T_a,V_xs)))))) # label(fact_distinct__butlast) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1549 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> c_List_Oremove1(T_b,V_x_2,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),V_xs_2)) = V_xs_2)) # label(fact_remove1__insort) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1550 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m_2)) -> (c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2 & c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m_2 <-> c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2)))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1551 (all V_ns_2 all V_n_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_n_2),hAPP(c_List_Oset(tc_Nat_Onat),V_ns_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_ns_2))))) # label(fact_elem__le__sum) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1552 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))))) # label(fact_less__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1553 (all V_A_2 all T_b c_Fun_Oinj__on(T_b,T_b,c_COMBI(T_b),V_A_2)) # label(fact_inj__on__id2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1554 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__self) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1555 (all V_lazy__sequence all V_a all T_a hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_a)),c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a,V_lazy__sequence)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_a)),V_lazy__sequence)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_lazy__sequence_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1556 (all V_B_2 all V_X2_2 all V_X1_2 all V_f_2 all T_d all V_r2_2 all V_A2_2 all T_c all V_r1_2 all V_A1_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_c,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_b,T_c,tc_fun(T_d,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X1_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A1_2),V_r1_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X2_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A2_2),V_r2_2))) -> ((all B_x1 (hBOOL(hAPP(hAPP(c_member(T_b),B_x1),V_A1_2)) -> (all B_x2 (hBOOL(hAPP(hAPP(c_member(T_c),B_x2),V_A2_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(V_f_2,B_x1),B_x2)),V_B_2)))))) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_d,tc_HOL_Obool)),V_X1_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_X2_2)),V_f_2))),V_B_2))))))))) # label(fact_UN__equiv__class__type2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1557 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_OInf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1558 (all V_r all V_q all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_a),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_a -> V_r = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_self__remainder) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1559 (all V_n all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))))))) # label(fact_power__mono) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1560 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) <-> c_Groups_Ozero__class_Ozero(T_b) != V_y_2 | c_Groups_Ozero__class_Ozero(T_b) != V_x_2))) # label(fact_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1561 (all V_y_2 all V_x_2 all T_b (class_Lattices_Osemilattice__sup(T_b) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2) = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1562 (all V_g_2 all V_x_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2))))) # label(fact_setsum_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1563 (all V_x_2 all V_g_2 all T_e all V_f_2 all T_b all T_c all T_d hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_b,V_g_2),V_x_2)) = hAPP(c_Product__Type_Oapsnd(T_e,T_c,T_b,hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_x_2)) # label(fact_apsnd__compose) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1564 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),V_A_2))))))) # label(fact_min__max_OInf__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1565 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_B_2)),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2))))))) # label(fact_Inf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1566 (all V_i_2 all V_f_2 all V_s_2 all T_b all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_s_2)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_i))))) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_s_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),V_s_2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(V_f_2,V_i_2))))))) # label(fact_setsum__nonneg__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1567 (all V_r_2 all V_A_2 all T_c all V_S_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> c_Relation_Orefl__on(T_c,hAPP(V_A_2,B_x),hAPP(V_r_2,B_x)))) -> c_Relation_Orefl__on(T_c,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_S_2),V_r_2)))) # label(fact_refl__on__UNION) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1568 (all V_z all V_y all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_z))) # label(fact_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1569 (all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))) # label(fact_neg__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1570 (all V_n_2 all T_b all V_x_2 all V_P_2 ((-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_Olist_ONil(T_b) = hAPP(c_List_Ofilter(T_b,V_P_2),c_List_Oreplicate(T_b,V_n_2,V_x_2))) & (hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(c_List_Ofilter(T_b,V_P_2),c_List_Oreplicate(T_b,V_n_2,V_x_2)) = c_List_Oreplicate(T_b,V_n_2,V_x_2)))) # label(fact_filter__replicate) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1571 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_inf__sup__aci_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1572 (all V_n all V_m all V_k ((V_k = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_k -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)))) # label(fact_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1573 (all T_b all T_a all V_t hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),hAPP(c_Product__Type_Ofst(T_a,T_b),V_t)),hAPP(c_Product__Type_Osnd(T_a,T_b),V_t)) = V_t) # label(fact_surjective__pairing) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1574 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1575 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_inj__imp__surj__inv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1576 (all V_xs_2 all V_x_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),c_List_Oremove1(T_b,V_x_2,V_xs_2))),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_set__remove1__subset) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1577 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (c_Fun_Oinj__on(T_d,T_b,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_d,T_c,hAPP(c_Fun_Ocomp(T_b,T_c,T_d,V_f_2),V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_inj__comp) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1578 (all V_xs all T_a all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Otake(T_a),V_j),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_xs))) # label(fact_take__add) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1579 (all V_ys_2 all V_y_2 all V_xs_2 all T_b (hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2) = hAPP(c_List_Orev(T_b),V_xs_2) <-> hAPP(hAPP(c_List_Oappend(T_b),hAPP(c_List_Orev(T_b),V_ys_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),c_List_Olist_ONil(T_b))) = V_xs_2)) # label(fact_rev__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1580 (all V_y all V_q_1 all V_b_1 all V_a_1 (V_a_1 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_q_1)),V_y) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_1),V_y))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_1)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_b_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a_1,V_b_1) = V_y)))) # label(fact_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1581 (all V_f_2 all V_X_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))))))) # label(fact_coinduct__lemma) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1582 (all V_k all V_a all V_b hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k)) # label(fact_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1583 (all V_xs all V_n all T_a V_xs = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_append__take__drop__id) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1584 (all V_n_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_b),hAPP(c_List_Olexn(T_b,V_r_2),V_n_2)))) # label(fact_wf__lexn) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1585 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2))) # label(fact_card__greaterThanAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1586 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)))) # label(fact_SUP2__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1587 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1588 (all V_g_2 all V_A_2 all T_c all V_I_2 all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_I_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_I_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_A_2,B_x))))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_I_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_I_2)) -> (B_x != B_xa -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_A_2,B_x)),hAPP(V_A_2,B_xa))))))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_g_2),c_Groups_Oone__class_Oone(T_d)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_b,c_Groups_Otimes__class_Otimes(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),T_d,T_b),hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_g_2),c_Groups_Oone__class_Oone(T_d))),V_A_2)),c_Groups_Oone__class_Oone(T_d)),V_I_2)))))) # label(fact_fold__image__UN__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1589 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)))))) # label(fact_Min_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1590 (all V_P_2 ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)))) <-> (exists B_x hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))))) # label(fact_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1591 (all V_x_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2)) -> c_Nat__Transfer_Ois__nat(V_x_2)))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1592 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2))))) # label(fact_fun__left__comm_Ofold__fun__comm) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1593 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olexord(T_b,V_r_2))) <-> (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_x_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_y_2)))) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(hAPP(c_List_Onth(T_b),V_x_2),B_i)),hAPP(hAPP(c_List_Onth(T_b),V_y_2),B_i))),V_r_2)) & hAPP(hAPP(c_List_Otake(T_b),B_i),V_x_2) = hAPP(hAPP(c_List_Otake(T_b),B_i),V_y_2))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_x_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_y_2))) & V_x_2 = hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_x_2)),V_y_2))) # label(fact_lexord__take__index__conv) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1594 (all V_r_2 all T_b c_Relation_Ototal__on(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_r_2)) # label(fact_total__on__empty) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1595 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))))) # label(fact_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1596 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))))) # label(fact_card_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1597 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg(V_a,V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1598 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Osemilattice__inf(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Osemilattice__inf) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1599 (all V_xs_2 all V_f_2 all T_b all T_c c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(T_c,tc_List_Olist(T_b)),V_f_2),V_xs_2)) = c_List_Omaps(T_c,T_b,V_f_2,V_xs_2)) # label(fact_maps__def) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1600 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_OnegDivAlg(V_a,V_b)))))) # label(fact_negDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1601 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = V_b)) # label(fact_minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1602 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) # label(fact_INT__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1603 (all V_x all V_z all V_y all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_mult__imp__less__div__pos) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1604 (all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z))))) # label(fact_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1605 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_u_2)),V_l_2)) # label(fact_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1606 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2)))))) # label(fact_gfp__lemma3) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1607 (all V_v_2 all V_n_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n_2)),c_Int_Onat)))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1608 (all V_k all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k)) # label(fact_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1609 (all V_A_2 all T_b all V_x_2 all V_B_2 (hBOOL(hAPP(V_B_2,V_x_2)) -> hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1I2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1610 (all V_l all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_l) -> c_List_Olinorder__class_Osorted(T_a,c_List_Oremdups(T_a,V_l))))) # label(fact_sorted__remdups) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1611 (all V_A_2 all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) -> c_Groups_Oone__class_Oone(T_b) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(c_COMBK(T_b,T_c),c_Groups_Oone__class_Oone(T_b))),V_A_2))) # label(fact_setprod__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1612 (all V_z all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)),V_x)))))) # label(fact_sup__least) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1613 (all V_A_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_x_2),V_A_2)))) # label(fact_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1614 (all V_A_2 all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),V_A_2)) # label(fact_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1615 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Lattices_Osemilattice__sup(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),V_y_2)),V_z_2))))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1616 (all V_R_2 all T_b hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_R_2) = V_R_2) # label(fact_rel__pow__1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1617 (all V_bs_2 all V_f_2 all T_d all V_as_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_d,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_d,T_c),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_fun(T_d,tc_HOL_Obool)),T_b),c_COMBB(T_c,tc_HOL_Obool,T_d)),V_P_2)),V_f_2),V_as_2,V_bs_2) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_as_2,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_bs_2)))) # label(fact_list__all2__map2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1618 (all V_S_2 all V_R_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_S_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_R_2),V_S_2)))) # label(fact_pred__subset__eq2) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1619 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__less__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1620 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_abs__minus__le__zero) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1621 (all V_xs_2 all V_c_2 all V_f_2 all T_c all T_b (class_Rings_Osemiring__0(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))),V_c_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Otimes__class_Otimes(T_b)),V_f_2)),V_c_2)),V_xs_2)))) # label(fact_listsum__mult__const) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1622 (all T_b c_Set_OBex(T_b) = c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_HOL_Obool)) # label(fact_SUPR__bool__eq) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1623 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2)) # label(fact_Int__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1624 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1625 (all V_xs_2 all T_c all T_b (class_Groups_Omonoid__add(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(c_COMBK(T_b,T_c),c_Groups_Ozero__class_Ozero(T_b))),V_xs_2)))) # label(fact_listsum__0) # label(axiom) # label(non_clause). [assumption]. 8.70/8.78 1626 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)))) # label(fact_DiffD2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1627 (all V_s_2 all V_r_2 all T_b hAPP(c_Relation_OField(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OField(T_b),V_r_2)),hAPP(c_Relation_OField(T_b),V_s_2))) # label(fact_Field__Un) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1628 (all V_m_2 all V_n_2 c_Nat_Onat_Onat__case(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n_2)),V_m_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) # label(fact_min__Suc1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1629 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Groups_Osgn__class_Osgn(T_b,V_a_2)))))) # label(fact_sgn__greater) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1630 (all V_g_2 all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_b))) # label(fact_setprod_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1631 (all V_fs_2 all T_b c_Wellfounded_Owf(T_b,c_List_Omeasures(T_b,V_fs_2))) # label(fact_wf__measures) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1632 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_a = V_b)))))) # label(fact_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1633 (all V_c all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))))))) # label(fact_divide__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1634 (all V_c all V_a all V_b all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))))))) # label(fact_divide__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1635 (all V_g_2 all V_x_2 all V_f_2 all T_b all T_c all T_d (hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_b,T_d),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_b,T_d),V_x_2)) <-> hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_g_2),V_x_2) = hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),V_x_2))) # label(fact_apsnd__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1636 (all V_u_2 all V_l_2 all V_less_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThanLessThan(T_b,V_less_2,V_l_2,V_u_2))) <-> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)))) # label(fact_ord_OgreaterThanLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1637 (all V_b all V_x all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x))))) # label(fact_min__max_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1638 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1639 (all V_n_2 all V_R_2 all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) # label(fact_rel__pow__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1640 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2),V_A_2))) # label(fact_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1641 (all V_R_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Wellfounded_Oacc(T_b,V_R_2))) -> -(all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_z),V_x_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),B_z),c_Wellfounded_Oacc(T_b,V_R_2))))))) # label(fact_not__acc__down) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1642 (all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_l_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_l_2))) # label(fact_ivl__disj__un_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1643 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2)) <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_b) & V_y_2 = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1644 (all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_2)))) # label(fact_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1645 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = V_x)) # label(fact_min__max_Oinf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1646 (all V_xs_2 all V_f_2 all T_c all T_b hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_set__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1647 (all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_a_2),c_List_Olists(T_b,V_A_2))) <-> (exists B_a exists B_l (V_a_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),B_a),B_l) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_l),c_List_Olists(T_b,V_A_2))) & hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2)))) | V_a_2 = c_List_Olist_ONil(T_b))) # label(fact_lists_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1648 (all V_xs all V_x all T_a c_List_Oremdups(T_a,c_List_Oinsert(T_a,V_x,V_xs)) = c_List_Oinsert(T_a,V_x,c_List_Oremdups(T_a,V_xs))) # label(fact_insert__remdups) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1649 (all V_r_2 all V_xs_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),c_List_Olist_ONil(T_b))),c_List_Olistrel1(T_b,V_r_2)))) # label(fact_not__listrel1__Nil) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1650 (all V_i_2 all V_xs_2 all T_b (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),c_List_Otranspose(T_b,V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2),V_i_2))))) # label(fact_transpose__column__length) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1651 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2))))) # label(fact_finite__Pow__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1652 (all V_a_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(hAPP(c_Set_Oinsert(T_b),c_HOL_OThe(T_b,hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),V_a_2))),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_inj__vimage__singleton) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1653 (all V_x all T_a (class_Orderings_Olinorder(T_a) -> c_List_Olinorder__class_Osorted(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))))) # label(fact_sorted__single) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1654 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_x)))) # label(fact_inf__le1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1655 (all V_F_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_x_2),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_x),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_x_2),V_F_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_xa),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_x_2),V_F_2))) -> (B_x != B_xa -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),B_x),B_xa) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_x_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_F_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_insert__partition) # label(axiom) # label(non_clause). [assumption]. 8.78/8.78 1656 (all T_a (class_Groups_Omonoid__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),c_List_Olist_ONil(T_a)))) # label(fact_listsum__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1657 (all V_P_2 all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_y_2),c_Int_OInteg)) -> ((all B_x hBOOL(hAPP(V_P_2,hAPP(c_Int_ORep__Integ,B_x)))) -> hBOOL(hAPP(V_P_2,V_y_2))))) # label(fact_Rep__Integ__induct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1658 (all V_n_2 all V_m_2 c_List_Oupt(V_m_2,V_n_2) = c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))) # label(fact_sorted__list__of__set__range) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1659 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_folding_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1660 (all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)))) # label(fact_gr0I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1661 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_z),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1662 (all V_xs_2 all V_f_2 all T_b all T_c c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_b)),hAPP(c_List_Omap(T_c,T_b),V_f_2)),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Oconcat(T_c,V_xs_2))) # label(fact_map__concat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1663 (all V_b_2 all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (V_a_2 = V_b_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))) # label(fact_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1664 (all V_fs_2 all T_b hAPP(hAPP(c_Relation_Oinv__image(tc_List_Olist(tc_Nat_Onat),T_b),c_List_Olex(tc_Nat_Onat,c_Wellfounded_Oless__than)),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(tc_fun(T_b,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_Nat_Onat),tc_Nat_Onat),tc_fun(tc_List_Olist(tc_fun(T_b,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),T_b),c_List_Omap(tc_fun(T_b,tc_Nat_Onat),tc_Nat_Onat)),hAPP(c_COMBC(tc_fun(T_b,tc_Nat_Onat),T_b,tc_Nat_Onat),c_COMBI(tc_fun(T_b,tc_Nat_Onat))))),V_fs_2)) = c_List_Omeasures(T_b,V_fs_2)) # label(fact_measures__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1665 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_a))))) # label(fact_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1666 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_z -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_x)),V_y)),V_z)))) # label(fact_diff__divide__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1667 (all V_B_2 all V_A_2 all V_R_2 all T_c all T_b (c_Relation_Osingle__valued(T_b,T_c,hAPP(c_Relation_Oconverse(T_c,T_b),V_R_2)) -> hAPP(c_Relation_OImage(T_c,T_b,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_B_2)))) # label(fact_Image__Int__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1668 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_List_Ofoldl(T_b,T_b,c_Orderings_Oord__class_Omax(T_b),V_x_2),V_xs_2) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_Max__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1669 (all V_n all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n))) # label(fact_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1670 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add__0__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1671 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_y)))) # label(fact_min__max_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1672 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2)))) # label(fact_ivl__disj__un_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1673 (all V_n all V_m ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))) -> c_List_Oupt(hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n)) = c_List_Olist_ONil(tc_Nat_Onat)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))) -> c_List_Oupt(hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n)) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),c_List_Oupt(hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n)))))) # label(fact_upt__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1674 (all V_A_2 all V_B_2 all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_B_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)))))) # label(fact_vimage__subsetD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1675 (all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,V_P_2),c_List_Olist_ONil(T_b)) = c_List_Olist_ONil(T_b)) # label(fact_filter_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1676 (all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))))) # label(fact_trancl_Or__into__trancl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1677 (all V_A_2 all V_g_2 all V_f_2 all T_c all T_b all T_d hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(c_Set_Oimage(T_d,T_b,V_f_2),V_A_2)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),c_Set_Oimage(T_d,T_c,V_g_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_b),c_Set_Ovimage(T_d,T_b,V_f_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),V_A_2))) = hAPP(c_Set_Oimage(T_d,tc_prod(T_b,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_prod(T_b,T_c)),T_d),c_Product__Type_OPair(T_b,T_c)),V_f_2)),V_g_2)),V_A_2)) # label(fact_image__split__eq__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1678 (all V_n all V_i all V_m hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_i)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n)),V_i)) # label(fact_min__diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1679 (all V_x_2 all V_f_2 all T_d all T_b all T_c hAPP(c_Product__Type_Osnd(T_d,T_b),V_x_2) = hAPP(c_Product__Type_Osnd(T_c,T_b),hAPP(c_Product__Type_Oapfst(T_d,T_c,T_b,V_f_2),V_x_2))) # label(fact_snd__apfst) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1680 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1681 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))) -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)))))) # label(fact_le__infE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1682 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2)),hAPP(c_Nat_OSuc,V_j_2)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2))) # label(fact_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1683 (all V_j_2 all V_i_2 all V_k_2 hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))) # label(fact_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1684 (all V_B_2 all V_C_2 all T_c all V_A_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1685 (all V_a all V_b all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_nonzero__minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1686 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z))) # label(fact_min__max_Osup__assoc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1687 (all V_g_2 all V_f_2 all V_d_2 all V_b_2 all V_c_2 all V_a_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) & class_Orderings_Oord(T_b) -> (V_a_2 = V_c_2 -> (V_b_2 = V_d_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),B_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),B_x),V_d_2)) -> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2))))))) # label(fact_setsum__ivl__cong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1688 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_finite__compl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1689 (all V_k_2 all V_less_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OlessThan(T_b,V_less_2,V_k_2))) <-> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)))) # label(fact_ord_OlessThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1690 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2))))) # label(fact_subset__Pow__Union) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1691 (all V_ys all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys) -> hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(c_List_Orev(T_a),V_xs)),hAPP(c_List_Orev(T_b),V_ys)) = hAPP(c_List_Orev(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)))) # label(fact_zip__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1692 (all V_f_2 all V_x_2 all T_b all T_c all V_i_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_i_2 -> hAPP(c_New__DSequence_Opos__decr__bind(T_c,T_b,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Oempty(T_b)) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_i_2 -> hAPP(c_Lazy__Sequence_Obind(T_c,T_b,hAPP(V_x_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b)),V_f_2),V_i_2)) = hAPP(c_New__DSequence_Opos__decr__bind(T_c,T_b,V_x_2,V_f_2),V_i_2)))) # label(fact_New__DSequence_Opos__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1693 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_finite__Diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1694 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1695 (all V_xs_2 all V_x_2 all T_b ((exists B_ys (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),B_ys))) & (exists B_zs V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_zs))))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_in__set__conv__decomp__first) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1696 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),T_d),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_INT__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1697 (all V_x_2 all V_f_2 all T_d all T_b all T_c hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_c,T_d),V_x_2)) = hAPP(c_Product__Type_Osnd(T_c,T_b),hAPP(c_Product__Type_Oapsnd(T_d,T_b,T_c,V_f_2),V_x_2))) # label(fact_snd__apsnd) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1698 (all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_zero__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1699 (all V_nat_2 all V_f2_2 all V_f1_2 all T_b hAPP(c_Nat_Onat_Onat__rec(T_b,V_f1_2,V_f2_2),hAPP(c_Nat_OSuc,V_nat_2)) = hAPP(hAPP(V_f2_2,V_nat_2),hAPP(c_Nat_Onat_Onat__rec(T_b,V_f1_2,V_f2_2),V_nat_2))) # label(fact_nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1700 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1701 (all V_r_2 all T_b (c_Relation_Oantisym(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) <-> c_Relation_Oantisym(T_b,V_r_2))) # label(fact_antisym__converse) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1702 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)))))) # label(fact_Max_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1703 (all V_v_H_2 all V_v_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),V_v_H_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))))) # label(fact_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1704 (all V_m all V_i hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_i))))) # label(fact_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1705 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> (hAPP(c_Nat_OSuc,V_m) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))))) # label(fact_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1706 (all V_fa_2 all T_b hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_b,V_fa_2),c_Lazy__Sequence_Olazy__sequence_OEmpty(T_b)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_lazy__sequence_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1707 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_y = V_x)))) # label(fact_order__antisym) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1708 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (V_y_2 = c_Groups_Ozero__class_Ozero(T_b) & V_x_2 = c_Groups_Ozero__class_Ozero(T_b) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1709 (all V_x_2 all V_k_2 all V_xs_2 all T_b (hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_k_2),V_x_2) = c_List_Olist_ONil(T_b) <-> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_list__update__nonempty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1710 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_sup__top__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1711 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Relation_OId__on(T_b,V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2))))) # label(fact_Id__on__subset__Times) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1712 (all V_P_2 all T_b all T_c (class_Enum_Oenum(T_b) & class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_Enum_Oenum__class_Oenum__all(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_HOL_Obool,T_c),V_P_2)),c_Product__Type_OPair(T_b,T_c))))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(tc_prod(T_b,T_c)),V_P_2))))) # label(fact_enum__all__prod__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1713 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1714 (all V_x_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Wellfounded_Oacc(T_b,V_r_2))))) # label(fact_acc__wfD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1715 (all V_X_2 all V_x_2 all V_Y_2 all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_Y_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__less)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),V_Y_2)),c_FunDef_Omax__strict)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_X_2)),V_Y_2)),c_FunDef_Omax__strict)))))) # label(fact_smax__insertI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1716 (all V_F_2 all V_h_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_h_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_Ovimage(T_b,T_c,V_h_2),V_F_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2))))) # label(fact_finite__vimage__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1717 (all V_i_1 all V_x all T_a hAPP(c_New__DSequence_Opos__single(T_a,V_x),V_i_1) = hAPP(c_Lazy__Sequence_Osingle(T_a),V_x)) # label(fact_New__DSequence_Opos__single__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1718 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))))) # label(fact_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1719 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_b (class_Lattices_Olattice(T_b) -> hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))) # label(fact_sup__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1720 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1721 (all T_b c_Relation_Otrans(T_b,c_Relation_OId(T_b))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1722 (all T_b hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_finite_OemptyI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1723 (all V_x all T_a all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_n)) -> hAPP(hAPP(c_List_Onth(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_i) = V_x)) # label(fact_nth__replicate) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1724 (all V_b_2 all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2) = V_a_2))) # label(fact_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1725 (all V_r_2 all T_b hAPP(c_List_Olexn(T_b,V_r_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool))) # label(fact_lexn_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1726 (all V_l all V_k (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l))) # label(fact_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1727 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2))) # label(fact_bij__betw__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1728 (all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oord(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_g_2),V_f_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,T_c)),V_f_2),V_g_2))))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1729 (all V_w_2 all V_m_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = V_w_2) <-> hAPP(c_Int_Onat,V_w_2) = V_m_2)) # label(fact_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1730 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_int__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1731 (all V_r_2 all V_xs_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_xs_2)),c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2))))) # label(fact_listrel__rtrancl__refl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1732 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_j),V_k))))) # label(fact_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1733 (all V_a5_2 all V_a1_2 all V_a2_2 all V_a4_2 all V_a3_2 all T_c all T_b (c_Nitpick_Ofold__graph_H(T_b,T_c,V_a3_2,V_a4_2,V_a2_2,hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2)) <-> (exists B_x exists B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_a2_2)) & c_Nitpick_Ofold__graph_H(T_b,T_c,V_a3_2,V_a4_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_a2_2),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),B_y) & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y))) | V_a4_2 = hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) & V_a2_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_fold__graph_H_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1734 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_l_2))))) # label(fact_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1735 (all V_y_2 all T_b (class_Int_Onumber__ring(T_b) -> (c_Groups_Oone__class_Oone(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2) <-> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2))))))) # label(fact_eq__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1736 (all V_x_2 all T_b (class_Int_Onumber__ring(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2) <-> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls))))))) # label(fact_eq__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1737 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_F_2,V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),V_A_2)))) # label(fact_folding__image_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1738 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__compl__bot) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1739 (all V_w hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_w))) # label(fact_zle__refl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1740 (all T_c all V_h_2 all V_f_2 all V_S_2 all T_b all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_S_2)) -> (B_x != B_y -> (hAPP(V_f_2,B_x) = hAPP(V_f_2,B_y) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_h_2,hAPP(V_f_2,B_x)))))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_h_2),V_f_2)),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_h_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_S_2)))))) # label(fact_setsum__reindex__nonzero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1741 (all V_B_2 all V_a_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_Oinsert(T_c),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_Oinsert(T_c),V_a_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_UN__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1742 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1743 (all V_zs all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Oappend(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys)),V_xs)),V_ys)),hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys)),V_xs)),V_zs)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),hAPP(hAPP(c_List_Oappend(T_b),V_ys),V_zs))) # label(fact_zip__append2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1744 (all V_l_2 all V_k_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(tc_Int_Oint,T_b),V_f_2),c_List_Oupto(V_k_2,V_l_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),V_f_2),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_k_2,V_l_2))))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1745 (all V_P_2 all T_b -c_List_Olist__ex(T_b,V_P_2,c_List_Olist_ONil(T_b))) # label(fact_list__ex__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1746 (all V_xs_2 all V_f_2 all T_c all T_b (class_Groups_Oab__group__add(T_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(c_Fun_Ocomp(T_b,T_b,T_c,c_Groups_Ouminus__class_Ouminus(T_b)),V_f_2)),V_xs_2)))) # label(fact_uminus__listsum__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1747 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) & V_B_2 != V_A_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1748 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Oplus__class_Oplus(T_c)),V_g_2),c_Groups_Ozero__class_Ozero(T_c)),V_A_2)))) # label(fact_setsum_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1749 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)))) # label(fact_of__int__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1750 (all V_y_2 all V_x_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (V_x_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),V_y_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1751 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2)) -> c_Divides_OposDivAlg(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> c_Divides_OposDivAlg(V_a_2,V_b_2) = hAPP(c_Divides_Oadjust(V_b_2),c_Divides_OposDivAlg(V_a_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b_2)))))) # label(fact_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1752 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Oring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))))) # label(fact_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1753 (all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Olist_ONil(T_c)) = c_List_Olist_ONil(T_b)) # label(fact_map_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1754 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) <-> c_Fun_Oinj__on(tc_List_Olist(T_b),tc_List_Olist(T_c),hAPP(c_List_Omap(T_b,T_c),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) # label(fact_inj__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1755 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)))))) # label(fact_card__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1756 (all V_g_2 all V_f_2 all T_d all T_b all T_c hAPP(c_Set_Oimage(T_c,T_b,hAPP(hAPP(c_COMBB(T_d,T_b,T_c),V_f_2),V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = hAPP(c_Set_Oimage(T_d,T_b,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_range__composition) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1757 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))) # label(fact_Suc__n__div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1758 (all V_x all V_y ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_y -> V_x = c_Nitpick_Onat__gcd(V_x,V_y)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_y -> c_Nitpick_Onat__gcd(V_x,V_y) = c_Nitpick_Onat__gcd(V_y,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y))))) # label(fact_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1759 (all V_b_2 all V_a_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_c))),V_S_2) = hAPP(V_b_2,V_a_2)) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_c))),V_S_2))))) # label(fact_setprod__delta) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1760 (all V_n_2 all V_v_2 all V_f_2 all V_a_2 all T_b hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_b),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_b,T_b),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),tc_Int_Oint),c_If(T_b)),c_Nat__Numeral_Oneg)),c_Nat_Onat_Onat__case(T_b,V_a_2,V_f_2,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),V_f_2),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Int_Onat)),V_n_2)))) = c_Nat_Onat_Onat__case(T_b,V_a_2,V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),V_n_2))) # label(fact_nat__case__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.79 1761 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)))))) # label(fact_le__supI1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1762 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1763 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = V_B_2 -> hAPP(c_Set_Oimage(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_c,T_b,V_f_2)),hAPP(c_Set_OPow(T_c),V_A_2)) = hAPP(c_Set_OPow(T_b),V_B_2))) # label(fact_image__Pow__surj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1764 (all V_y_2 all V_x_2 all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_R_2,V_y_2),V_x_2))))) # label(fact_equivp__symp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1765 (all V_n all V_m all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)))) # label(fact_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1766 (all V_r_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),B_x),V_r_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2)))) <-> c_Relation_Orefl__on(T_b,V_A_2,V_r_2))) # label(fact_refl__on__def_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1767 (all V_R_2 all V_n_2 all T_b hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2)) # label(fact_relpow_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1768 (all V_w all T_b (class_Int_Onumber(T_b) & class_Groups_Omonoid__mult(T_b) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)))) # label(fact_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1769 (all V_B_2 all V_C_2 all V_a_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_C_2)) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2)))) # label(fact_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1770 (all V_n_2 all V_m_2 hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = c_Nat_Onat_Onat__case(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Orderings_Oord__class_Omin(tc_Nat_Onat)),V_n_2)),V_m_2)) # label(fact_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1771 (all V_xs_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_y_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),V_xs_2)))) # label(fact_insort__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1772 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1773 (all V_n all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) # label(fact_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1774 (all V_Q_2 all V_P_2 (hBOOL(V_P_2) | hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)))) # label(help_c__fimplies__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1775 (all V_ys all T_a V_ys = c_List_Osplice(T_a,c_List_Olist_ONil(T_a),V_ys)) # label(fact_splice_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1776 (all V_n all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))) # label(fact_power__divide) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1777 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2))),c_Int_Ointrel)))) # label(fact_intrel__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1778 (all V_x all V_ys all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_i),V_x) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_ys))) # label(fact_list__update__append1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1779 (all V_m c_Divides_Odivmod__nat(V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) # label(fact_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1780 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> c_Fun_Obij__betw(T_c,T_b,c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),V_B_2,V_A_2))) # label(fact_bij__betw__the__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1781 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_minus__divide__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1782 (all V_y_2 all V_A_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2)) -> V_y_2 = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)))) # label(fact_fun__left__comm_Ofold__equality) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1783 (all V_x_2 all V_A_2 all T_c all T_b (class_Groups_Ouminus(T_b) -> hAPP(hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,T_b)),V_A_2),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(V_A_2,V_x_2)))) # label(fact_uminus__apply) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1784 (all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) # label(fact_expand__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1785 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N))))))) # label(fact_power__increasing) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1786 (all V_x_2 all V_f_2 all T_b -hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_x_2))) # label(fact_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1787 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))) # label(fact_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1788 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)))) # label(fact_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1789 (all V_xs_2 all T_b all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_n_2)),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_nth__mem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1790 (all V_v all V_b all V_a all T_b (class_Int_Onumber(T_b) & class_Rings_Osemiring(T_b) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a),V_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)))) # label(fact_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1791 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_UN__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1792 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_A_2)) = hAPP(c_Set_Oimage(tc_prod(T_c,T_b),tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_b,tc_prod(T_b,T_c)),hAPP(c_COMBC(T_b,T_c,tc_prod(T_b,T_c)),c_Product__Type_OPair(T_b,T_c)))),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_B_2)))) # label(fact_swap__product) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1793 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l))) # label(fact_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1794 (all V_r_2 all T_b c_Relation_Osym(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) # label(fact_sym__Un__converse) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1795 (all V_XS_2 all V_A_2 all T_b hAPP(c_Set_OCollect(tc_List_Olist(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_List_Olist(T_b)),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_List_Olist(T_b)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),tc_List_Olist(T_b)),c_COMBS(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),T_b),c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))),tc_List_Olist(T_b)),c_COMBB(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_List_Olist(T_b),tc_HOL_Obool,tc_List_Olist(T_b))),c_fequal))),c_List_Olist_OCons(T_b)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_b))),V_XS_2)))))) = c_List_Oset__Cons(T_b,V_A_2,V_XS_2)) # label(fact_set__Cons__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1796 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)))) # label(fact_transfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1797 (all V_g_2 all T_b all T_c V_g_2 = hAPP(c_Fun_Ocomp(T_c,T_c,T_b,c_Fun_Oid(T_c)),V_g_2)) # label(fact_id__o) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1798 (all V_x_2 all V_A_2 all T_b (class_Lattices_Odistrib__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))))))))) # label(fact_sup__Inf1__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1799 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y)))))) # label(fact_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1800 (all T_b (class_Finite__Set_Ofinite(T_b) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_finite__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1801 (all V_A_2 all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_a_2),c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)) = c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))) # label(fact_Inf__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1802 (all T_b (class_Orderings_Olinorder(T_b) -> c_Orderings_Oord_Omin(T_b,hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_b))) = c_Orderings_Oord__class_Omax(T_b))) # label(fact_dual__min) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1803 (all V_xs_2 all V_n_2 all T_b (c_List_Olist_ONil(T_b) = hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_take__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1804 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),c_List_Oset(T_b)),V_A_2))) -> c_Fun_Oinj__on(tc_List_Olist(T_b),tc_List_Olist(T_c),hAPP(c_List_Omap(T_b,T_c),V_f_2),V_A_2))) # label(fact_inj__on__mapI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1805 (all T_a (class_Int_Onumber__ring(T_a) & class_Rings_Olinordered__idom(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1806 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Max_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1807 (all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2))))) # label(fact_atLeastatMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1808 (all V_n all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n) = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))))) # label(fact_hd__drop__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1809 (all V_s_2 all V_s_H_2 all T_d all V_r_2 all V_r_H_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_H_2),V_r_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_s_H_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_H_2),V_s_H_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_2),V_s_2)))))) # label(fact_rel__comp__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1810 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_B_2))),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Inter__Un__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1811 (all V_u_2 all V_l_2 hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2))) # label(fact_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1812 (all V_X_2 all T_b c_HOL_OThe(T_b,hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),hAPP(c_fequal,V_X_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = c_Set_Othe__elem(T_b,V_X_2)) # label(fact_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1813 (all V_b all V_a all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a))))) # label(fact_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1814 (all T_1 all T_2 class_Nat_Osize(tc_prod(T_2,T_1))) # label(arity_prod__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1815 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_a_2),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))))) # label(fact_min__max_OInf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1816 (all V_xs all V_n all T_a hAPP(c_List_Orotate1(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_List_Orotate(T_a,hAPP(c_Nat_OSuc,V_n)),V_xs)) # label(fact_rotate__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1817 (all V_w_2 all V_x_2 all V_z_2 all V_y_2 all T_b (class_Fields_Ofield(T_b) -> (V_y_2 != c_Groups_Ozero__class_Ozero(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_z_2 -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_y_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_z_2) <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_w_2),V_z_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_x_2),V_y_2)))))) # label(fact_frac__eq__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1818 (all V_s_2 all V_r_2 all T_d all V_xz_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),V_xz_2),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_r_2),V_s_2))) -> -(all B_x all B_y all B_z (V_xz_2 = hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_x),B_z) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_d),B_x),B_y)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_y),B_z)),V_s_2))))))) # label(fact_rel__compE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1819 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))) # label(fact_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1820 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1821 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c (class_Lattices_Osemilattice__sup(T_c) & class_Lattices_Osemilattice__sup(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2))),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_A_2),V_B_2))))))) # label(fact_mono__sup) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1822 (all V_n all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1823 (all T_b hAPP(c_Finite__Set_Ocard(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1824 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit1(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1825 (all V_s_2 all V_g_2 all V_h_2 all V_f_2 all T_c all T_d all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_h_2),hAPP(V_g_2,B_x)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_c,hAPP(V_f_2,B_x)),V_h_2))) -> hAPP(c_List_Ofoldl(T_d,T_b,hAPP(c_COMBC(T_b,T_d,T_d),V_f_2),hAPP(V_h_2,V_s_2)),V_xs_2) = hAPP(V_h_2,hAPP(c_List_Ofoldl(T_c,T_b,hAPP(c_COMBC(T_b,T_c,T_c),V_g_2),V_s_2),V_xs_2)))) # label(fact_foldl__apply) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1826 (all V_f_2 all T_d all V_g_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),T_c,T_d),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Set_OCollect(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2)))),hAPP(c_COMBC(T_b,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_b),c_fequal),V_f_2)))))),hAPP(c_Set_Oimage(T_b,T_d,V_f_2),V_S_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_S_2)))) # label(fact_setsum__image__gen) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1827 (all V_x all T_a (class_Orderings_Otop(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),c_Orderings_Otop__class_Otop(T_a))))) # label(fact_top__greatest) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1828 (all V_l_2 all V_xs_2 all T_b all V_x_2 all V_P_2 (-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_OtakeWhile(T_b,V_P_2,V_xs_2) = c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_l_2))))) # label(fact_takeWhile__tail) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1829 (all V_ys_092_060_094isub_0622_2 all V_ys_092_060_094isub_0621_2 all V_xs_092_060_094isub_0622_2 all V_xs_092_060_094isub_0621_2 all T_b (hAPP(hAPP(c_List_Oappend(T_b),V_ys_092_060_094isub_0621_2),V_ys_092_060_094isub_0622_2) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_092_060_094isub_0621_2),V_xs_092_060_094isub_0622_2) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_092_060_094isub_0621_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_092_060_094isub_0621_2))) -> V_ys_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_b),hAPP(hAPP(c_List_Odrop(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2)),V_xs_092_060_094isub_0622_2) & V_ys_092_060_094isub_0621_2 = hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_092_060_094isub_0621_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_092_060_094isub_0621_2))) -> V_xs_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_b),hAPP(hAPP(c_List_Odrop(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_092_060_094isub_0621_2)),V_ys_092_060_094isub_0621_2)),V_ys_092_060_094isub_0622_2) & V_xs_092_060_094isub_0621_2 = hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_092_060_094isub_0621_2)),V_ys_092_060_094isub_0621_2)))) # label(fact_append__eq__append__conv__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1830 (all V_y_2 all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> (hAPP(V_Rep_2,V_x_2) = hAPP(V_Rep_2,V_y_2) <-> V_x_2 = V_y_2))) # label(fact_type__definition_ORep__inject) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1831 (all V_zs_2 all V_ys_2 all V_xs_2 all V_x_2 all T_b (hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),V_zs_2) <-> c_List_Olist_ONil(T_b) = V_ys_2 & hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2) = V_zs_2 | (exists B_ys_H (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),B_ys_H),V_zs_2) & V_ys_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_ys_H))))) # label(fact_Cons__eq__append__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1832 (all V_B_2 all V_A_2 all V_a_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_B_2,V_a_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),V_B_2)) # label(fact_INT__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1833 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> V_g_2 = hAPP(c_Fun_Ocomp(T_c,T_d,T_b,hAPP(c_Fun_Ocomp(T_b,T_d,T_c,V_g_2),c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2))),V_f_2))) # label(fact_o__inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1834 (all V_Q_2 all V_P_2 (-hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2)) | hBOOL(V_Q_2) | hBOOL(V_P_2))) # label(help_c__fdisj__3) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1835 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2))))))) # label(fact_Max_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1836 (all V_fs_2 all V_f_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) | hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,V_fs_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_b,tc_Nat_Onat)),V_f_2),V_fs_2)))))) # label(fact_in__measures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1837 (all V_qr_2 all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)) = V_m_2 & (V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2)))) & (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)) <-> hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m_2,V_n_2),V_qr_2)))) # label(fact_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1838 (all V_f_2 all T_b all T_c (c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) <-> hAPP(c_Fun_Ocomp(T_c,T_b,T_b,V_f_2),c_Hilbert__Choice_Oinv__into(T_c,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)) = c_Fun_Oid(T_b))) # label(fact_surj__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1839 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_add__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1840 (all V_M_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_M_2)) -> (exists B_h c_Fun_Obij__betw(T_b,tc_Nat_Onat,B_h,V_M_2,hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_M_2)))))) # label(fact_ex__bij__betw__finite__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1841 (all V_A_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Set_Oinsert(T_b)),V_f_2)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)) # label(fact_image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1842 (all V_d_2 c_Wellfounded_Owf(tc_Int_Oint,c_Int_Oint__ge__less__than(V_d_2))) # label(fact_wf__int__ge__less__than) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1843 (all V_B_2 all V_A_2 all T_b (hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2) <-> V_A_2 = V_B_2)) # label(fact_Compl__eq__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1844 (all V_A_2 all V_C_2 all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Int__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1845 (all V_i all V_b ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b)) -> c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = c_Random_Olog(V_b,V_i)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Random_Olog(V_b,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_i),V_b))) = c_Random_Olog(V_b,V_i)))) # label(fact_log_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1846 (all V_c_2 all T_b -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1847 (all V_xs all T_b all V_ys all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_ys)) -> hBOOL(hAPP(c_List_Odistinct(tc_prod(T_b,T_a)),hAPP(hAPP(c_List_Ozip(T_b,T_a),V_xs),V_ys))))) # label(fact_distinct__zipI2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1848 (all V_A_2 all V_f_2 all V_g_2 all T_d all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_g_2),V_f_2),V_A_2) -> c_Fun_Oinj__on(T_d,T_c,V_g_2,hAPP(c_Set_Oimage(T_b,T_d,V_f_2),V_A_2)))) # label(fact_inj__on__imageI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1849 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_A_2)))))) # label(fact_Min__in) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1850 (all V_z all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_z))) # label(fact_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1851 (all V_b_2 all V_a_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_equiv__class__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1852 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_List_Olast(tc_Nat_Onat,c_List_Oupt(V_i,V_j)))) # label(fact_last__upt) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1853 (all V_a_2 all V_f_2 all T_d all V_r2_2 all V_A2_2 all T_c all V_r1_2 all V_A1_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_c,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_b,T_c,tc_fun(T_d,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A2_2)) -> c_Equiv__Relations_Ocongruent(T_b,tc_fun(T_d,tc_HOL_Obool),V_r1_2,hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))),V_f_2))))))) # label(fact_congruent2__implies__congruent__UN) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1854 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> c_List_Olast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = c_List_Olast(T_a,V_xs))) # label(fact_last__drop) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1855 (all V_r_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Relation_ORange(T_b,T_c),V_r_2))))) # label(fact_finite__Range) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1856 (all V_n all V_m hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) # label(fact_zadd__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1857 (all T_c all T_b all V_b_2 all V_a_2 all V_f_2 (hBOOL(hAPP(hAPP(V_f_2,V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))))) # label(fact_splitI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1858 (all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_x_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)))) # label(fact_image__uminus__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1859 (all V_A_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OAll(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_fimplies),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_b,tc_HOL_Obool))),V_A_2)))),c_member(T_b)))) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)) # label(fact_Inter__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1860 (all V_M_2 all T_b all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> V_M_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),hAPP(c_COMBK(T_c,T_b),V_M_2))))) # label(fact_SUP__const) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1861 (all V_f_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_b,V_f_2),V_A_2)),V_A_2)) -> (c_Fun_Oinj__on(T_b,T_b,V_f_2,V_A_2) -> V_A_2 = hAPP(c_Set_Oimage(T_b,T_b,V_f_2),V_A_2))))) # label(fact_endo__inj__surj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1862 (all T_b (class_Lattices_Osemilattice__sup(T_b) -> c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)))) # label(fact_fun__left__comm__idem__sup) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1863 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_B_2))))) # label(fact_Pow__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1864 (all T_b c_Relation_Orefl__on(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Relation_OId(T_b))) # label(fact_refl__Id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1865 (all V_C_2 all V_B_2 all T_d all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_UN__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1866 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.78/8.80 1867 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1868 (all V_ys_2 all V_xs_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_List_Oset(T_c),V_ys_2))) = hAPP(c_List_Oset(tc_prod(T_b,T_c)),c_Enum_Oproduct(T_b,T_c,V_xs_2,V_ys_2))) # label(fact_product__list__set) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1869 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 | hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1870 (all V_s_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2)),hAPP(c_Relation_ODomain(T_b,T_c),V_s_2))))) # label(fact_Domain__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1871 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) -> (hAPP(c_Finite__Set_Ocard(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2) -> V_A_2 = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1872 (all V_r_2 all V_A_2 all V_X_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> -(all B_x (hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_X_2 -> -hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)))))) # label(fact_quotientE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1873 (all V_M_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_x),V_M_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_x)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_M_2))))) # label(fact_finite__Inter) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1874 (all V_A_2 all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) -> V_A_2 = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_A_2)))) # label(fact_surj__image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1875 (all V_xss_2 all T_b hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xss_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oconcat(T_b,V_xss_2))) # label(fact_length__concat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1876 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z) = V_z) # label(fact_zadd__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1877 (all V_A_2 all T_b hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,T_b),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_A_2)) # label(fact_card__eq__setsum) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1878 (all T_a c_List_Obutlast(T_a,c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_butlast_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1879 (all V_n all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))))) # label(fact_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1880 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) # label(fact_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1881 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1882 (all V_i_2 all V_a_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_i_2)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) -> hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_a_2)))))) # label(fact_distinct__list__update) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1883 (all V_n hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) # label(fact_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1884 (all T_b c_Relation_Orefl__on(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)))) # label(fact_refl__on__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1885 (all V_G_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_G_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_G_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)))) # label(fact_finite__Un) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1886 (all V_z V_z = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z))) # label(fact_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1887 (all V_b_2 all V_a_2 all T_b (class_Rings_Oring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))))))) # label(fact_Ints__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1888 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hAPP(c_Nat_OSuc,V_n) = V_m))) # label(fact_le__SucE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1889 (all V_R_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_R_2))) & V_a_2 != V_b_2 | V_a_2 = V_b_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1890 (all V_zs_2 all V_x_2 all T_b (class_Groups_Omonoid__add(T_b) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_x_2),hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_zs_2)) = hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_x_2),V_zs_2))) # label(fact_foldl__absorb0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1891 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> -(all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_y)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_z_2)),V_R_2)))))) # label(fact_rel__pow__Suc__E) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1892 (all V_z_2 all T_b (class_Rings_Oring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Int_Oring__1__class_OInts(T_b))))) # label(fact_Ints__of__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1893 (all V_a_2 all V_b_2 all V_r_2 all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_b_2)) -> (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2))))) # label(fact_accp__downward) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1894 (all V_a_2 all T_b (class_Rings_Oring__1__no__zero__divisors(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1895 (all V_m all V_y all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_y,V_m)),V_m)) # label(fact_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1896 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)) -> -(hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)) -> -hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2))))) # label(fact_inf2E) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1897 (all V_B_2 all V_A_2 all V_I_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_B_2))) # label(fact_UN__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1898 (all V_j_2 all V_i_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_j_2) = hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_i_2) <-> V_i_2 = V_j_2))))) # label(fact_nth__eq__iff__index__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1899 (all V_r_2 all V_v_2 all V_u_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_x_2),V_u_2)),hAPP(hAPP(c_List_Oappend(T_b),V_x_2),V_v_2))),c_List_Olexord(T_b,V_r_2))) -> ((all B_a -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_a),B_a)),V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_u_2),V_v_2)),c_List_Olexord(T_b,V_r_2)))))) # label(fact_lexord__append__leftD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1900 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)))) # label(fact_of__nat__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1901 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2)),V_A_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))))) # label(fact_vimage__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1902 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))) # label(fact_inf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1903 (all V_xs_2 all T_b all V_ys_2 (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) <-> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_self__append__conv2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1904 (all V_x_2 all V_f_2 all T_c all T_b V_f_2 = c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,hAPP(V_f_2,V_x_2))) # label(fact_fun__upd__triv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1905 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1906 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_n)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_n)))))) # label(fact_number__of1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1907 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1908 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)))) # label(fact_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1909 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),V_A_2) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2))) # label(fact_bij__vimage__eq__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1910 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2)))) # label(fact_finite__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1911 (all V_B_2 all V_A_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_B_2)) = hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))))) # label(fact_fold1__Un) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1912 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)),V_B_2)))) # label(fact_Inter__lower) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1913 (all V_b_2 all V_B_2 all V_A_2 all T_c all T_b ((exists B_x (hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)))) <-> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_SUP1__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1914 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2))))) # label(fact_refl__onD2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1915 (all V_a1_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(c_Predicate_ODomainP(T_b,T_c,V_r_2),V_a1_2)) <-> (exists B_x1 hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_x1))))) # label(fact_DomainP_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1916 (all V_q all V_r all V_b all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_r),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))))))) # label(fact_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1917 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1918 (all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Relation_OId(T_b))) -> -(all B_x hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x) != V_p_2))) # label(fact_IdE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1919 (all V_l_2 all V_k_2 all T_b (class_Orderings_Oorder(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),V_l_2))))) # label(fact_greaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1920 (all V_v_2 all V_f_2 all T_c all V_xs_2 all V_y_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_List_Omap(T_b,T_c),c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_y_2,V_v_2)),V_xs_2) = hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2))) # label(fact_map__fun__upd) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1921 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)),V_z_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1922 (all V_k_2 all V_i_2 all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_k_2)))))) # label(fact_greaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1923 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_p),V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1924 (all V_l all V_k ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l)) -> V_l = c_Quickcheck_Obeyond(V_k,V_l)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l)) -> c_Quickcheck_Obeyond(V_k,V_l) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))) # label(fact_beyond__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1925 (all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_a_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_a_2),V_B_2))) # label(fact_Union__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1926 (all V_ys_2 all V_P_2 all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))) -> c_List_OdropWhile(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = c_List_OdropWhile(T_b,V_P_2,V_ys_2))) # label(fact_dropWhile__append2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1927 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))))) # label(fact_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1928 (all V_Q_2 all V_P_2 all T_c all T_b hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2))),V_Q_2))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(c_Set_OCollect(T_b),V_P_2)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),c_Set_OCollect(T_c)),V_Q_2))) # label(fact_SetCompr__Sigma__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1929 (all V_b_2 all V_a_2 all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_b_2)) <-> V_b_2 = V_a_2)) # label(fact_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1930 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2))))))))) # label(fact_min__max_OInf__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1931 (all V_i_2 all V_M_2 (-hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_M_2)) -> hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2)))))) # label(fact_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1932 (all V_x_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),hAPP(c_List_Orev(T_b),c_List_OtakeWhile(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_x_2)),V_xs_2))) = c_List_OdropWhile(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_x_2)),hAPP(c_List_Orev(T_b),V_xs_2))))) # label(fact_dropWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1933 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_p_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2),V_p_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_p_2)))))) # label(fact_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1934 (all V_x all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i) = V_x)) # label(fact_nth__list__update__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1935 (all V_n all V_m all V_i (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_i)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n))))) # label(fact_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1936 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)))))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1937 (all V_i_2 all V_j_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),V_i_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2))))) # label(fact_simp__from__to) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1938 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1939 (all V_ys all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Olist_ONil(T_a)),V_ys) = V_ys) # label(fact_append__Nil) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1940 (all V_y_2 all V_x_2 all V_z_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_x_2))))) # label(fact_min__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1941 (all V_w all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) -> V_z = V_w))) # label(fact_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1942 (all V_n all V_i c_Code__Numeral_Onat__of__aux(V_i,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_i)),V_n)) # label(fact_nat__of__aux__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1943 (all V_ys all V_xs all V_x all T_a hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_OremoveAll(T_a,V_x),V_xs)),hAPP(c_List_OremoveAll(T_a,V_x),V_ys))) # label(fact_removeAll__append) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1944 (all V_f_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(V_f_2,B_x) & (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2)) -> (B_xa != B_x -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))))))) # label(fact_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1945 (all V_h_2 all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_f_2,hAPP(hAPP(V_F_2,V_g_2),V_A_2)),hAPP(hAPP(V_F_2,V_h_2),V_A_2)) = hAPP(hAPP(V_F_2,hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),V_f_2),V_g_2)),V_h_2)),V_A_2)))) # label(fact_folding__image_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1946 (all V_B_2 all V_A_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))) # label(fact_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1947 (all V_B_2 all V_A_2 hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2)))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1948 (all V_v_2 all V_c_2 all V_b_2 all V_a_2 all T_b all T_c all T_d (hAPP(c_Fun_Ocomp(T_c,T_c,T_b,c_Fun_Oid(T_c)),V_c_2) = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_a_2),V_b_2) -> hAPP(V_c_2,V_v_2) = hAPP(V_a_2,hAPP(V_b_2,V_v_2)))) # label(fact_o__eq__id__dest) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1949 (all V_n all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1950 (all V_Y_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_Y_2)) -> (V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_Y_2)),c_FunDef_Omax__strict))))) # label(fact_smax__emptyI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1951 (all V_no_2 all V_yes_2 all V_xs_2 all V_P_2 all T_b (hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_yes_2),V_no_2) = c_List_Opartition(T_b,V_P_2,V_xs_2) -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_no_2))) -> -hBOOL(hAPP(V_P_2,B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_yes_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_partition__P) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1952 (all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Olinorder__class_Osort__key(T_c,T_b,V_f_2,V_xs_2))))) # label(fact_sorted__sort__key) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1953 (all V_i_2 all V_xq_2 hAPP(c_New__DSequence_Opos__not__seq(V_xq_2),V_i_2) = c_Lazy__Sequence_Ohb__not__seq(hAPP(V_xq_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_i_2)))) # label(fact_New__DSequence_Opos__not__seq__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1954 (all T_b (class_Orderings_Oord(T_b) -> c_Orderings_Oord_Omin(T_b,c_Orderings_Oord__class_Oless__eq(T_b)) = c_Orderings_Oord__class_Omin(T_b))) # label(fact_min__ord__min) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1955 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_OInf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1956 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_l)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_l)))))) # label(fact_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1957 (all V_m all V_n all V_a all T_a (class_Fields_Ofield(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_power__diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1958 (all V_v all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) # label(fact_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1959 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),V_b) = V_a))) # label(fact_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1960 (all V_y all V_x (c_Nat__Transfer_Ois__nat(V_x) -> (c_Nat__Transfer_Ois__nat(V_y) -> c_Nat__Transfer_Ois__nat(c_Nat__Transfer_Otsub(V_x,V_y))))) # label(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1961 (all V_v_H all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1962 (all V_xs_2 all T_b all V_P_2 all V_j_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_j_2)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))))))) # label(fact_length__takeWhile__less__P__nth) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1963 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBS(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal))),c_Orderings_Oord__class_Omin(T_b)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)))))))) # label(fact_min__max_Oinf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1964 (all V_r_2 all T_b (c_Wellfounded_Oacyclic(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) <-> c_Wellfounded_Oacyclic(T_b,V_r_2))) # label(fact_acyclic__converse) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1965 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2) <-> V_A_2 = V_B_2))) # label(fact_inj__on__Un__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1966 (all V_x_2 all V_n_2 all T_b (c_List_Oreplicate(T_b,V_n_2,V_x_2) = c_List_Olist_ONil(T_b) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_replicate__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1967 (all V_z_2 all V_w_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1968 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1969 (all V_P_2 all T_b (class_Enum_Oenum(T_b) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_b),V_P_2))))) # label(fact_exists__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1970 (all V_tsa_2 all V_zs_2 all V_ys_2 all V_xs_2 all T_b ((exists B_us (hAPP(hAPP(c_List_Oappend(T_b),B_us),V_ys_2) = V_tsa_2 & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),V_zs_2),B_us) | V_zs_2 = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),B_us) & V_ys_2 = hAPP(hAPP(c_List_Oappend(T_b),B_us),V_tsa_2))) <-> hAPP(hAPP(c_List_Oappend(T_b),V_zs_2),V_tsa_2) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) # label(fact_append__eq__append__conv2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1971 (all T_c all V_b_2 all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)))) -> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_INF1__I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1972 (all V_xss_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_b)),c_List_Otranspose__rel(T_b)),hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_xss_2))) -> hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_Olist__case(tc_List_Olist(T_b),T_b,c_List_Olist_ONil(T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_b),c_COMBK(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b))))),V_xss_2)))),c_List_Otranspose(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),V_xs_2),c_List_Oconcat(tc_List_Olist(T_b),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_b)),T_b,c_List_Olist_ONil(tc_List_Olist(T_b)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_OCons(tc_List_Olist(T_b))),c_List_Olist_ONil(tc_List_Olist(T_b)))))),V_xss_2))))) = c_List_Otranspose(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_xss_2)))) # label(fact_transpose_Opsimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1973 (all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m)))) # label(fact_le__square) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1974 (all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremdups(T_a,V_xs) = V_xs)) # label(fact_distinct__remdups__id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1975 (all V_xs_2 all V_P_2 all T_b hAPP(hAPP(c_List_Oappend(T_b),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)),c_List_OdropWhile(T_b,V_P_2,V_xs_2)) = V_xs_2) # label(fact_takeWhile__dropWhile__id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1976 (all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),V_A_2) = hAPP(V_F_2,V_A_2)))) # label(fact_folding__image__simple_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1977 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1978 (all V_xs_2 all V_P_2 all T_b hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),V_P_2)),V_xs_2)))) # label(fact_sum__length__filter__compl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1979 (all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1980 (all V_x all V_y (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y))) # label(fact_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1981 (all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)))) # label(fact_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1982 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.81 1983 (all V_A_2 all V_B_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_A_2))) # label(fact_Inter__image__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1984 (all V_ys_2 all V_f_2 all T_d all V_xs_2 all T_c all T_b hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_d),tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_d,tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_d,T_c),tc_fun(T_d,tc_prod(T_b,T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(tc_fun(T_d,T_c),tc_fun(T_d,tc_prod(T_b,T_c))),T_b),c_COMBB(T_c,tc_prod(T_b,T_c),T_d)),c_Product__Type_OPair(T_b,T_c))),V_f_2))),hAPP(hAPP(c_List_Ozip(T_b,T_d),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_ys_2))) # label(fact_zip__map2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1985 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y)))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1986 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_min__0R) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1987 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))))) # label(fact_set__mp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1988 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1989 (all V_b_2 all V_a_2 all T_c all T_b all V_f_2 (hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))) -> hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_b,T_c,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)))) # label(fact_curryI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1990 (all V_y all V_x all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)))))) # label(fact_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1991 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1992 (all V_nat_H hAPP(c_Nat_OSuc,V_nat_H) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1993 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_not__leE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1994 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Int_Oiszero(T_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_iszero__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1995 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b)) = c_Groups_Osgn__class_Osgn(T_b,V_a_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_sgn__1__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1996 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1997 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremove1(T_a,V_x,V_xs) = hAPP(c_List_OremoveAll(T_a,V_x),V_xs))) # label(fact_distinct__remove1__removeAll) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1998 (all V_i_2 all V_M_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_M_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))))) # label(fact_card__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 1999 (all V_a_2 all V_xs_2 all V_f_2 all T_d all V_g_2 all T_b all T_c c_List_Ofoldr(T_d,T_b,hAPP(c_Fun_Ocomp(T_c,tc_fun(T_b,T_b),T_d,V_g_2),V_f_2),V_xs_2,V_a_2) = c_List_Ofoldr(T_c,T_b,V_g_2,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2),V_a_2)) # label(fact_foldr__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2000 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> c_Groups_Osgn__class_Osgn(T_a,V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))))) # label(fact_sgn__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2001 (all V_Q_2 all V_P_2 (hBOOL(V_P_2) | -hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2)))) # label(help_c__fconj__2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2002 (all V_n_2 all V_P_2 (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> (hBOOL(hAPP(V_P_2,V_n_2)) -> (exists B_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_k),V_n_2)) & hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_k),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & (all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_k)) -> -hBOOL(hAPP(V_P_2,B_i))))))))) # label(fact_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2003 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))))) # label(fact_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2004 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))))) # label(fact_sup__ge1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2005 (all V_C_2 all V_J_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_J_2),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2)) # label(fact_Sigma__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2006 (all V_A_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) != hAPP(c_Set_OPow(T_b),V_A_2)) # label(fact_Pow__not__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2007 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c))) # label(fact_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2008 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),hAPP(c_List_OremoveAll(T_b,V_x_2),V_xs_2)) = hAPP(c_List_OremoveAll(T_c,hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)))) # label(fact_map__removeAll__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2009 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) # label(fact_min__max_OSup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2010 (all V_R_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2))) -> V_x_2 = V_y_2)) # label(fact_rel__pow__0__E) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2011 (all V_n_2 all V_k_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2) <-> V_m_2 = V_n_2 | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2)) # label(fact_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2012 (all V_j_2 all V_i_2 all V_P_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_Oall__interval__nat(V_P_2,V_i_2,V_j_2))) # label(fact_all__interval__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2013 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),B_x),hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2),B_x)))) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2))) # label(fact_list__all2__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2014 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b) -> (V_a_2 != V_b_2 & V_c_2 != V_d_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_d_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_d_2))))) # label(fact_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2015 (all V_C_2 all V_B_2 all V_A_2 all T_b (hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) -> (hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) -> hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_B_2),V_C_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))))) # label(fact_union__comp__emptyR) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2016 (all V_b c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b)) # label(fact_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2017 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2018 (all V_lazy__sequence_2 all V_a_2 all V_f2_2 all V_f1_2 all T_c all T_b hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2) = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_b,T_c,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2019 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__ex1(T_b,V_P_2,V_xs_2) <-> (exists B_x ((all B_y (hBOOL(hAPP(V_P_2,B_y)) & hBOOL(hAPP(hAPP(c_member(T_b),B_y),hAPP(c_List_Oset(T_b),V_xs_2))) -> B_x = B_y)) & hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))))))) # label(fact_list__ex1__iff__raw) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2020 (all V_m_2 hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_int__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2021 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) # label(fact_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2022 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2))))) # label(fact_Pow__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2023 (all V_l all V_k c_Int_OBit1(V_l) != c_Int_OBit0(V_k)) # label(fact_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2024 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_card_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2025 (all V_P_2 all V_n_2 ((exists B_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2)) & hBOOL(hAPP(V_P_2,B_m)))) <-> (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))))))) # label(fact_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2026 (all V_n hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n)) # label(fact_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2027 (all V_a2_2 all V_a1_2 all V_R_2 all T_b (c_Wellfounded_Omax__extp(T_b,V_R_2,V_a1_2,V_a2_2) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a2_2)) & c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_a2_2 & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)) & hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_a2_2)))))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_a1_2)))) # label(fact_max__extp_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2028 (all T_b c_Lazy__Sequence_Oanamorph(tc_Lazy__Sequence_Olazy__sequence(T_b),T_b,c_Lazy__Sequence_Oyield(T_b)) = c_Lazy__Sequence_Oyieldn(T_b)) # label(fact_yieldn__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2029 (all V_n all V_m ((V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n))) & (V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2030 (all V_c all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))))))) # label(fact_divide__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2031 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_List_Olists(T_b,V_A_2)))) # label(fact_lists_ONil) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2032 (all V_xss_2 all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_b)),c_List_Otranspose__rel(T_b)),hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_xss_2))) -> c_List_Otranspose(T_b,V_xss_2) = c_List_Otranspose(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_xss_2)))) # label(fact_transpose_Opsimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2033 (all V_B_2 all V_a_2 all T_b hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)))) # label(fact_insertI1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2034 (all V_i_2 all V_P_2 all V_p_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_p_2)) -> (hBOOL(hAPP(V_P_2,B_i)) -> hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_i),V_p_2)))))) -> (hBOOL(hAPP(V_P_2,V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_p_2)) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))))) # label(fact_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2035 (all V_z_2 all V_f_2 all T_c all T_b c_Nitpick_Ofold__graph_H(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_z_2)) # label(fact_fold__graph_H_Ointros_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2036 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_A_2)) -> V_t_2 != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xa),B_x))))) -> V_t_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),V_D_2))))) # label(fact_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2037 (all V_c_2 all V_b_2 all V_a_2 all V_R_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_fun(T_d,tc_HOL_Obool)),V_R_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_c_2)) -> hBOOL(hAPP(hAPP(hAPP(V_R_2,V_a_2),V_b_2),V_c_2)))) # label(fact_splitD_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2038 (all T_b hAPP(c_List_Omap(T_b,T_b),c_Fun_Oid(T_b)) = c_Fun_Oid(tc_List_Olist(T_b))) # label(fact_List_Omap_Oid) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2039 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> -(hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))))) # label(fact_DiffE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2040 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2041 (all V_b_2 all V_a_2 all V_f_2 all T_b all T_d all T_c hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)) = hAPP(hAPP(c_Product__Type_Ocurry(T_c,T_d,T_b,V_f_2),V_a_2),V_b_2)) # label(fact_curry__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2042 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> V_r_2 = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)),V_r_2))) # label(fact_equiv__comp__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2043 (all V_a_2 all V_f_2 all V_r2_2 all T_d all T_c all V_r1_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r1_2) -> (c_Equiv__Relations_Ocongruent2(T_b,T_c,T_d,V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> c_Equiv__Relations_Ocongruent(T_c,T_d,V_r2_2,hAPP(V_f_2,V_a_2)))))) # label(fact_congruent2__implies__congruent) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2044 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z))),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)))))) # label(fact_min__max_Odistrib__sup__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2045 (all V_xs_2 all T_b hAPP(c_List_Orev(T_b),V_xs_2) = hAPP(c_List_Ofoldl(tc_List_Olist(T_b),T_b,hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b)),V_xs_2)) # label(fact_rev__foldl__cons) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2046 (all V_b_2 all V_a_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) <-> -(-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2))))) # label(fact_dvd_OatLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2047 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs))) # label(fact_take__tl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2048 (all V_l all V_k c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_Opred(V_k)),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2049 (all V_f_2 all V_n_2 all T_c all T_b c_Fun_Ocomp(T_c,T_c,T_b,hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2)) = hAPP(hAPP(c_Nat_Ocompow(tc_fun(T_b,T_c),tc_fun(T_b,T_c)),V_n_2),c_Fun_Ocomp(T_c,T_c,T_b,V_f_2))) # label(fact_comp__funpow) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2050 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x_2),V_u_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y_2),V_v_2))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y_2),V_u_2)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2051 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2)) <-> hBOOL(c_Nat_Onat_Onat__case(tc_HOL_Obool,c_fFalse,hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2052 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2053 (all T_b all V_R_2 ((exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) -> (c_Predicate_Osymp(T_b,V_R_2) -> (c_Predicate_Otransp(T_b,V_R_2) -> c_Equiv__Relations_Opart__equivp(T_b,V_R_2))))) # label(fact_part__equivpI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2054 (all V_xs_2 all V_f_2 all T_c all V_P_2 all T_b hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_OdropWhile(T_c,hAPP(c_Fun_Ocomp(T_b,tc_HOL_Obool,T_c,V_P_2),V_f_2),V_xs_2)) = c_List_OdropWhile(T_b,V_P_2,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_dropWhile__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2055 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2056 (all V_m_2 all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(hAPP(c_Nat_OSuc,V_n_2),hAPP(c_Nat_OSuc,V_m_2))) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_n_2,V_m_2)) # label(fact_greaterThanAtMost__upt) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2057 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))))) # label(fact_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2058 (all V_S_2 all V_R_2 all T_b (c_Wellfounded_Owf(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_S_2)),V_R_2)) -> c_FunDef_Oreduction__pair(T_b,hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_reduction__pairI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2059 (all V_A_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2)) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2060 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_trancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2061 (all V_c_2 all V_b_2 all V_a_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_c_2),hAPP(c_Finite__Set_Ocard(T_b),V_S_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),V_c_2)),V_S_2)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_b_2,V_a_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_S_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),V_c_2)),V_S_2))))) # label(fact_setprod__gen__delta) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2062 (all V_y_2 all V_x_2 all V_z_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),V_y_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_z_2),V_x_2))))) # label(fact_le__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2063 (all V_A_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 | -hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2064 (all V_xs_2 all V_r_2 all T_c all T_b (class_Rings_Osemiring__1(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))),V_r_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(c_COMBK(T_b,T_c),V_r_2)),V_xs_2)))) # label(fact_listsum__triv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2065 (all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Nat_OSuc,hAPP(c_Int_Onat,V_z)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)))) # label(fact_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2066 (all V_M_2 all V_n_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)),V_M_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n_2)),V_M_2)),hAPP(V_M_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_UN__le__eq__Un0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2067 (all V_y_2 all V_n_2 all V_x_2 all T_b (V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_x_2 = V_y_2 <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_y_2)))))) # label(fact_in__set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2068 (all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b))) # label(fact_mod__minus__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2069 (all V_P_2 all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_b),V_P_2)))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) & hBOOL(hAPP(V_P_2,V_x_2)))) # label(fact_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2070 (all V_B_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))))) # label(fact_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2071 (all V_x_2 all T_b hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_UNIV__I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2072 (all V_f_2 all V_xs_2 all V_a_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)) -> (V_a_2 = hAPP(c_List_Ohd(T_b),hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_b),hAPP(c_fequal,hAPP(V_f_2,V_a_2))),V_f_2)),V_xs_2)) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_a_2),c_List_Oremove1(T_b,V_a_2,V_xs_2)) = V_xs_2))))) # label(fact_insort__key__remove1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2073 (all V_xs all T_a c_List_Obutlast(T_a,V_xs) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs)) # label(fact_butlast__conv__take) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2074 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2) <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2075 (all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2076 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),V_f_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_f_2)),V_B_2),V_A_2)))) # label(fact_inf__INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2077 (all V_k_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_k_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_k_2,V_l_2)))) # label(fact_greaterThanAtMost__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2078 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_List_Oupto__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_i all B_j (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_List_Oupto__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_i),B_j))) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_i),B_j)) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_i),c_Groups_Oone__class_Oone(tc_Int_Oint))),B_j))) -> hBOOL(hAPP(hAPP(V_P_2,B_i),B_j))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2079 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) # label(fact_length__take) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2080 (all T_c all T_b all V_B_2 all V_y_2 all V_x_2 all V_A_2 (hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2))))) # label(fact_inf2I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2081 (all V_f_2 all T_c all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> c_Relation_Osym(T_c,hAPP(hAPP(c_Relation_Oinv__image(T_b,T_c),V_r_2),V_f_2)))) # label(fact_sym__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2082 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2083 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))))) # label(fact_converse__rtrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2084 (all V_x all T_a (class_Rings_Osemiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x)))))) # label(fact_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2085 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(V_A_2,B_x)),V_B_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)),V_B_2)))) # label(fact_UN__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2086 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) <-> V_n_2 = V_m_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2087 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_divide__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2088 (all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olinorder__class_Osort__key(T_b,T_b,c_COMBI(T_b),c_List_Oremdups(T_b,V_xs_2)) = c_List_Olinorder__class_Osorted__list__of__set(T_b,hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_sorted__list__of__set__sort__remdups) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2089 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))) # label(fact_min__max_Osup__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.82 2090 (all V_b_2 all V_a_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2091 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Oacyclic(T_b,V_r_2))) # label(fact_wf__acyclic) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2092 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2093 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c))) # label(fact_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2094 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_a)))))) # label(fact_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2095 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__ab__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> V_c = V_b))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2096 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2097 (all V_g_2 all V_f_2 ((all B_x hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)) -> V_g_2 = V_f_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2098 (all V_R_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_R_2) = c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),c_Relation_OId(T_b)))) # label(fact_rtrancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2099 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))) # label(fact_diff__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2100 (all V_P_2 all T_c all T_b c_Hilbert__Choice_OEps(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2)) = c_Hilbert__Choice_OEps(tc_prod(T_b,T_c),hAPP(hAPP(c_COMBS(tc_prod(T_b,T_c),T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_HOL_Obool),tc_prod(T_b,T_c)),V_P_2),c_Product__Type_Ofst(T_b,T_c))),c_Product__Type_Osnd(T_b,T_c)))) # label(fact_Eps__split) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2101 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> c_Groups_Oone__class_Oone(T_a) = c_Groups_Osgn__class_Osgn(T_a,V_a)))) # label(fact_sgn__pos) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2102 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)))))) # label(fact_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2103 (all V_r_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_List_Olists(T_b,c_Wellfounded_Oacc(T_b,V_r_2)))))) # label(fact_lists__accI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2104 (all V_a_2 all T_b c_HOL_OThe(T_b,hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2)) = V_a_2) # label(fact_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2105 (all V_b hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2106 (all V_xs all T_a hBOOL(hAPP(c_List_Odistinct(T_a),c_List_Oremdups(T_a,V_xs)))) # label(fact_distinct__remdups) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2107 (all V_x_2 all V_A_2 all V_L_2 all T_d all T_b all T_c hAPP(V_L_2,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_fun(T_d,T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),T_c),hAPP(c_COMBS(tc_fun(T_d,T_c),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_d,T_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_d,T_c))),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_fun(T_d,T_c))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),T_d,T_c),c_COMBI(tc_fun(T_d,T_c))),V_x_2)))))) = c_Partial__Function_Ofun__lub(T_c,T_b,T_d,V_L_2,V_A_2,V_x_2)) # label(fact_fun__lub__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2108 (all V_r_2 all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_r_2),c_Transitive__Closure_Ortrancl(T_b,V_r_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_r_2)) # label(fact_r__comp__rtrancl__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2109 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_b all T_c c_New__Random__Sequence_Opos__decr__bind(T_c,T_b,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__decr__bind(T_c,T_b,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b)))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_New__Random__Sequence_Opos__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2110 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hAPP(V_f_2,V_x_2) = V_y_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),V_y_2) = V_x_2)))) # label(fact_the__inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2111 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2))) # label(fact_Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2112 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (V_x != V_y -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2113 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c))))))) # label(fact_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2114 (all T_b c_Relation_OId(T_b) = hAPP(c_Relation_Oconverse(T_b,T_b),c_Relation_OId(T_b))) # label(fact_converse__Id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2115 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2116 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2117 (all V_k_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),V_k_2))))) # label(fact_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2118 (all V_xs_2 all V_P_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))) # label(fact_length__filter__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2119 (all V_m all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v))))) # label(fact_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2120 (all T_2 all T_1 (class_Groups_Ominus(T_1) -> class_Groups_Ominus(tc_fun(T_2,T_1)))) # label(arity_fun__Groups_Ominus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2121 (all V_ys_2 all V_P_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = c_List_OtakeWhile(T_b,V_P_2,V_xs_2)))) # label(fact_takeWhile__append1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2122 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2123 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Union__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2124 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (V_n_2 = V_m_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2125 (all V_b_2 all V_a_2 all V_f1_2 all T_b all T_d all T_c c_Product__Type_Oprod_Oprod__rec(T_c,T_d,T_b,V_f1_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2126 (all V_n all V_x (c_Nat__Transfer_Ois__nat(V_x) -> c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)))) # label(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2127 (all V_k_2 all V_n_2 all V_P_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hBOOL(hAPP(V_P_2,V_n_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) -> hBOOL(hAPP(V_P_2,B_j))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (all B_i all B_j (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)) -> hBOOL(hAPP(V_P_2,B_j))))) <-> hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))))) # label(fact_split__zmod) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2128 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2129 (all V_n_2 all V_m_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Orderings_Olinorder(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hAPP(V_f_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_m_2),V_n_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),hAPP(V_f_2,V_m_2)),hAPP(V_f_2,V_n_2))))) # label(fact_min__of__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2130 (all V_ys_2 all T_c all V_xs_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) -> hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_b),c_Product__Type_Ofst(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)) = V_xs_2)) # label(fact_map__fst__zip) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2131 (all V_a_2 all T_b (class_Rings_Oring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Int_Oring__1__class_OInts(T_b)))))) # label(fact_Ints__minus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2132 (all V_a_2 all T_b c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) # label(fact_hoare__derivs_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2133 (all V_n all V_m ((hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = V_n -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) != V_n -> hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)))) # label(fact_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2134 (all V_x_2 all T_c all T_b (class_Orderings_Otop(T_b) -> c_Orderings_Otop__class_Otop(T_b) = hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_c,T_b)),V_x_2))) # label(fact_top__apply) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2135 (all V_zs_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_zs_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) & c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_zs_2))))) # label(fact_sorted__many__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2136 (all V_z_2 all V_w_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z_2))),c_COMBI(tc_Int_Oint))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))))) # label(fact_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2137 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> V_x_2 = c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)))))) # label(fact_Sup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2138 (all V_a_2 all V_r_2 all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Wellfounded_Oacc(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Wellfounded_Oacc(T_b,V_r_2)))))) # label(fact_acc__downward) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2139 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)))) # label(fact_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2140 (all V_A_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))))) # label(fact_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2141 (all V_r_2 all V_x_2 all V_y_2 all T_b (c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_x_2)),V_r_2)) <-> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) & c_Wellfounded_Owf(T_b,V_r_2))) # label(fact_wf__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2142 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))))) # label(fact_div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2143 (all V_r_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Relation_OId(T_b)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_r_2))) # label(fact_rtrancl__unfold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2144 (all V_y all V_z all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2145 (all V_y_2 all V_x_2 (V_x_2 != V_y_2 | hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)))) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2146 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2147 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2)))) # label(fact_setprod_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2148 (all V_r_2 all T_b c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)) = c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2))) # label(fact_listrel__rtrancl__eq__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2149 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Groups_Oordered__ab__semigroup__add__imp__le(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_b_2)))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2150 (all V_n V_n = hAPP(c_Int_Onat,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) # label(fact_nat__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2151 (all V_t_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> V_t_2 = c_Inductive_Ocomplete__lattice__class_Olfp(T_b,hAPP(c_COMBK(T_b,T_b),V_t_2)))) # label(fact_lfp__const) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2152 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2153 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_u_2)) # label(fact_atLeastPlusOneLessThan__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2154 (all T_1 (class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_Option_Ooption(T_1)))) # label(arity_Option__Ooption__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2155 (all V_B_2 all V_a_2 all V_f_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2))) # label(fact_vimage__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2156 (all V_c all V_a all V_b all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b)))))))) # label(fact_divide__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2157 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(T_c,T_d),T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_b),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_b)),V_f_2)),V_B_2)),V_A_2))) # label(fact_setsum__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2158 (all V_f_2 all V_A_2 all V_B_2 all T_b all T_c (class_Groups_Oordered__ab__semigroup__add(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2)))))))) # label(fact_setsum__mono3) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2159 (all V_y all V_x all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)))) # label(fact_minus__min__eq__max) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2160 (all V_R_2 all T_b all V_y_2 all V_x_2 all V_f_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2))))) # label(fact_mlex__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2161 (all V_xs_2 all V_f_2 all T_c all V_P_2 all T_b hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,hAPP(c_Fun_Ocomp(T_b,tc_HOL_Obool,T_c,V_P_2),V_f_2)),V_xs_2))) # label(fact_length__filter__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2162 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_diff__card__le__card__Diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2163 (all T_b (class_Orderings_Olinorder(T_b) & class_Lattices_Osemilattice__inf(T_b) -> c_Lattices_Osemilattice__inf__class_Oinf(T_b) = c_Orderings_Oord__class_Omin(T_b))) # label(fact_inf__min) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2164 (all V_xs_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_A_2,B_x)))) <-> hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_xs_2)))) # label(fact_in__listsp__conv__set) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2165 (all V_x all T_a all V_xs ((c_List_Olist_ONil(T_a) = V_xs -> V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) & (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_last_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2166 (all V_n all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n))))) # label(fact_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2167 (all V_B_2 all V_A_2 all T_c all T_b hAPP(c_Relation_ODomain(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_c),V_A_2)),hAPP(c_Relation_ODomain(T_b,T_c),V_B_2))) # label(fact_Domain__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2168 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),V_k_2)) # label(fact_atLeast__Suc__greaterThan) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2169 (all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),T_b),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),V_B_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),T_c),c_Set_Oinsert(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)) # label(fact_Sigma__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2170 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_y)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2171 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z))),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)))))) # label(fact_min__max_Odistrib__inf__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2172 (all V_B_2 all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_insert__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2173 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2))))) # label(fact_Union__upper) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2174 (all V_c all V_a all V_b all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_c)))) # label(fact_min__max_Osup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2175 (all V_r_2 all V_yx_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),V_yx_2),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2))) -> -(all B_x all B_y (hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_y),B_x) = V_yx_2 -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),B_y)),V_r_2)))))) # label(fact_converseE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2176 (all V_b all V_a all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))))))) # label(fact_min__max_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2177 (all V_m_2 all V_n_2 all V_i_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_i_2),V_n_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_n_2),V_m_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_i_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_i_2),V_n_2))))) # label(fact_ivl__diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2178 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2179 (all V_y_2 all V_x_2 all T_b (class_Orderings_Opreorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2180 (all V_f_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) <-> (exists B_x (hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat) & (all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_A_2)) -> (B_xa != B_x -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))))))) # label(fact_setsum__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2181 (all V_v_2 all V_d_2 all V_c_2 all T_e all V_b_2 all V_a_2 all T_b all T_c all T_d (hAPP(c_Fun_Ocomp(T_e,T_c,T_b,V_c_2),V_d_2) = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_a_2),V_b_2) -> hAPP(V_c_2,hAPP(V_d_2,V_v_2)) = hAPP(V_a_2,hAPP(V_b_2,V_v_2)))) # label(fact_o__eq__dest) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2182 (all V_B_2 all V_A_2 (V_A_2 = V_B_2 <-> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) # label(fact_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2183 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2))))) # label(fact_Inf__fin__Inf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2184 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_b (class_Lattices_Olattice(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) = hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,T_b)),V_f_2),V_g_2),V_x_2))) # label(fact_inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2185 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_a_2)))))) # label(fact_min__max_Ofold1__belowI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2186 (all V_y_2 all V_x_2 all V_g_2 all V_f_2 all T_b all T_c all T_d c_Fun_Ofun__upd(T_b,T_c,hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_2),V_g_2),V_x_2,hAPP(V_f_2,V_y_2)) = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_2),c_Fun_Ofun__upd(T_b,T_d,V_g_2,V_x_2,V_y_2))) # label(fact_fun__upd__comp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2187 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__idem_Osubset__comp__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2188 (all V_v ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2189 (all V_l all V_k c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l))) # label(fact_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2190 (all V_k_2 all V_f_2 all V_n_2 all V_m_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),B_i)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,V_m_2)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2))) -> (exists B_i (V_k_2 = hAPP(V_f_2,B_i) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),B_i))))))))) # label(fact_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2191 (all V_Q_2 all V_P_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),V_P_2)),hAPP(c_Set_OCollect(T_b),V_Q_2)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),V_P_2)),V_Q_2))) # label(fact_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2192 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))))))) # label(fact_le__infI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2193 (all V_i_1 all T_a c_Lazy__Sequence_Oempty(T_a) = hAPP(c_New__DSequence_Opos__empty(T_a),V_i_1)) # label(fact_New__DSequence_Opos__empty__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2194 (all V_xs all T_a c_Enum_On__lists(T_a,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_xs) = hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a)))) # label(fact_n__lists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.83 2195 (all V_a_2 all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) -> V_b_2 = V_a_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2196 (all V_x_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)) <-> hBOOL(hAPP(c_Quotient_ORespects(T_b,V_R_2),V_x_2)))) # label(fact_Respects__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2197 (all V_l all V_k c_Int_OBit1(V_k) != c_Int_OBit0(V_l)) # label(fact_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2198 (all V_x_2 all V_a_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_x_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_a_2)),V_r_2))))) # label(fact_wf__not__sym) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2199 (all V_R_2 all T_b all V_y_2 all V_x_2 all V_f_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2)))))) # label(fact_mlex__leq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2200 (all T_1 (class_Enum_Oenum(T_1) -> class_Enum_Oenum(tc_Option_Ooption(T_1)))) # label(arity_Option__Ooption__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2201 (all V_x_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2))))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2202 (all V_b_2 all V_a_2 all V_f_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)) = hAPP(hAPP(V_f_2,V_a_2),V_b_2)) # label(fact_split__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2203 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Fields_Ofield(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hAPP(V_f_2,B_x) != c_Groups_Ozero__class_Ozero(T_c))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_setprod__Un) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2204 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = V_x)) # label(fact_sup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2205 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2206 (all V_y_2 all V_x_2 all V_r_2 all T_b (c_Predicate_Osymp(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_r_2,V_y_2),V_x_2))))) # label(fact_sympE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2207 (all V_n_2 all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Nat_OSuc,V_n_2))) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),c_Int_Onat)),V_n_2)))))) # label(fact_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2208 (all V_B_2 all V_C_2 all V_a_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2209 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_b all T_c c_New__DSequence_Opos__bind(T_c,T_b,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b)))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2)) = hAPP(hAPP(hAPP(c_New__Random__Sequence_Opos__bind(T_c,T_b,V_R_2,V_f_2),V_nrandom_2),V_size_2),V_seed_2)) # label(fact_pos__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2210 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_folding__image__simple__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2211 (all V_n all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> c_List_Olinorder__class_Osorted(T_a,hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))))) # label(fact_sorted__take) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2212 (all V_xs all T_a all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_n)) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),V_i))) # label(fact_nth__take) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2213 (all V_x_2 all V_r_2 all T_b (c_Wellfounded_OwfP(T_b,V_r_2) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_x_2)))) # label(fact_accp__wfPD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2214 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2))))))))) # label(fact_Max_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2215 (all V_n_2 all V_P_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> ((all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) -> hBOOL(hAPP(hAPP(V_P_2,B_i),B_j)))) <-> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2)))))) # label(fact_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2216 (all V_k_2 all V_n_2 all V_P_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_P_2,B_i)))))) <-> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_split__div) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2217 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z))))))) # label(fact_inf__greatest) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2218 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2219 (all V_n_2 all V_m_2 c_HOL_OThe(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat__rel(V_m_2,V_n_2)) = c_Divides_Odivmod__nat(V_m_2,V_n_2)) # label(fact_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2220 (all T_b (class_Nat_Osize(T_b) -> c_FunDef_Ois__measure(T_b,c_Nat_Osize__class_Osize(T_b)))) # label(fact_measure__size) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2221 (all V_B_2 all V_A_2 all T_c all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_b,T_c),tc_HOL_Obool)))) # label(fact_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2222 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)))) # label(fact_of__int__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2223 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> (c_Relation_Otrans(T_b,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)),V_r_2)),V_r_2))))) # label(fact_sym__trans__comp__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2224 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_subset__psubset__trans) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2225 (all V_k_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_single__Diff__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2226 (all V_B_2 all V_b_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_UN__I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2227 (all V_Y_2 all T_b hAPP(c_Set_Oimage(T_b,T_b,c_Fun_Oid(T_b)),V_Y_2) = V_Y_2) # label(fact_image__id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2228 (all V_ys_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Obutlast(T_b,V_xs_2)))) | hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Obutlast(T_b,V_ys_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Obutlast(T_b,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))))))) # label(fact_in__set__butlast__appendI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2229 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n))) # label(fact_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2230 (all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2231 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2232 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a))))) # label(fact_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2233 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2234 (all V_B_2 all V_A_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_A_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_c,hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2))))))) # label(fact_Image__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2235 (all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)))) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))))) # label(fact_finite__cartesian__productD1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2236 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2237 (all V_y_2 all V_x_2 (c_Nat__Transfer_Ois__nat(V_x_2) -> c_Nat__Transfer_Onat__set(c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_x_2,V_y_2)))) # label(fact_SetInterval_Otransfer__int__nat__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2238 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2239 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_b,tc_HOL_Obool),T_d),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),V_B_2)) # label(fact_INT__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2240 (all T_b all V_c_2 all V_P_2 ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) -> (V_c_2 = c_Hilbert__Choice_OEps(T_b,V_P_2) -> hBOOL(hAPP(V_P_2,V_c_2))))) # label(fact_exE__some) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2241 (all V_B_2 all V_a_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2242 (all V_t_2 all V_s_2 all V_b_2 all V_a_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_t_2))),c_FunDef_Opair__less)))) # label(fact_pair__lessI1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2243 (all V_d_2 all V_c_2 all T_e all V_b_2 all V_a_2 all T_b all T_c all T_d (hAPP(c_Fun_Ocomp(T_e,T_c,T_b,V_c_2),V_d_2) = hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_a_2),V_b_2) -> (all B_v hAPP(V_c_2,hAPP(V_d_2,B_v)) = hAPP(V_a_2,hAPP(V_b_2,B_v))))) # label(fact_o__eq__elim) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2244 (all V_n_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_b)),hAPP(c_Set_OCollect(tc_List_Olist(T_b)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),c_List_Oset(T_b))),V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),V_n_2))))))) # label(fact_finite__lists__length__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2245 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)))) # label(fact_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2246 (all V_P_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (-hBOOL(hAPP(V_P_2,V_x_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))))) # label(fact_length__filter__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2247 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Int_Osucc(V_k)) # label(fact_succ__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2248 (all V_a_2 all V_b_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))))) # label(fact_neg__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2249 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))))) # label(fact_le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2250 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_H),V_b_H),V_c))))) # label(fact_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2251 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_n = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2252 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)) <-> (exists B_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n)) = V_z_2))) # label(fact_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2253 (all V_m all V_n hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,V_n),V_m) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_minus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2254 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice__bot(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2255 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_a_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)))))) # label(fact_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2256 (all V_A_2 all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,V_A_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2)),V_A_2))))) # label(fact_lfp__lowerbound) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2257 (all V_f_2 all V_B_2 all T_c all V_A_2 all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_prod(T_b,T_c),T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_d),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_d),tc_fun(tc_fun(T_c,tc_HOL_Obool),T_d),T_b),c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d)),V_f_2)),V_B_2)),V_A_2))))) # label(fact_setprod__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2258 (all V_xs_2 all T_b (hAPP(c_List_Oset(T_b),V_xs_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_set__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2259 (all V_B_2 all V_A_2 all T_b c_List_Olistsp(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_b))),c_List_Olists(T_b,V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_b))),c_List_Olists(T_b,V_B_2)))) # label(fact_lists__Int__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2260 (all V_r_2 all T_b hAPP(hAPP(c_Relation_Oinv__image(tc_prod(tc_Nat_Onat,tc_List_Olist(T_b)),tc_List_Olist(T_b)),c_Wellfounded_Olex__prod(tc_Nat_Onat,tc_List_Olist(T_b),c_Wellfounded_Oless__than,c_List_Olex(T_b,V_r_2))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_prod(tc_Nat_Onat,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_List_Olist(T_b),tc_prod(tc_Nat_Onat,tc_List_Olist(T_b))),tc_List_Olist(T_b)),c_Product__Type_OPair(tc_Nat_Onat,tc_List_Olist(T_b))),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),c_COMBI(tc_List_Olist(T_b)))) = c_List_Olenlex(T_b,V_r_2)) # label(fact_lenlex__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2261 (all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_B_2) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Un__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2262 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2263 (all V_zs all V_y all V_x all T_a c_List_Oremove1(T_a,V_y,c_List_Oremove1(T_a,V_x,V_zs)) = c_List_Oremove1(T_a,V_x,c_List_Oremove1(T_a,V_y,V_zs))) # label(fact_remove1__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2264 (all V_k_2 all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_i ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)))) -> hBOOL(hAPP(V_P_2,B_i))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (all B_i ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)))) -> hBOOL(hAPP(V_P_2,B_i))))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2265 (all V_M_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_b)),V_M_2)) -> (exists B_n all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_x),V_M_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),B_x)),B_n)))))) # label(fact_finite__maxlen) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2266 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k))) # label(fact_number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2267 (all V_l all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_l)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_l))))) # label(fact_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2268 (all V_p_2 all V_r_2 all T_b (c_Wellfounded_OwfP(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),V_p_2),V_r_2)) -> c_Wellfounded_OwfP(T_b,V_p_2)))) # label(fact_wfP__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2269 (all V_v_2 all V_n_2 (hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2)),c_Int_Onat))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))))) # label(fact_less__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2270 (all V_lazy__sequence_2 all V_a_2 all V_f2_2 all V_f1_2 all T_c all T_b hAPP(hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_b,T_c,V_f1_2,V_f2_2,V_lazy__sequence_2)) = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_b,T_c,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2271 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__geq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2272 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) -> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_inf1D1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2273 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2274 (all V_f_2 all T_c all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_insert__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2275 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_setsum__shift__bounds__cl__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2276 (all V_xs_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(c_List_Onth(T_b),V_xs_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),V_xs_2))) # label(fact_listsum__setsum__nth) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2277 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_b (class_Groups_Oab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_p_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_p_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2),V_p_2)))))) # label(fact_setsum__diff__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2278 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = c_List_Olast(T_a,V_xs))) # label(fact_last__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2279 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2))) & hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2))))) # label(fact_mem__Sigma__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2280 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k_2),V_l_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2281 (all V_I_2 all V_xs_2 all T_b hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_I_2)))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Osublist(T_b,V_xs_2,V_I_2))) # label(fact_length__sublist) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2282 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),V_f_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_f_2)),V_B_2),V_A_2)))) # label(fact_sup__SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2283 (all V_r_2 all T_b (c_Predicate_Oreflp(T_b,V_r_2) <-> c_Relation_Orefl__on(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))))) # label(fact_reflp__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2284 (all V_n_2 all T_b (class_Rings_Osemiring__1(T_b) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_b,hAPP(hAPP(c_COMBC(T_b,T_b,T_b),c_Groups_Oplus__class_Oplus(T_b)),c_Groups_Oone__class_Oone(T_b)),V_n_2,c_Groups_Ozero__class_Ozero(T_b)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2))) # label(fact_semiring__1__class_Oof__nat__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2285 (all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_a_2)) # label(fact_Sup__singleton) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2286 (all V_lazy__sequence_2 all V_a_2 all V_fa_2 all T_b hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_b,V_fa_2),c_Lazy__Sequence_Olazy__sequence_OInsert(T_b,V_a_2,V_lazy__sequence_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_fa_2,V_a_2)),hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_b,V_fa_2),V_lazy__sequence_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_lazy__sequence_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2287 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) <-> V_xs_2 = c_List_Oremdups(T_b,V_xs_2))) # label(fact_remdups__id__iff__distinct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2288 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2289 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_b,tc_HOL_Obool),T_c),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_B_2)) # label(fact_INT__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2290 (all V_m all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(c_Nat_OSuc,V_j)))) # label(fact_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2291 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,hAPP(c_Fun_Ocomp(tc_prod(T_c,T_d),T_c,T_b,c_Product__Type_Ofst(T_c,T_d)),V_f_2)),V_A_2)),hAPP(c_Set_Ovimage(T_b,T_d,hAPP(c_Fun_Ocomp(tc_prod(T_c,T_d),T_d,T_b,c_Product__Type_Osnd(T_c,T_d)),V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_b,tc_prod(T_c,T_d),V_f_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2)))) # label(fact_vimage__Times) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2292 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2293 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2)))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2294 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2)))))) # label(fact_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2295 (all V_B_2 all V_A_2 all T_b all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_INTER__eq__Inter__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2296 (all V_size_2 all V_nrandom_2 all V_x_2 all T_b hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_DSequence_Osingle(T_b),V_x_2)) = hAPP(hAPP(hAPP(c_Random__Sequence_Osingle(T_b),V_x_2),V_nrandom_2),V_size_2)) # label(fact_Random__Sequence_Osingle__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2297 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> V_n != V_m)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2298 (all V_n_2 all V_m_2 all V_a_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2)) -> (V_m_2 = V_n_2 <-> hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_m_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2))))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2299 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_divide__Numeral0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2300 (all V_b all V_a all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)))))) # label(fact_min__max_Ole__supI1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2301 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))))) # label(fact_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2302 (all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_ab__left__minus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2303 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OMin) = c_Int_Opred(V_k)) # label(fact_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2304 (all V_z_2 c_Set_Othe__elem(tc_Nat_Onat,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Int_ORep__Integ,V_z_2)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_Set_Oinsert(tc_Nat_Onat))),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)))),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))) = hAPP(c_Int_Onat,V_z_2)) # label(fact_nat__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.84 2305 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Divides_Osemiring__div(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_a_2 -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_a_2),V_b_2)) -> (V_c_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_b),V_b_2),V_a_2) <-> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)))))) # label(fact_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2306 (all V_x_2 all V_y_2 all V_c_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_c_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_c_2),V_y_2)) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_c_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_y_2),V_c_2))))) # label(fact_image__minus__const__atLeastLessThan__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2307 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),c_List_Olistsp(T_b,V_A_2)),c_List_Olistsp(T_b,V_B_2)) = c_List_Olistsp(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_listsp__inf__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2308 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2309 (all V_f_2 all T_b c_Wellfounded_Owf(T_b,hAPP(c_Wellfounded_Omeasure(T_b),V_f_2))) # label(fact_wf__measure) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2310 (all V_b all V_a all T_a (class_Rings_Olinordered__ring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2311 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Otake(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))) # label(fact_take__1__Cons) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2312 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> c_Fun_Oinj__on(T_c,T_b,c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_inj__on__the__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2313 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2314 (all V_s_2 all V_r_2 all T_d all T_b all T_c c_Predicate_Oconversep(T_c,T_b,c_Predicate_Opred__comp(T_c,T_d,T_b,V_r_2,V_s_2)) = c_Predicate_Opred__comp(T_b,T_d,T_c,c_Predicate_Oconversep(T_d,T_b,V_s_2),c_Predicate_Oconversep(T_c,T_d,V_r_2))) # label(fact_converse__pred__comp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2315 (all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_b)),c_List_Otranspose__rel(T_b)),c_List_Olist_ONil(tc_List_Olist(T_b)))) -> c_List_Olist_ONil(tc_List_Olist(T_b)) = c_List_Otranspose(T_b,c_List_Olist_ONil(tc_List_Olist(T_b))))) # label(fact_transpose_Opsimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2316 (all V_q all V_p all T_b all T_a (hAPP(c_Product__Type_Ofst(T_a,T_b),V_q) = hAPP(c_Product__Type_Ofst(T_a,T_b),V_p) -> (hAPP(c_Product__Type_Osnd(T_a,T_b),V_q) = hAPP(c_Product__Type_Osnd(T_a,T_b),V_p) -> V_p = V_q))) # label(fact_prod__eqI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2317 (all V_z (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> V_z = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)))) # label(fact_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2318 (all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_b),B_x),hAPP(hAPP(c_Set_Oinsert(T_b),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),V_A_2))))))) # label(fact_Inf__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2319 (all V_l c_Int_OMin != c_Int_OBit0(V_l)) # label(fact_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2320 (all V_y all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2321 (all V_c_2 all V_a_2 all V_b_2 all T_b (class_Groups_Ocancel__semigroup__add(T_b) -> (V_b_2 = V_c_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_a_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2322 (all V_R_2 all V_n_2 all V_m_2 all T_b hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_m_2),V_R_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) # label(fact_rel__pow__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2323 (all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_of__int__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2324 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_a_2),V_A_2)))) # label(fact_Min_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2325 (all V_B_2 all T_c all V_d_2 all V_A_2 all V_c_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_d_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_c,T_b,hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),T_c),c_If(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_c_2)),V_d_2)),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_d_2),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Ovimage(T_c,T_b,hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),T_c),c_If(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_c_2)),V_d_2)),V_A_2))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_d_2),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Ovimage(T_c,T_b,hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),T_c),c_If(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_c_2)),V_d_2)),V_A_2)) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_d_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_c,T_b,hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),T_c),c_If(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_c_2)),V_d_2)),V_A_2) = V_B_2)))) # label(fact_vimage__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2326 (all V_x all V_xs all T_a all V_j all V_i (V_i != V_j -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j))) # label(fact_nth__list__update__neq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2327 (all V_A_2 all T_b c_Fun_Oinj__on(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Orev(T_b),V_A_2)) # label(fact_inj__on__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2328 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z))))))) # label(fact_min__max_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2329 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)))))) # label(fact_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2330 (all V_B_2 all V_A_2 all T_b (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2331 (all V_r_2 all V_s_2 all T_b (c_Wellfounded_Oacyclic(T_b,V_s_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> c_Wellfounded_Oacyclic(T_b,V_r_2)))) # label(fact_acyclic__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2332 (all V_R_2 all V_Y_2 all V_X_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_X_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_Y_2)) -> (V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_X_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)) & hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_Y_2)))))) -> c_Wellfounded_Omax__extp(T_b,V_R_2,V_X_2,V_Y_2)))))) # label(fact_max__extp_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2333 (all V_a1_2 all V_a2_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_c,V_r_2),V_a2_2),V_a1_2)) <-> hBOOL(hAPP(hAPP(V_r_2,V_a1_2),V_a2_2)))) # label(fact_conversep_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2334 (all V_n all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n)))) # label(fact_add__leD2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2335 (all V_R_2 all T_b (c_Predicate_Oreflp(T_b,V_R_2) & c_Predicate_Osymp(T_b,V_R_2) & c_Predicate_Otransp(T_b,V_R_2) <-> c_Equiv__Relations_Oequivp(T_b,V_R_2))) # label(fact_equivp__reflp__symp__transp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2336 (all V_xs_2 all T_b hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_b)))),V_xs_2)) = c_List_Ofoldr(tc_List_Olist(T_b),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_b)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_transpose__max__length) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2337 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),V_d))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2338 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n) # label(fact_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2339 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)))) # label(fact_SigmaD1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2340 (all V_n_2 all T_b (class_Rings_Ocomm__semiring__1(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),c_Groups_Oone__class_Oone(T_b))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),c_Nat_Osemiring__1__class_Oof__nat(T_b)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),V_n_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),c_Groups_Oone__class_Oone(T_b))))) # label(fact_gauss__sum) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2341 (all V_A_2 all V_g_2 all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Otimes__class_Otimes(T_b)),V_f_2)),V_g_2)),V_A_2))) # label(fact_setprod__timesf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2342 (all V_n_2 all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2),hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_ys_2)))) # label(fact_list__all2__takeI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2343 (all V_B_2 all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))))) # label(fact_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2344 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2345 (all V_g_2 all V_f_2 all V_A_2 all T_b all T_c (class_Fields_Ofield__inverse__zero(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Rings_Oinverse__class_Odivide(T_c)),V_f_2)),V_g_2)),V_A_2)))) # label(fact_setprod__dividef) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2346 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Nat__Transfer_Otsub(V_x,V_y)))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2347 (all V_A_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_A_2)) # label(fact_Compl__disjoint2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2348 (all V_b_2 all V_f_2 all T_c all V_a_2 all T_b (V_b_2 = hAPP(V_f_2,V_a_2) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))) # label(fact_vimage__singleton__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2349 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2350 (all V_X_2 all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2))))))) # label(fact_coinduct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2351 (all V_A_2 all V_C_2 all V_X_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),V_X_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_C_2)))))) # label(fact_UnionI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2352 (all V_ys_2 all V_g_2 all T_e all V_xs_2 all V_f_2 all T_d all T_c all T_b hAPP(hAPP(c_List_Ozip(T_b,T_c),hAPP(hAPP(c_List_Omap(T_d,T_b),V_f_2),V_xs_2)),hAPP(hAPP(c_List_Omap(T_e,T_c),V_g_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_e),tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,T_c),tc_fun(T_e,tc_prod(T_b,T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(tc_fun(T_e,T_c),tc_fun(T_e,tc_prod(T_b,T_c))),T_d),c_COMBB(T_c,tc_prod(T_b,T_c),T_e)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_prod(T_b,T_c)),T_d),c_Product__Type_OPair(T_b,T_c)),V_f_2))),V_g_2))),hAPP(hAPP(c_List_Ozip(T_d,T_e),V_xs_2),V_ys_2))) # label(fact_zip__map__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2353 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_sum(T_2,T_1)))) # label(arity_sum__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2354 (all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_Rep_2,V_x_2)),V_A_2)))) # label(fact_type__definition_ORep) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2355 (all V_j_2 all V_i_2 hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2))) # label(fact_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2356 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding_Ocommute__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2357 (all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),V_m)),V_k) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))))) # label(fact_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2358 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_ys)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_drop__append) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2359 (all V_k_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_k_2)))) # label(fact_finite__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2360 (all V_A_2 all V_a_2 all T_b hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_Oimage(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oinsert(T_b),V_a_2)),hAPP(c_Set_OPow(T_b),V_A_2)))) # label(fact_Pow__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2361 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),hAPP(c_Nat_OSuc,V_u_2))) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2362 (all T_b -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_b)),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) # label(fact_infinite__UNIV__listI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2363 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_atLeastAtMostSuc__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2364 (all V_N_2 all V_h_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x all B_y hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_N_2))))))) # label(fact_min__max_OSup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2365 (all V_n all V_m hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) # label(fact_zmult__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2366 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_x)) | V_x = V_y | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),V_y)))) # label(fact_zless__linear) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2367 (all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (V_b_2 = V_a_2 -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)))) # label(fact_atLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2368 (all V_y_2 all V_x_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,c_List_Olist_ONil(tc_fun(T_b,tc_Nat_Onat)))))) # label(fact_in__measures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2369 (all V_rb_2 all T_c all V_ra_2 all T_b (c_Wellfounded_Owf(T_b,V_ra_2) -> (c_Wellfounded_Owf(T_c,V_rb_2) -> c_Wellfounded_Owf(tc_prod(T_b,T_c),c_Wellfounded_Olex__prod(T_b,T_c,V_ra_2,V_rb_2))))) # label(fact_wf__lex__prod) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2370 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mod__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2371 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit1(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2372 (all V_A_2 all T_b all V_y_2 all V_x_2 (V_x_2 = V_y_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2373 (all T_d all T_c all T_b all V_c_2 all V_s_2 all V_b_2 all V_a_2 all V_r_2 (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(V_s_2,V_b_2),V_c_2)) -> hBOOL(hAPP(hAPP(c_Predicate_Opred__comp(T_b,T_c,T_d,V_r_2,V_s_2),V_a_2),V_c_2))))) # label(fact_pred__comp_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2374 (all V_f_2 all T_c all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> c_Wellfounded_Owf(T_c,hAPP(c_Set_Oimage(tc_prod(T_b,T_b),tc_prod(T_c,T_c),c_Product__Type_Omap__pair(T_b,T_c,T_b,T_c,V_f_2,V_f_2)),V_r_2))))) # label(fact_wf__map__pair__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2375 (all V_xq_2 all V_x_2 all V_f_2 all T_b all T_c c_Lazy__Sequence_Omap(T_c,T_b,V_f_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_x_2,V_xq_2)) = c_Lazy__Sequence_Olazy__sequence_OInsert(T_b,hAPP(V_f_2,V_x_2),c_Lazy__Sequence_Omap(T_c,T_b,V_f_2,V_xq_2))) # label(fact__06) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2376 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2377 (all V_y_2 all V_x_2 (hAPP(c_Int_ORep__Integ,V_x_2) = hAPP(c_Int_ORep__Integ,V_y_2) <-> V_y_2 = V_x_2)) # label(fact_Rep__Integ__inject) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2378 (all V_c all V_b all V_v all T_b (class_Rings_Oring(T_b) & class_Int_Onumber(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_c)))) # label(fact_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2379 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))))) # label(fact_odd__0__le__power__imp__0__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2380 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (c_Fun_Obij__betw(T_d,T_b,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Fun_Ocomp(T_b,T_d,T_c,c_Hilbert__Choice_Oinv__into(T_d,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_g_2)),c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)) = c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Fun_Ocomp(T_b,T_c,T_d,V_f_2),V_g_2))))) # label(fact_o__inv__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2381 (all V_n_2 all V_v_2 hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Int_Oint),c_Orderings_Oord__class_Omin(tc_Nat_Onat)),c_Int_Onat)),V_n_2))))) # label(fact_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2382 (all V_s_2 all V_r_2 all T_b all T_c hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)),hAPP(c_Relation_Oconverse(T_c,T_b),V_s_2))) # label(fact_converse__Int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2383 (all V_b_2 all V_a_2 all V_A_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_b_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_a_2),V_b_2))))))) # label(fact_fold__inf__le__inf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2384 (all V_w_2 ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(hAPP(c_HOL_OLet(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_COMBI(tc_Nat_Onat)))) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2385 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),c_Predicate_OPowp(T_b,V_A_2)),c_Predicate_OPowp(T_b,V_B_2))))) # label(fact_Powp__mono) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2386 (all V_d_2 hAPP(c_Set_OCollect(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_d_2)))),c_Orderings_Oord__class_Oless(tc_Int_Oint)))) = c_Int_Oint__ge__less__than2(V_d_2)) # label(fact_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2387 (all V_b_2 all V_a_2 all T_b (class_Rings_Oring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))))))) # label(fact_Ints__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2388 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n))) # label(fact_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2389 (all V_A_2 all T_b (class_Finite__Set_Ofinite(T_b) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_finite__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2390 (all V_A_2 all V_f_2 all T_c all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Oabs__class_Oabs(T_b)),V_f_2)),V_A_2))))) # label(fact_setsum__abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2391 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)))) # label(fact_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2392 (all V_n all V_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),V_n)))) # label(fact_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2393 (all V_r_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_ORange(T_c,T_b),V_r_2))) -> -(all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),V_b_2)),V_r_2))))) # label(fact_RangeE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2394 (all V_A_2 all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(c_COMBK(T_b,T_c),c_Groups_Ozero__class_Ozero(T_b))),V_A_2))) # label(fact_setsum__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2395 (all V_d_2 all V_m_2 ((exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_d_2),B_q)) <-> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2396 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)) = hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_j),c_List_Olist_ONil(tc_Nat_Onat))))) # label(fact_upt__Suc__append) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2397 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_transfer__int__nat__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2398 (all T_b all V_A_2 all T_c all V_x_2 all V_f_2 all V_b_2 (hAPP(V_f_2,V_x_2) = V_b_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)))))) # label(fact_image__eqI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2399 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2))))) # label(fact_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2400 (all V_x_2 all V_A_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2))))))) # label(fact_fold1__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2401 (all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_q = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2402 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))))) # label(fact_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2403 (all V_z_2 all V_y_2 all V_x_2 all V_f_2 all T_c all T_b c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_x_2,V_z_2) = c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_z_2)) # label(fact_fun__upd__upd) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2404 (all V_R_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_R_2) = c_Transitive__Closure_Ortrancl(T_b,c_Transitive__Closure_Otrancl(T_b,V_R_2))) # label(fact_trancl__rtrancl__absorb) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2405 (all T_c all V_f_2 all V_r_2 all T_b ((all B_y all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_z)),V_r_2)) -> hAPP(V_f_2,B_z) = hAPP(V_f_2,B_y))) -> c_Equiv__Relations_Ocongruent(T_b,T_c,V_r_2,V_f_2))) # label(fact_congruentI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2406 (all V_xs_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Ofoldl(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Orderings_Otop__class_Otop(T_b)),V_xs_2))) # label(fact_Inf__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2407 (all V_m all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_j)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))),V_m))) # label(fact_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2408 (all V_n all V_m ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)))) # label(fact_mod__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2409 (all V_xss_2 all T_b hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xss_2)),c_List_Oset(T_b))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oremdups(T_b,c_List_Oconcat(T_b,V_xss_2)))) # label(fact_length__remdups__concat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2410 (all V_f_2 all V_r_2 all T_c all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))),V_r_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Oinv__image(T_c,T_b),V_r_2),V_f_2))))) # label(fact_in__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2411 (all V_R2_2 all V_R1_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R1_2),V_R2_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Wellfounded_Oacc(T_b,V_R2_2)),c_Wellfounded_Oacc(T_b,V_R1_2))))) # label(fact_acc__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2412 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_B_2)))))) # label(fact_setsum__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2413 (all V_y all V_x (c_Nat__Transfer_Ois__nat(V_x) -> (c_Nat__Transfer_Ois__nat(V_y) -> c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y))))) # label(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2414 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(tc_List_Olist(T_b)),V_xs_2)) -> ((all B_ys (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_ys),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(c_List_Odistinct(T_b),B_ys)))) -> ((all B_ys (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_ys),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xs_2))) -> (all B_zs (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_zs),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xs_2))) -> (B_zs != B_ys -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),B_ys)),hAPP(c_List_Oset(T_b),B_zs))))))) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_Oconcat(T_b,V_xs_2))))))) # label(fact_distinct__concat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2415 (all V_x_2 all T_c all T_b (class_Orderings_Obot(T_b) -> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,T_b)),V_x_2) = c_Orderings_Obot__class_Obot(T_b))) # label(fact_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2416 (all T_c all V_f_2 all T_b (c_FunDef_Ois__measure(T_b,V_f_2) -> c_FunDef_Ois__measure(tc_prod(T_b,T_c),hAPP(hAPP(c_COMBB(T_b,tc_Nat_Onat,tc_prod(T_b,T_c)),V_f_2),c_Product__Type_Ofst(T_b,T_c))))) # label(fact_measure__fst) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2417 (all V_xs_2 all V_x_2 all T_b c_Random_Oselect__weight(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b)),hAPP(hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_b),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),V_x_2)),V_xs_2)) = c_Random_Oselect__weight(T_b,V_xs_2)) # label(fact_select__weight__cons__zero) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2418 (all V_n_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_setsum__shift__bounds__cl__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2419 (all V_A_2 all T_b (class_Finite__Set_Ofinite(T_b) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_finite) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2420 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2421 (all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = V_a) # label(fact_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2422 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))))))) # label(fact_pos__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.85 2423 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2)))) # label(fact_ivl__disj__un_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2424 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)))))) # label(fact_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2425 (all V_r_2 all T_b (c_Wellfounded_Oacyclic(T_b,V_r_2) -> c_Relation_Oantisym(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) # label(fact_acyclic__impl__antisym__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2426 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_y)))) # label(fact_inf__le2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2427 (all V_y_2 all V_z_2 all V_f_2 all T_c all V_A_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),hAPP(hAPP(V_f_2,V_x_2),V_y_2)))))) # label(fact_fold__graph_OinsertI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2428 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n) = c_Groups_Oone__class_Oone(tc_Int_Oint) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = V_m)) # label(fact_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2429 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_b),c_List_Olenlex(T_b,V_r_2)))) # label(fact_wf__lenlex) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2430 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olistrel1(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olistrel1(T_b,V_r_2)))))) # label(fact_in__listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2431 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_List_Orev(T_a),hAPP(c_List_Orotate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)))),V_xs))) # label(fact_rotate__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2432 (all V_y_2 all V_x_2 all V_Q_2 all V_P_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_P_2),V_Q_2)) -> (hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2))))) # label(fact_predicate2D) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2433 (all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))))) # label(fact_sup__ge2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2434 (all V_k_2 all V_i_2 all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_k_2),V_i_2))))) # label(fact_atLeast__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2435 (all V_v_H all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2436 (all V_f_2 all V_x_2 all T_b all T_c all V_i_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_i_2 -> c_Lazy__Sequence_Ohit__bound(T_b) = hAPP(c_New__DSequence_Oneg__decr__bind(T_c,T_b,V_x_2,V_f_2),V_i_2)) & (V_i_2 != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(c_New__DSequence_Oneg__decr__bind(T_c,T_b,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Ohb__bind(T_c,T_b,hAPP(V_x_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b))),V_f_2),V_i_2))))) # label(fact_New__DSequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2437 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_i_2,V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) # label(fact_set__upt) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2438 (all V_A_2 all T_b V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2439 (all V_xs_2 all V_x_2 all V_n_2 all T_b hAPP(hAPP(c_List_Otake(T_b),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = c_Nat_Onat_Onat__case(tc_List_Olist(T_b),c_List_Olist_ONil(T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_b),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Otake(T_b)),V_xs_2)),V_n_2)) # label(fact_take__Cons) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2440 (all V_tsa_2 all V_ts_H_2 all V_G_2 all T_b (c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_ts_H_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)),V_tsa_2),V_ts_H_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_tsa_2)))) # label(fact_weaken) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2441 (all V_xs all V_i all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__drop) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2442 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_a all B_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b))) -> ((-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_a),B_b)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B_b)))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2443 (all V_g_2 all V_f_2 all T_c all T_b c_Finite__Set_Ofold(T_c,T_b,hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),V_f_2),V_g_2)) = hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2)) # label(fact_fold__image__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2444 (all V_A_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__mult(T_b) & class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),V_A_2))) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2445 (all V_A_2 all T_b ((exists B_n exists B_f V_A_2 = hAPP(c_Set_Oimage(tc_Nat_Onat,T_b,B_f),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),B_n)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_finite__conv__nat__seg__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2446 (all V_P_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))))) # label(fact_distinct__takeWhile) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2447 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))) # label(fact_card__insert) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2448 (all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) <-> c_List_Olist_ONil(T_b) != V_xs_2)) # label(fact_length__greater__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2449 (all V_x_2 all V_f_2 all T_b all T_c all V_k_2 ((V_k_2 = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(hAPP(c_Lazy__Sequence_Oanamorph(T_c,T_b,V_f_2),V_k_2),V_x_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),T_c),c_List_Olist_ONil(T_b)),V_x_2)) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> c_Option_Ooption_Ooption__case(tc_prod(tc_List_Olist(T_b),T_c),tc_prod(T_b,T_c),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),T_c),c_List_Olist_ONil(T_b)),V_x_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_prod(tc_List_Olist(T_b),T_c)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c)),tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_List_Olist(T_b),T_c),tc_prod(tc_List_Olist(T_b),T_c)),tc_fun(tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c)),tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c))),T_b),c_COMBB(tc_prod(tc_List_Olist(T_b),T_c),tc_prod(tc_List_Olist(T_b),T_c),T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c))),tc_fun(tc_prod(tc_List_Olist(T_b),T_c),tc_prod(tc_List_Olist(T_b),T_c)),T_b),c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),T_c,tc_prod(tc_List_Olist(T_b),T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c))),T_b),hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(T_c,tc_prod(tc_List_Olist(T_b),T_c)),tc_List_Olist(T_b)),c_Product__Type_OPair(tc_List_Olist(T_b),T_c))),c_List_Olist_OCons(T_b))))),hAPP(c_Lazy__Sequence_Oanamorph(T_c,T_b,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_k_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))))),hAPP(V_f_2,V_x_2)) = hAPP(hAPP(c_Lazy__Sequence_Oanamorph(T_c,T_b,V_f_2),V_k_2),V_x_2)))) # label(fact_anamorph_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2450 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2451 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c))) # label(fact_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2452 (all V_r_2 all T_b ((all B_x hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Wellfounded_Oacc(T_b,V_r_2)))) <-> c_Wellfounded_Owf(T_b,V_r_2))) # label(fact_wf__acc__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2453 (all V_I_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Osublist(T_b,V_xs_2,V_I_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_in__set__sublistD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2454 (all V_xs_2 all T_b hAPP(hAPP(c_List_Omap(T_b,tc_prod(T_b,T_b)),hAPP(hAPP(c_COMBS(T_b,T_b,tc_prod(T_b,T_b)),c_Product__Type_OPair(T_b,T_b)),c_COMBI(T_b))),V_xs_2) = hAPP(hAPP(c_List_Ozip(T_b,T_b),V_xs_2),V_xs_2)) # label(fact_zip__same__conv__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2455 (all V_B_2 all V_A_2 all V_c_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),V_c_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),hAPP(V_B_2,B_x))) -> V_c_2 != hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_x),B_y))))))) # label(fact_SigmaE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2456 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)))) # label(fact_div__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2457 (all V_g_2 all V_f_2 all T_c all T_b all T_d all T_e hAPP(c_Fun_Ocomp(T_c,T_d,tc_prod(T_b,T_c),V_g_2),c_Product__Type_Osnd(T_b,T_c)) = hAPP(c_Fun_Ocomp(tc_prod(T_e,T_d),T_d,tc_prod(T_b,T_c),c_Product__Type_Osnd(T_e,T_d)),c_Product__Type_Omap__pair(T_b,T_e,T_c,T_d,V_f_2,V_g_2))) # label(fact_snd__comp__map__pair) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2458 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_w)))) # label(fact_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2459 (all V_P_2 all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Quotient_ORespects(T_b,V_R_2)))))))) # label(fact_bex__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2460 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2461 (all V_f_2 all V_x_2 all T_c all T_d all T_b hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_d,T_b,T_c),hAPP(c_Product__Type_OPair(T_d,T_b),V_x_2)),V_f_2) = hAPP(V_f_2,V_x_2)) # label(fact_Pair__scomp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2462 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__mult__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2463 (all V_xs_2 all T_b all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),c_List_Otranspose(T_b,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),c_List_Otranspose(T_b,V_xs_2)),V_i_2) = hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,T_b),c_List_Onth(T_b)),V_i_2)),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),V_xs_2)))) # label(fact_nth__transpose) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2464 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2465 (all V_R_2 all T_b (c_Equiv__Relations_Opart__equivp(T_b,V_R_2) -> -((exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) -> (c_Predicate_Osymp(T_b,V_R_2) -> -c_Predicate_Otransp(T_b,V_R_2))))) # label(fact_part__equivpE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2466 (all V_xs_2 all V_P_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> -hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))))))) # label(fact_nth__length__takeWhile) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2467 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)),V_z_2))))) # label(fact_max__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2468 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2469 (all T_c all T_b all V_t_2 all V_s_2 (V_s_2 = V_t_2 <-> hAPP(c_Product__Type_Osnd(T_b,T_c),V_s_2) = hAPP(c_Product__Type_Osnd(T_b,T_c),V_t_2) & hAPP(c_Product__Type_Ofst(T_b,T_c),V_t_2) = hAPP(c_Product__Type_Ofst(T_b,T_c),V_s_2))) # label(fact_Pair__fst__snd__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2470 (all V_x all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(c_List_Otl(T_a),c_List_Oreplicate(T_a,V_n,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_x))) # label(fact_tl__replicate) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2471 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z))) # label(fact_inf__sup__aci_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2472 (all V_b_2 all V_a_2 all V_A_2 all T_b (class_Lattices_Osemilattice__sup(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),V_A_2))))))) # label(fact_sup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2473 (all V_ys_2 all V_vs_2 all V_us_2 all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> (hAPP(hAPP(c_List_Oappend(T_b),V_us_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_vs_2)) = V_xs_2 -> (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_b),V_us_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_vs_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))))))) # label(fact_listrel1I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2474 (all V_l c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_l))) # label(fact_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2475 (all V_xss all T_a c_List_Otranspose(T_a,V_xss) = c_List_Otranspose(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),V_xss))) # label(fact_transpose_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2476 (all V_y_2 all V_x_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)) | V_x_2 = V_y_2))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2477 (all T_d all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_b,tc_fun(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,T_c),T_b),c_COMBK(T_c,T_d)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_inj__fun) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2478 (all V_a_2 all V_B_2 all V_A_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,V_a_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2))))) # label(fact_INT__E) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2479 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2480 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Compl__empty__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2481 (all V_l all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l))),V_l) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit1(V_k)),V_l)) # label(fact_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2482 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_inj__image__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2483 (all T_a hBOOL(hAPP(c_List_Odistinct(T_a),c_List_Olist_ONil(T_a)))) # label(fact_distinct_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2484 (all V_n_2 all V_m_2 (V_m_2 = V_n_2 <-> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))) # label(fact_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2485 (all V_u_2 hAPP(c_Int_Onat,V_u_2) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))) # label(fact_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2486 (all V_n all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_m),V_n))))))) # label(fact_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2487 (all V_A_2 all V_f_2 all T_b all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2488 (all V_d_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_d_2)))) # label(fact_Int__atLeastAtMostL1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2489 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_b all T_c c_New__DSequence_Oneg__decr__bind(T_c,T_b,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b)))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b))))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b)))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2)) = c_New__Random__Sequence_Oneg__decr__bind(T_c,T_b,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_New__Random__Sequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2490 (all V_v_2 all V_n_2 (hAPP(c_Nat_OSuc,V_n_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_fequal),c_Int_Onat)),V_n_2))))))) # label(fact_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2491 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2492 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_image__set__diff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2493 (all V_n_2 all V_x_2 all T_b (class_Fields_Ofield(T_b) -> (V_x_2 != c_Groups_Oone__class_Oone(T_b) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),V_n_2)),c_Groups_Oone__class_Oone(T_b))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_x_2),c_Groups_Oone__class_Oone(T_b))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(c_Power_Opower__class_Opower(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))))) # label(fact_geometric__sum) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2494 (all V_z all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)))) # label(fact_of__int__minus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2495 (all V_h_2 all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hAPP(V_h_2,B_x) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(V_F_2,V_h_2),V_A_2) = hAPP(hAPP(V_F_2,V_g_2),V_A_2))))) # label(fact_folding__image_Ocong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2496 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2)))) # label(fact_subset__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2497 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> (exists B_k hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k)) = V_n_2))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2498 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n)),hAPP(c_Code__Numeral_Oof__nat,V_m))) # label(fact_plus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2499 (all T_1 all T_2 class_Nat_Osize(tc_Nitpick_Opair__box(T_2,T_1))) # label(arity_Nitpick__Opair____box__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2500 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_A_2),V_C_2))),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_C_2)))) # label(fact_Un__Union__image) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2501 (all V_xs_2 all V_f_2 all T_c all V_n_2 all T_b hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2)) = hAPP(c_List_Orotate(T_b,V_n_2),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_rotate__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2502 (all V_list_H all V_a_H all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H) != c_List_Olist_ONil(T_a)) # label(fact_list_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2503 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_List_Olinorder__class_Osorted__list__of__set(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = c_List_Oremove1(T_b,V_x_2,c_List_Olinorder__class_Osorted__list__of__set(T_b,V_A_2))))) # label(fact_sorted__list__of__set__remove) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2504 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))))) # label(fact_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2505 (all V_n all V_m hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n))) # label(fact_int__power) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2506 (all V_a_2 all V_A_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b)),V_a_2),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))))) # label(fact_fold1__eq__fold) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2507 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2508 (all V_n_2 all V_d_2 all V_a_2 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),V_d_2))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),V_d_2))))) # label(fact_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2509 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2510 (all V_n all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) = hAPP(c_List_Ohd(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs)))) # label(fact_hd__rotate__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2511 (all V_n_2 all V_r_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_prod(tc_prod(T_b,tc_List_Olist(T_b)),tc_prod(T_b,tc_List_Olist(T_b))),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_Product__Type_Omap__pair(tc_prod(T_b,tc_List_Olist(T_b)),tc_List_Olist(T_b),tc_prod(T_b,tc_List_Olist(T_b)),tc_List_Olist(T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)))),c_Wellfounded_Olex__prod(T_b,tc_List_Olist(T_b),V_r_2,hAPP(c_List_Olexn(T_b,V_r_2),V_n_2)))),hAPP(c_Set_OCollect(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),hAPP(c_Nat_OSuc,V_n_2))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),hAPP(c_Nat_OSuc,V_n_2)))))) = hAPP(c_List_Olexn(T_b,V_r_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_lexn_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2512 (all V_R_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_b,V_R_2))) -> (exists B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_z)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_z),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_R_2))))))) # label(fact_tranclD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2513 (all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) = V_a)) # label(fact_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2514 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Max__Un) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2515 (all V_N_2 all V_h_2 all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> ((all B_x all B_y hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_N_2))))))) # label(fact_hom__fold1__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2516 (all T_a all V_n ((V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_List_Olist_ONil(tc_List_Olist(T_a)) = c_Enum_On__lists(T_a,V_n,c_List_Olist_ONil(T_a))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a))) = c_Enum_On__lists(T_a,V_n,c_List_Olist_ONil(T_a))))) # label(fact_n__lists__Nil) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2517 (all V_xs all T_a hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Oremdups(T_a,V_xs))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)))) # label(fact_length__remdups__leq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2518 (all V_n all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> c_List_Olinorder__class_Osorted(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))))) # label(fact_sorted__drop) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2519 (all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_x) = V_x)) # label(fact_Max_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2520 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2521 (all V_xs_2 all V_f_2 all T_b all T_c hAPP(c_List_Otl(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Otl(T_c),V_xs_2))) # label(fact_map__tl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2522 (all V_y_2 all V_x_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_y_2)))))) # label(fact_atLeast__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2523 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))) # label(fact_rtrancl__converseD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2524 (all V_s_2 all T_d all V_r_2 all T_c all T_b (c_Relation_Osingle__valued(T_b,T_c,V_r_2) -> (c_Relation_Osingle__valued(T_c,T_d,V_s_2) -> c_Relation_Osingle__valued(T_b,T_d,hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_2),V_s_2))))) # label(fact_single__valued__rel__comp) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2525 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_y_2),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_min__max_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2526 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_arith__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2527 (all V_k_2 all V_M_2 all V_n_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool),tc_Nat_Onat),V_M_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_k_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))),V_M_2)) # label(fact_UN__le__add__shift) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2528 (all V_M_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_M_2)) -> (exists B_h c_Fun_Obij__betw(tc_Nat_Onat,T_b,B_h,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_M_2)),V_M_2)))) # label(fact_ex__bij__betw__nat__finite__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2529 (all V_x all V_z all V_y all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_z),V_x)))) # label(fact_inf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2530 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),V_A_2))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2531 (all V_b_2 all V_c_2 all V_a_2 all T_b (class_Groups_Oordered__ab__semigroup__add__imp__le(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_c_2)))))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2532 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)) <-> V_y_2 = V_x_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_y_2),V_x_2))))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2533 (all V_A_2 all V_f_2 all T_c all T_b (class_Rings_Oring__1(T_b) -> hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Int_Oint),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_b,T_c),c_Int_Oring__1__class_Oof__int(T_b)),V_f_2)),V_A_2))) # label(fact_of__int__setsum) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2534 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_b,T_b),c_Relation_OId(T_b))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2535 (all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olist_ONil(T_b) = c_List_Olinorder__class_Osorted__list__of__set(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_sorted__list__of__set__empty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.86 2536 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__mult) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2537 (all V_n all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_m))) # label(fact_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2538 (all V_A_2 all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_List_Olists(T_b,V_A_2))))) # label(fact_in__lists__conv__set) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2539 (all V_ys all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Ohd(T_a),V_xs) = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_hd__append2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2540 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)))) # label(fact_min__max_Osup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2541 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),V_b_2))))) # label(fact_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2542 (all V_a (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2543 (all V_w all V_z hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),V_w)) # label(fact_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2544 (all V_A_2 all T_b all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))) # label(fact_eq__mem__trans) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2545 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_UN__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2546 (all V_y all V_x (V_y != V_x -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_y),V_x))))) # label(fact_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2547 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_OSuc,V_n)))) # label(fact_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2548 (all V_a all V_x all V_b all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x))))) # label(fact_min__max_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2549 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Sup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2550 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)))))) # label(fact_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2551 (all V_k_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_k_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_k_2))) # label(fact_Compl__atMost) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2552 (all V_y_2 all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_y_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),hAPP(hAPP(V_f_2,V_x_2),V_y_2)))))) # label(fact_Diff1__fold__graph) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2553 (all V_y_2 all V_x_2 all V_c_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Ocurry(T_c,T_d,T_b,V_c_2),V_x_2),V_y_2) = hAPP(V_c_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2))) # label(fact_curry__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2554 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2555 (all V_size_2 all V_g_2 all T_b all V_nrandom_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_nrandom_2),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) -> hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_b)) = c_Random__Sequence_ORandom(T_b,V_g_2,V_nrandom_2,V_size_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_nrandom_2),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) -> c_Random__Sequence_ORandom(T_b,V_g_2,V_nrandom_2,V_size_2) = hAPP(hAPP(c_Product__Type_Oscomp(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm)),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),hAPP(V_g_2,V_size_2)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)))),tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_prod(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),hAPP(c_Product__Type_Oscomp(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random__Sequence_ORandom(T_b,V_g_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_nrandom_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),V_size_2))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b))))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)))),tc_prod(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b))))),c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b))))),tc_prod(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),c_DSequence_Ounion(T_b)),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),c_DSequence_Osingle(T_b)),c_Product__Type_Ofst(T_b,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm)))))))))) # label(fact_Random_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2556 (all V_B_2 all V_a_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2)) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2557 (all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2)))) # label(fact_insert__compr) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2558 (all V_B_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__image__simple__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2559 (all V_i_1 all V_x all T_a hAPP(c_New__DSequence_Oneg__single(T_a,V_x),V_i_1) = c_Lazy__Sequence_Ohb__single(T_a,V_x)) # label(fact_New__DSequence_Oneg__single__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2560 (all V_c all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_c)))) # label(fact_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2561 (all V_B_2 all V_A_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_c),V_A_2)),hAPP(c_Relation_ODomain(T_b,T_c),V_B_2))),hAPP(c_Relation_ODomain(T_b,T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Domain__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2562 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2563 (all V_n V_n = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) # label(fact_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2564 (all V_P_2 all T_b ((all B_x -hBOOL(hAPP(V_P_2,B_x))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_b),V_P_2))) # label(fact_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2565 (all T_2 all T_1 (class_Groups_Ouminus(T_1) -> class_Groups_Ouminus(tc_fun(T_2,T_1)))) # label(arity_fun__Groups_Ouminus) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2566 (all V_ys_2 all V_xs_2 all V_f_2 all T_b all T_c (hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2) -> (c_Fun_Oinj__on(T_c,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_ys_2 = V_xs_2))) # label(fact_map__injective) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2567 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> c_Fun_Obij__betw(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),V_B_2,V_A_2))) # label(fact_bij__betw__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2568 (all T_c all T_b all V_b_2 all V_a_2 all V_r_2 (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> hBOOL(hAPP(c_Predicate_ORangeP(T_b,T_c,V_r_2),V_b_2)))) # label(fact_RangeP_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2569 (all V_y_2 all V_x_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_r_2)),c_fequal),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))))) # label(fact_reflcl__set__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2570 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> -(all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_y)),V_R_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))))))) # label(fact_rel__pow__Suc__E2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2571 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2572 (all V_A_2 all V_a_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))))) # label(fact_inj__image__mem__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2573 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = V_x)) # label(fact_double__compl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2574 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = V_m)) # label(fact_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2575 (all V_B_2 all V_f_2 all T_c all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,B_x)),V_B_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),V_B_2)))) # label(fact_image__subsetI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2576 (all V_A_2 all T_c all V_I_2 all V_k_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_UN__absorb) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2577 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r))))))) # label(fact_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2578 (all V_x_2 all V_n_2 all T_b all V_f_2 hAPP(V_f_2,hAPP(hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_n_2),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_n_2),V_f_2),hAPP(V_f_2,V_x_2))) # label(fact_funpow__swap1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2579 (all V_r all V_q all T_a c_Nitpick_Otimes__frac(T_a,V_q,V_r) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_a),V_q)),hAPP(c_Nitpick_Onum(T_a),V_r))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Odenom(T_a),V_q)),hAPP(c_Nitpick_Odenom(T_a),V_r)))) # label(fact_Nitpick_Otimes__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2580 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setprod__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2581 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2))) # label(fact_Diff__Compl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2582 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_y_2 | V_x_2 != c_Groups_Ozero__class_Ozero(T_b) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2))))))) # label(fact_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2583 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(c_List_Orev(T_b),V_xs_2),hAPP(c_List_Orev(T_c),V_ys_2)) <-> c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2))) # label(fact_list__all2__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2584 (all V_R_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_R_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2))) # label(fact_rtrancl__is__UN__rel__pow) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2585 (all V_I_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_Osublist(T_b,V_xs_2,V_I_2))))) # label(fact_distinct__sublistI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2586 (all V_f2_2 all V_f1_2 all T_c all T_b V_f1_2 = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_b,T_c,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c))) # label(fact_lazy__sequence_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2587 (all V_a_2 all T_b -hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_emptyE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2588 (all V_g_2 all V_f_2 all T_b all T_c all T_d hAPP(c_List_Omap(T_b,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_2),V_g_2)) = hAPP(c_Fun_Ocomp(tc_List_Olist(T_d),tc_List_Olist(T_c),tc_List_Olist(T_b),hAPP(c_List_Omap(T_d,T_c),V_f_2)),hAPP(c_List_Omap(T_b,T_d),V_g_2))) # label(fact_map__comp__map) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2589 (all V_r_2 all V_A_2 all V_f_2 all T_c all T_b (class_Fields_Ofield(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Rings_Oinverse__class_Odivide(T_b)),V_f_2)),V_r_2)),V_A_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)),V_r_2))) # label(fact_setsum__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2590 (all V_A_2 all V_I_2 all T_c all V_B_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2)),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2))) # label(fact_Int__UN__distrib) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2591 (all V_r_2 all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))))) # label(fact_quotientI) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2592 (all V_xs_2 all V_Q_2 all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(c_List_Ofilter(T_b,V_Q_2),V_xs_2)) = hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_Q_2)),V_P_2)),V_xs_2)) # label(fact_filter__filter) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2593 (all V_m all V_i hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m))))) # label(fact_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2594 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_x_2),hAPP(c_Set_OPow(T_b),V_A_2))) <-> hBOOL(hAPP(c_Predicate_OPowp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)),V_x_2)))) # label(fact_Powp__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2595 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2596 (all V_k c_Int_OBit1(c_Int_Opred(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(V_k))) # label(fact_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2597 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_B_2),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))))) # label(fact_Int__eq__Inter) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2598 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2599 (all V_R_2 all V_x_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2)))) # label(fact_rel__pow__0__I) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2600 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n_2)),hAPP(c_Code__Numeral_Onat__of,V_m_2))))) # label(fact_less__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2601 (all V_a c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_a)) # label(fact_zmod__self) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2602 (all V_c_2 all T_d all T_c all T_b all V_q_2 all V_p_2 (V_q_2 = V_p_2 -> hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_c_2),V_p_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_c_2),V_q_2))) # label(fact_split__weak__cong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2603 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2604 (all V_i_2 all V_y_2 all V_x_2 all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) -> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_x_2),hAPP(hAPP(c_List_Olist__update(T_c,V_ys_2),V_i_2),V_y_2)))))) # label(fact_list__all2__update__cong2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2605 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)))) # label(fact_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2606 (all V_z_2 all T_b (class_Int_Oring__char__0(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2) <-> V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_of__int__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2607 (all V_f_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (exists B_x1 hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,V_A_2),B_x1)))))) # label(fact_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2608 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> -(c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (c_Relation_Osym(T_b,V_r_2) -> -c_Relation_Otrans(T_b,V_r_2))))) # label(fact_equivE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2609 (all V_f_2 all T_c all T_b hAPP(c_Fun_Ocomp(T_b,T_c,T_b,V_f_2),c_Fun_Oid(T_b)) = V_f_2) # label(fact_o__id) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2610 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_Compl__subset__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2611 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__left__absorb) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2612 (all V_xs all V_x all T_a hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = V_x) # label(fact_hd_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2613 (all V_P_H_2 all V_P_2 ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> (hBOOL(hAPP(V_P_2,B_x)) <-> hBOOL(hAPP(V_P_H_2,B_x))))) -> hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),V_P_2)) = hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),V_P_H_2)))) # label(fact_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2614 (all V_xs all T_a hAPP(c_List_Orev(T_a),hAPP(c_List_Orev(T_a),V_xs)) = V_xs) # label(fact_rev__rev__ident) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2615 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)))) # label(fact_INT__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2616 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(tc_fun(T_b,tc_HOL_Obool),c_Wellfounded_Omin__ext(T_b,V_r_2)))) # label(fact_min__ext__wf) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2617 (all V_x_2 all T_b (class_Groups_Oone(T_b) -> (c_Groups_Oone__class_Oone(T_b) = V_x_2 <-> c_Groups_Oone__class_Oone(T_b) = V_x_2))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2618 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2619 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_b (c_Predicate_Otransp(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(V_r_2,V_y_2),V_z_2)) -> hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_z_2)))))) # label(fact_transpE) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2620 (all V_r_2 all T_b ((all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_x)),V_r_2)))) <-> c_Relation_Osym(T_b,V_r_2))) # label(fact_sym__def) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2621 (all V_k_2 all V_N_2 all T_b (class_Groups_Olinordered__ab__semigroup__add(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_N_2 -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_k_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_N_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(T_b),V_k_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_N_2))))))))) # label(fact_add__Min__commute) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2622 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2623 (all V_A_2 all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_b,tc_HOL_Obool))),V_A_2)))),c_member(T_b))))) # label(fact_Union__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2624 (all V_xs_2 all T_b (class_Groups_Omonoid__add(T_b) -> hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_xs_2) = c_List_Ofoldr(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_xs_2,c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_foldl__foldr1) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2625 (all V_n_2 all V_m_2 all V_k_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2))) # label(fact_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2626 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_B_2)) -> V_t_2 != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xa),B_x))))) -> V_t_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),V_D_2))))) # label(fact_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2627 (all V_n_2 all V_a_2 all T_b (class_Rings_Oring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2)),c_Int_Oring__1__class_OInts(T_b)))))) # label(fact_Ints__power) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2628 (all V_x_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(c_List_Orev(T_b),hAPP(c_List_Otl(T_b),c_List_OdropWhile(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_x_2)),V_xs_2))) = c_List_OtakeWhile(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_x_2)),hAPP(c_List_Orev(T_b),V_xs_2))))) # label(fact_takeWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2629 (all V_f_2 all V_S_2 all V_T_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_S_2)))))) # label(fact_setsum__mono__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2630 (all V_g_2 all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_b))) # label(fact_setsum_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2631 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2))),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_image__diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2632 (all V_n all V_a all V_b all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))))) # label(fact_nonzero__power__divide) # label(axiom) # label(non_clause). [assumption]. 8.78/8.87 2633 (all V_n_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_x_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2634 (all T_2 all T_1 (class_Orderings_Otop(T_1) -> class_Orderings_Otop(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Otop) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2635 (all V_code__numeral_2 all V_f2_2 all V_f1_2 all T_b c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_b,V_f1_2,V_f2_2,c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2)) = hAPP(V_f2_2,V_code__numeral_2)) # label(fact_code__numeral_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2636 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> c_SetInterval_Oord__class_OatLeastAtMost(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2),V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_c_2)))) # label(fact_Int__atLeastAtMostL2) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2637 (all T_b all B_P ((exists B_x hBOOL(hAPP(B_P,B_x))) -> hBOOL(hAPP(B_P,c_Hilbert__Choice_OEps(T_b,B_P))))) # label(fact_tfl__some) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2638 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omax(T_b)),V_A_2)))) # label(fact_Max_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2639 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2640 (all V_a_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_2)),V_r_2)))) # label(fact_wf__not__refl) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2641 (all V_x_H all V_x all V_xs all T_a all V_i_H all V_i (V_i_H != V_i -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i_H),V_x_H) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i_H),V_x_H)),V_i),V_x))) # label(fact_list__update__swap) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2642 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))) # label(fact_min__max_Oinf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2643 (all V_a_2 all T_b c_FunDef_Ois__measure(T_b,V_a_2)) # label(fact_is__measure_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2644 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)))))) # label(fact_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2645 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2646 (all V_b_2 all V_a_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_c))),V_S_2) = c_Groups_Oone__class_Oone(T_c)) & (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBC(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_c))),V_S_2) = hAPP(V_b_2,V_a_2))))) # label(fact_setprod__delta_H) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2647 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_n)))) # label(fact_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2648 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2)))) # label(fact_finite__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2649 (all V_X_2 all V_f_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (V_A_2 = c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_X_2),V_A_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),V_A_2))))))) # label(fact_def__coinduct) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2650 (all V_xs_2 all V_x_2 all V_y_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)))) -> V_x_2 = V_y_2 | hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_set__ConsD) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2651 (all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = V_i)) # label(fact_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.78/8.88 2652 (all V_r_H all V_q_H all V_b_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_q_H)))))) # label(fact_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2653 (all V_u_2 all V_l_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),V_l_2) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_l_2,V_u_2))) # label(fact_card__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2654 (all V_xs_2 all V_f_2 all T_c all V_P_2 all T_b hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Ofilter(T_c,hAPP(c_Fun_Ocomp(T_b,tc_HOL_Obool,T_c,V_P_2),V_f_2)),V_xs_2))) # label(fact_filter__map) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2655 (all V_r_2 all T_b c_Relation_Otrans(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2))) # label(fact_trans__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2656 (all V_A_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) # label(fact_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2657 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) = V_j_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2) = V_k_2))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2658 (all V_k_2 all V_i_2 all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_i_2),V_k_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2)))))) # label(fact_lessThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2659 (all V_P_2 all T_b c_List_Opartition(T_b,V_P_2,c_List_Olist_ONil(T_b)) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_List_Olist_ONil(T_b))) # label(fact_partition_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2660 (all V_k all V_l ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_SMT_Oz3div(V_k,V_l) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_SMT_Oz3div(V_k,V_l) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)))))) # label(fact_z3div__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2661 (all V_a_2 all V_w_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_a_2))))) # label(fact_divide__less__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2662 (all V_r_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))) # label(fact_r__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2663 (all V_f_H_2 all T_d all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_c,T_d,V_f_H_2,hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) <-> c_Fun_Oinj__on(T_b,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_f_H_2),V_f_2),V_A_2)))) # label(fact_comp__inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2664 (all V_xs all V_x all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs)))) # label(fact_drop__Cons_H) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2665 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2666 (all V_t_2 all V_B_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_t_2))))))) # label(fact_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2667 (all V_c_2 all V_z_2 all T_d all T_c all T_b all V_p_2 ((all B_a all B_b (hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b) = V_p_2 -> hBOOL(hAPP(hAPP(c_member(T_d),V_z_2),hAPP(hAPP(V_c_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(c_member(T_d),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_fun(T_d,tc_HOL_Obool)),V_c_2),V_p_2))))) # label(fact_mem__splitI2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2668 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)))))) # label(fact_Sup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2669 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2)))) # label(fact_finite__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2670 (all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = V_m) # label(fact_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2671 (all V_S_2 all V_R_2 all T_c all T_b (V_S_2 = V_R_2 <-> hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_S_2) = hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_R_2))) # label(fact_pred__equals__eq2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2672 (all V_R_2 all V_Y_2 all V_X_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_X_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_Y_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_Y_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_X_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_xa)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_Y_2)))))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_Y_2)),c_Wellfounded_Omax__ext(T_b,V_R_2)))))))) # label(fact_max__ext_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2673 (all V_B_2 all V_A_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_c,T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_c,T_b),V_A_2)),hAPP(c_Relation_ORange(T_c,T_b),V_B_2))))) # label(fact_Range__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2674 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))))) # label(fact_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2675 (all V_B_2 all V_b_2 all T_c all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,B_x))))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_INT__I) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2676 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_Odivmod__int(V_a,V_b))))) # label(fact_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2677 (all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))))) # label(fact_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2678 (all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Sum__Type_OPlus(T_b,T_c,V_A_2,V_B_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)))) # label(fact_finite__PlusD_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2679 (all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_neg__0__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2680 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2681 (all V_r_2 all T_b (c_Relation_Oantisym(T_b,V_r_2) <-> c_Relation_Oantisym(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b))))) # label(fact_antisym__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2682 (all V_A_2 all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = V_A_2) # label(fact_double__complement) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2683 (all V_B_2 all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(T_c)),V_f_2)),V_B_2))) # label(fact_vimage__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2684 (all V_k all V_b all V_a all T_a (class_Rings_Odvd(T_a) -> (V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_k) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_b),V_a))))) # label(fact_dvdI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2685 (all V_xs_2 all V_n_2 all T_b hAPP(c_List_Oset(tc_List_Olist(T_b)),c_Enum_On__lists(T_b,V_n_2,V_xs_2)) = hAPP(c_Set_OCollect(tc_List_Olist(T_b)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),V_n_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),c_List_Oset(T_b))),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_set__n__lists) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2686 (all V_y_2 all V_x_2 all V_s_2 all V_r_2 all T_d all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_d),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_2),V_s_2))) <-> hBOOL(hAPP(hAPP(c_Predicate_Opred__comp(T_b,T_c,T_d,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_r_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),V_s_2)),V_x_2),V_y_2)))) # label(fact_pred__comp__rel__comp__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2687 (all V_ys_2 all T_b all V_xs_2 (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) <-> c_List_Olist_ONil(T_b) = V_ys_2)) # label(fact_self__append__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2688 (all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Orderings_Obot__class_Obot(T_b) = c_Complete__Lattice_OInf__class_OInf(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_Inf__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2689 (all V_i hAPP(c_Code__Numeral_Onat__of,V_i) = c_Code__Numeral_Onat__of__aux(V_i,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__of__code) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2690 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_x)))) # label(fact_min__max_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2691 (all V_A_2 all V_r_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_A_2))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),V_b_2)),V_r_2)))))) # label(fact_Image__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2692 (all V_w all V_z2 all V_z1 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z1),V_z2)),V_w)) # label(fact_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2693 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))))) # label(fact_less__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2694 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2695 (all V_nat hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_nat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_nat)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2696 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_folding__image__simple_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2697 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_c_2),V_d_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2698 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x) = V_x)) # label(fact_inf__top__left) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2699 (all T_c all T_b all V_A_2 all V_y_2 all V_x_2 all V_B_2 ((-hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2))) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2CI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2700 (all V_n all V_m hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n))) # label(fact_zpower__int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2701 (all V_R_2 all T_c all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_c),c_Relation_OId(T_b)),V_R_2) = V_R_2) # label(fact_Id__O__R) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2702 (all T_b c_Transitive__Closure_Ortrancl(T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) = c_Relation_OId(T_b)) # label(fact_rtrancl__empty) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2703 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Opred(V_w)))) # label(fact_number__of__pred) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2704 (all V_xs_2 all T_b (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) <-> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_length__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2705 (all V_Q_2 all T_c all T_b all V_y_2 all V_x_2 all V_P_2 (hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_P_2),V_Q_2)) -> hBOOL(hAPP(hAPP(V_Q_2,V_x_2),V_y_2))))) # label(fact_rev__predicate2D) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2706 (all V_t_2 all V_A_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)))))))) # label(fact_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2707 (all V_code__numeral_H c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) # label(fact_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2708 (all V_b_2 all V_a_2 all V_f1_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_prod_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2709 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_x))))))))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2710 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> c_Relation_Osym(T_b,c_Transitive__Closure_Otrancl(T_b,V_r_2)))) # label(fact_sym__trancl) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2711 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z))) # label(fact_max__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2712 (all V_k_2 all V_less__eq_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeast(T_b,V_less__eq_2,V_k_2))))) # label(fact_ord_OatLeast__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2713 (all V_n_2 all V_m_2 (c_Nat__Transfer_Ois__nat(V_m_2) -> (c_Nat__Transfer_Ois__nat(V_n_2) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Int_Onat,V_m_2),hAPP(c_Int_Onat,V_n_2))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_m_2,V_n_2)))) # label(fact_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2714 (all V_r_2 all V_x_2 all T_b hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool))) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_r_2)) # label(fact_singleton__quotient) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2715 (all V_Q_2 all V_P_2 all T_c all T_b hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2))),V_Q_2))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(c_Set_OCollect(T_b),V_P_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_OCollect(T_c),V_Q_2)))) # label(fact_Collect__split) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2716 (all V_ys_2 all V_xs_2 all T_b (V_xs_2 = hAPP(c_List_Orev(T_b),V_ys_2) <-> hAPP(c_List_Orev(T_b),V_xs_2) = V_ys_2)) # label(fact_rev__swap) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2717 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = V_a)) # label(fact_minus__minus) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2718 (all T_a (class_Rings_Osemiring__1(T_a) -> -c_Int_Oiszero(T_a,c_Groups_Oone__class_Oone(T_a)))) # label(fact_not__iszero__1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2719 (all V_u_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)))) # label(fact_finite__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2720 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> V_A_2 = hAPP(c_Set_Oimage(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_image__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2721 (all T_b (class_Rings_Oring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Ozero__class_Ozero(T_b)),c_Int_Oring__1__class_OInts(T_b))))) # label(fact_Ints__0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2722 (all V_j all V_i hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),c_List_Oupto(V_i,V_j)))) # label(fact_distinct__upto) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2723 (all V_x all V_xs all T_a V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))))) # label(fact_last__snoc) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2724 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_a)),c_Groups_Osgn__class_Osgn(T_a,V_b)) = c_Groups_Osgn__class_Osgn(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_sgn__times) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2725 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2))))) # label(fact_card__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2726 (all V_B_2 all V_A_2 all V_a_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2727 (all V_xs_2 all T_b (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2))) -> c_List_OtakeWhile(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_b))),V_xs_2) = c_List_Otranspose(T_b,c_List_Otranspose(T_b,V_xs_2)))) # label(fact_transpose__transpose) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2728 (all V_s_2 all V_c_2 all T_d all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_b_2),V_c_2)),V_s_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_d),V_a_2),V_c_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_d),V_r_2),V_s_2)))))) # label(fact_rel__compI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2729 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),c_Orderings_Otop__class_Otop(T_b)),V_A_2) = c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)))) # label(fact_Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2730 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_plus__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2731 (all V_br all V_bq all V_b all V_ar all V_aq all V_c all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_aq),V_ar))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_bq),V_br))) -> (V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br),V_c)))))))) # label(fact_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2732 (all V_a_2 all V_b_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)) & (-(c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | V_a_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Nitpick_Oint__gcd,V_a_2),V_b_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2))),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Nitpick_Onorm__frac(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b_2)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)))) # label(fact_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2733 (all V_n_2 all T_b (class_Rings_Oring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),c_Int_Oring__1__class_OInts(T_b))))) # label(fact_Ints__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2734 (all V_P_2 all T_b all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> hBOOL(hAPP(V_P_2,B_x)))) # label(fact_ball__empty) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2735 (all V_f_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_f_2)),c_Orderings_Otop__class_Otop(T_c)),V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),V_f_2)))) # label(fact_INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2736 (all V_x_2 all V_A_2 all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)) = hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_fold1__insert__idem) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2737 (all V_xs all V_x all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olist_ONil(T_a)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs))))) # label(fact_take__Cons_H) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2738 (all V_B_2 all V_J_2 all T_d all V_A_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2))),V_B_2)))) # label(fact_Int__UN__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2739 (all V_w all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w)))) # label(fact_zle__linear) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2740 (all V_seed_2 all V_size_2 all V_nrandom_2 all T_b c_New__DSequence_Opos__empty(T_b) = c_New__Random__Sequence_Opos__empty(T_b,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__empty__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2741 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_divide__zero) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2742 (all V_xs_2 all V_x_2 all T_b hAPP(c_List_Ofilter(T_b,hAPP(c_fequal,V_x_2)),V_xs_2) = c_List_Oreplicate(T_b,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,hAPP(c_fequal,V_x_2)),V_xs_2)),V_x_2)) # label(fact_replicate__length__filter) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2743 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))))) # label(fact_card__Diff1__less) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2744 (all V_c_2 all T_c all T_b all V_A_2 ((V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(T_b,T_c,hAPP(c_COMBK(T_c,T_b),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(c_Set_Oimage(T_b,T_c,hAPP(c_COMBK(T_c,T_b),V_c_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2745 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__le__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2746 (all V_w all V_c all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_c),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))))) # label(fact_add__number__of__diff2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2747 (all V_P_2 all T_c all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> c_Quotient_OBabs(T_b,T_c,c_Quotient_ORespects(T_b,V_R_2),V_P_2) = V_P_2)) # label(fact_babs__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2748 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) | V_B_2 = V_A_2)) # label(fact_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2749 (all V_w_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_w_2)))))) # label(fact_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2750 (all V_C_2 all V_B_2 all T_c all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_B_2,B_x)),V_C_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)))) # label(fact_UN__least) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2751 (all V_B_2 all V_x_2 all V_A_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2))))) # label(fact_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2752 (all V_z all V_x all V_y all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y)))) # label(fact_add__frac__num) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2753 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_B_2)) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Inter__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2754 (all V_S_2 all V_R_2 all T_b (c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2)) <-> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_R_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_S_2),V_R_2))),V_S_2)))) # label(fact_wf__union__merge) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2755 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)))) # label(fact_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2756 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n))) # label(fact_power__abs) # label(axiom) # label(non_clause). [assumption]. 8.88/8.88 2757 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_y_2),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2))))) # label(fact_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2758 (all V_a all V_b (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))))) # label(fact_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2759 (all V_b all V_a all T_b all T_a V_a = hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b))) # label(fact_fst__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2760 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_b),hAPP(c_COMBK(tc_HOL_Obool,T_b),c_fTrue))) # label(fact_UNIV__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2761 (all V_v_H all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H)))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2762 (all V_va all V_v all T_a c_List_Osplice(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_v),V_va),c_List_Olist_ONil(T_a)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_v),V_va)) # label(fact_splice_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2763 (all V_z all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),V_z))) # label(fact_minus__number__of__mult) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2764 (all V_n_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_b)),hAPP(c_Set_OCollect(tc_List_Olist(T_b)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),c_List_Oset(T_b))),V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),V_n_2))))))) # label(fact_finite__lists__length__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2765 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__abs) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2766 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_A_2),V_C_2))),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_C_2))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Int__Inter__image) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2767 (all V_xs all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) -> hAPP(c_List_Orotate(T_a,V_n),V_xs) = V_xs)) # label(fact_rotate__id) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2768 (all V_n_2 all V_k_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)))) # label(fact_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2769 (all V_t_2 all V_D_2 all V_d_2 all T_b (class_Rings_Ocomm__ring(T_b) & class_Rings_Odvd(T_b) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),V_D_2)) -> (all B_x all B_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),B_x),V_t_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_k),V_D_2))),V_t_2)))))))) # label(fact_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2770 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_u_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2771 (all V_l all V_k c_Divides_Opdivmod(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l)))) # label(fact_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2772 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)))))) # label(fact_le__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2773 (all V_C_2 all V_J_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_J_2),V_C_2))) # label(fact_Sigma__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2774 (all V_ys_2 all V_xs_2 all T_b all V_y_2 all V_x_2 all V_r_2 (hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_y_2)) -> (c_List_Olistrelp(T_b,V_r_2,V_xs_2,V_ys_2) -> c_List_Olistrelp(T_b,V_r_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))))) # label(fact_listrelp_OCons) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2775 (all V_n all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),V_n)) -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n))))) # label(fact_add__leE) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2776 (all V_b_2 all V_a_2 all V_f1_2 all T_b all T_d all T_c hAPP(hAPP(V_f1_2,V_a_2),V_b_2) = c_Nitpick_Opair__box_Opair__box__case(T_c,T_d,T_b,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2))) # label(fact_pair__box_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2777 (all V_xs all V_m all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_xs) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Odrop(T_a),V_m),V_xs))) # label(fact_drop__drop) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2778 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__mono) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2779 (all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2)))) # label(fact_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2780 (all V_z all V_x all V_d (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_z))),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d))),V_z)))) # label(fact_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2781 (all V_x_2 all T_b (c_List_Olist_ONil(T_b) = V_x_2 <-> c_List_Olist_ONil(T_b) = c_List_Oremdups(T_b,V_x_2))) # label(fact_remdups__eq__nil__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2782 (all V_A_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) -> (c_Fun_Oinj__on(T_c,T_b,V_f_2,V_A_2) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_finite__imageD) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2783 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Oset(T_b),c_List_Olinorder__class_Oinsort__insert__key(T_b,T_b,c_COMBI(T_b),V_x_2,V_xs_2)))) # label(fact_set__insort__insert) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2784 (all V_w_2 all T_b (class_Int_Oring__char__0(T_b) & class_Int_Onumber__ring(T_b) -> (c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) <-> c_Int_Oiszero(T_b,hAPP(c_Int_Onumber__class_Onumber__of(T_b),c_Int_OBit0(V_w_2)))))) # label(fact_iszero__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2785 (all V_a_2 all T_b (class_Rings_Oring__1__no__zero__divisors(T_b) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(T_b) <-> V_a_2 = c_Groups_Oone__class_Oone(T_b) | hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Groups_Oone__class_Oone(T_b)) = V_a_2))) # label(fact_power2__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2786 (all V_c_2 all V_b_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_c_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_c_2),V_A_2))))))) # label(fact_min__max_Oinf__le__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2787 (all V_C_2 c_Nat__Transfer_Onat__set(hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_C_2))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2788 (all V_k_2 all V_i_2 all V_r_2 all V_f_2 all T_b ((all B_i hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(V_f_2,hAPP(c_Nat_OSuc,B_i))),hAPP(V_f_2,B_i))),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2))),hAPP(V_f_2,V_i_2))),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))) # label(fact_sequence__trans) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2789 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2790 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> V_b = V_a))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2791 (all V_R_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_b,V_R_2))) -> (exists B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_z),V_y_2)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_z)),c_Transitive__Closure_Ortrancl(T_b,V_R_2))))))) # label(fact_tranclD2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2792 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) <-> hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2) = V_r_2)) # label(fact_sym__conv__converse__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2793 (all V_xs_2 all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> (c_List_Olist_ONil(T_b) != V_xs_2 -> hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),hAPP(c_List_Ohd(T_b),V_xs_2)),hAPP(c_List_Otl(T_b),V_xs_2))))) # label(fact_fold1__set) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2794 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_b),V_c))))) # label(fact_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2795 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2796 (all V_M_2 all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_M_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_M_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_M_2)) # label(fact_INT__Un) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2797 (all V_c all V_a all V_b all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))))) # label(fact_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2798 (all V_P_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(V_P_2,V_A_2)) -> ((all B_a all B_A (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_A)) -> (hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A)) -> (hBOOL(hAPP(V_P_2,B_A)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),B_A),hAPP(hAPP(c_Set_Oinsert(T_b),B_a),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))))) -> hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))))) # label(fact_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2799 (all V_B_2 all V_l_2 all V_A_2 all T_b (hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_l_2)) -> (hBOOL(hAPP(c_List_Olistsp(T_b,V_B_2),V_l_2)) -> hBOOL(hAPP(c_List_Olistsp(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_l_2))))) # label(fact_listsp__infI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2800 (all V_a_2 all V_ys_2 all V_xs_2 all V_f_2 all T_b all T_c c_List_Ofoldr(T_c,T_b,V_f_2,V_xs_2,c_List_Ofoldr(T_c,T_b,V_f_2,V_ys_2,V_a_2)) = c_List_Ofoldr(T_c,T_b,V_f_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2),V_a_2)) # label(fact_foldr__append) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2801 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oring(T_b) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2) = V_d_2))) # label(fact_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2802 (all V_b c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2803 (all V_xs_2 all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_b,T_b),c_Groups_Oabs__class_Oabs(T_b)),V_xs_2)))))) # label(fact_listsum__abs) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2804 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_A_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),V_B_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),c_Groups_Oone__class_Oone(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_fold__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2805 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(V_f_2,V_x_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2806 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (V_y_2 = V_x_2 <-> hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2807 (all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)))) # label(fact_card__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2808 (all V_xs all T_a all V_i all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_i)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),V_i) = hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_i)))) # label(fact_nth__drop) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2809 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2810 (all V_P_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> (hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> ((all B_x all B_F (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_F)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_F)) -> (hBOOL(hAPP(V_P_2,B_F)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Set_Oinsert(T_b),B_x),B_F))))))) -> hBOOL(hAPP(V_P_2,V_F_2)))))) # label(fact_finite__induct) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2811 (all V_x_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Relation_ORange(T_b,T_c),V_r_2))) <-> hBOOL(hAPP(c_Predicate_ORangeP(T_b,T_c,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_r_2)),V_x_2)))) # label(fact_RangeP__Range__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2812 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_B_2))))))))) # label(fact_Inf__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2813 (all V_P_2 all V_xs_2 all T_b ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))))) <-> c_List_Olist__ex(T_b,V_P_2,V_xs_2))) # label(fact_Bex__set__list__ex) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2814 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)) # label(fact_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2815 (all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,T_c),tc_HOL_Obool)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_finite__fun__UNIVD2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2816 (all V_xs_2 all V_f_2 all T_c all T_b hAPP(c_List_Orev(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__map) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2817 (all V_h_2 all T_c all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_h_2),V_F_2))))) # label(fact_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2818 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))))) # label(fact_le__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2819 (all V_ys_2 all V_xs_2 all V_a_2 all V_f_2 all T_c all T_b hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,V_a_2),V_xs_2)),V_ys_2) = hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,V_a_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_foldl__append) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2820 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(tc_List_Olist(T_b),tc_List_Olist(T_c),hAPP(c_List_Omap(T_b,T_c),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_inj__mapD) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2821 (all V_M_2 all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_M_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_M_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_M_2)) # label(fact_UN__Un) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2822 (all V_P_2 all V_f_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_f_2))) -> (c_Orderings_Oorder__class_Omono(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_f_2) -> ((all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b)),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_f_2)),hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2)))))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)))) -> hBOOL(hAPP(hAPP(V_P_2,V_a_2),V_b_2)))))) # label(fact_lfp__induct2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2823 (all T_b all V_xs_2 (c_List_Olist_ONil(T_b) != V_xs_2 -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_List_Ohd(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_hd__in__set) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2824 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2825 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa))))) -> (B_x != V_t_2 -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2) != V_t_2)))))) # label(fact_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2826 (all V_x_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_x)))) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)))))) # label(fact_sorted_OCons) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2827 (all V_n_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))),hAPP(V_f_2,V_n_2)))) # label(fact_setsum__lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2828 (all V_w all V_z all V_y all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_z),V_w)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_w)))) # label(fact_times__divide__times__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2829 (all V_n_2 all V_A_2 all V_f_2 all T_b (hAPP(c_Nat_OSuc,V_n_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2) -> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))))))) # label(fact_setsum__SucD) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2830 (all V_A_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),V_A_2)) # label(fact_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2831 (all V_w all V_z hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2832 (all V_t_2 all V_A_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)))))))) # label(fact_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2833 (all V_b all V_a all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))))))) # label(fact_less__half__sum) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2834 (all V_f_2 all V_B_H_2 all V_A_H_2 all T_c all V_B_2 all V_A_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_H_2),V_B_H_2) -> (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_B_2,V_B_H_2) -> c_Fun_Obij__betw(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_H_2),V_B_H_2))))))) # label(fact_bij__betw__Disj__Un) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2835 (all V_xs all V_x all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs)))) # label(fact_listsum__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2836 (all V_N_2 all V_M_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_M_2),V_N_2)) -> (V_M_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_M_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_N_2)))))))) # label(fact_Max__mono) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2837 (all V_b all V_c all V_a all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b))))) # label(fact_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2838 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2839 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2840 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> (hAPP(c_Code__Numeral_Oof__nat,V_y_2) = hAPP(c_Code__Numeral_Oof__nat,V_x_2) <-> V_x_2 = V_y_2)))) # label(fact_code__numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2841 (all V_y_2 all V_x_2 all T_b (class_Lattices_Oboolean__algebra(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2)))))) # label(fact_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2842 (all V_z2_2 all V_y2_2 all V_z1_2 all V_y1_2 all V_f_2 all V_r2_2 all V_r1_2 all T_d all T_c all T_b (c_Equiv__Relations_Ocongruent2(T_b,T_c,T_d,V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y1_2),V_z1_2)),V_r1_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y2_2),V_z2_2)),V_r2_2)) -> hAPP(hAPP(V_f_2,V_z1_2),V_z2_2) = hAPP(hAPP(V_f_2,V_y1_2),V_y2_2))))) # label(fact_congruent2D) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2843 (all V_g2_2 all V_g1_2 all T_g all V_f2_2 all V_f1_2 all T_f all T_e all T_c all T_d all T_b hAPP(c_Fun_Ocomp(tc_prod(T_f,T_g),tc_prod(T_d,T_e),tc_prod(T_b,T_c),c_Product__Type_Omap__pair(T_f,T_d,T_g,T_e,V_f1_2,V_g1_2)),c_Product__Type_Omap__pair(T_b,T_f,T_c,T_g,V_f2_2,V_g2_2)) = c_Product__Type_Omap__pair(T_b,T_d,T_c,T_e,hAPP(c_Fun_Ocomp(T_f,T_d,T_b,V_f1_2),V_f2_2),hAPP(c_Fun_Ocomp(T_g,T_e,T_c,V_g1_2),V_g2_2))) # label(fact_map__pair__compose) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2844 (all V_r_2 all T_b all T_c hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(c_Relation_Oconverse(T_b,T_c),V_r_2)) = V_r_2) # label(fact_converse__converse) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2845 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OPls)))) # label(fact_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2846 (all T_c all T_b hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_prod(T_b,T_c)),c_Product__Type_OPair(T_b,T_c)) = c_Fun_Oid(tc_prod(T_b,T_c))) # label(fact_split__Pair) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2847 (all V_y_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) -> V_y_2 = hAPP(V_Rep_2,hAPP(V_Abs_2,V_y_2))))) # label(fact_type__definition_OAbs__inverse) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2848 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2)))) # label(fact_ivl__disj__un_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2849 (all V_y all V_m all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y),V_m)) # label(fact_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2850 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2851 (all V_f_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(T_c)),V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),V_f_2)))) # label(fact_SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2852 (all V_i_2 all V_x_2 all T_b c_List_Oreplicate(T_b,V_i_2,V_x_2) = hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_b),hAPP(c_COMBK(T_b,tc_Nat_Onat),V_x_2)),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_i_2))) # label(fact_map__replicate__trivial) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2853 (all V_ys_2 all V_xs_2 all V_x_2 all T_c all T_b hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_b,T_c)),T_c,c_List_Olist_ONil(tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_b,T_c)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_b,T_c)))),T_c),c_COMBB(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c))),T_c),c_List_Olist_OCons(tc_prod(T_b,T_c))),hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2)))),hAPP(c_List_Ozip(T_b,T_c),V_xs_2))),V_ys_2) = hAPP(hAPP(c_List_Ozip(T_b,T_c),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_ys_2)) # label(fact_zip__Cons1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2854 (all V_z all V_z_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H))))) # label(fact_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2855 (all V_c_2 all V_b_2 all V_a_2 all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),V_r_2)))))) # label(fact_transD) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2856 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setprod__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2857 (all V_b all V_a all V_c all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_mult__divide__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2858 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b)),V_c)) # label(fact_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2859 (all V_xs_2 all T_b hAPP(c_List_Orotate1(T_b),V_xs_2) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(T_b),T_b,c_List_Olist_ONil(T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_b),hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Oappend(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b)))),V_xs_2)) # label(fact_rotate1__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2860 (all V_xs_2 all T_b hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_b),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_Nat_Onat,T_b),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_b),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_b),V_xs_2))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))))) # label(fact_set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.88/8.89 2861 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_t_2)))))))) # label(fact_aset_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2862 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_a))),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)))) # label(fact_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2863 (all T_b all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_b) -> c_Random_Oselect__weight(T_b,hAPP(hAPP(c_List_Omap(T_b,tc_prod(tc_Code__Numeral_Ocode__numeral,T_b)),hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),V_xs_2)) = c_Random_Oselect(T_b,V_xs_2))) # label(fact_select__weigth__select) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2864 (all V_w all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)))))) # label(fact_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2865 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin))) # label(fact_of__int__m1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2866 (all V_x all V_y all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_x)),V_y) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))) # label(fact_abs__div__pos) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2867 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2)))) # label(fact_finite__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2868 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n)),hAPP(c_Code__Numeral_Oof__nat,V_m))) # label(fact_times__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2869 (all V_x_2 all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,T_b),V_A_2),V_x_2) = c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(tc_fun(T_c,T_b))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),T_b),hAPP(c_COMBS(tc_fun(T_c,T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,T_b))),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,T_b),T_b),tc_fun(tc_fun(T_c,T_b),tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,tc_fun(T_c,T_b))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_c,T_b),T_c,T_b),c_COMBI(tc_fun(T_c,T_b))),V_x_2)))))))) # label(fact_Sup__apply) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2870 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Orderings_Oorder(T_c) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> (V_y_2 = V_x_2 <-> hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2))))) # label(fact_strict__mono__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2871 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2872 (all T_a c_List_Olist_ONil(T_a) = hAPP(c_List_Orev(T_a),c_List_Olist_ONil(T_a))) # label(fact_rev_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2873 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_b,tc_HOL_Obool),T_c),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2874 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) != c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2875 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_x_2),V_A_2) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_min__max_OSup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2876 (all V_z_2 all V_g_2 all V_a_2 all V_A_2 all T_b all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_a_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Otimes__class_Otimes(T_c)),V_g_2),V_z_2),V_A_2)))))) # label(fact_fold__image__insert) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2877 (all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) = c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))) # label(fact_UNIV__Times__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2878 (all V_R_2 all V_Q_2 all V_P_2 all T_b all T_d all T_c hAPP(hAPP(hAPP(c_COMBS(T_c,T_d,T_b),V_P_2),V_Q_2),V_R_2) = hAPP(hAPP(V_P_2,V_R_2),hAPP(V_Q_2,V_R_2))) # label(help_c__COMBS__1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2879 (all V_y_2 all V_f_2 all T_c all T_b all V_x_2 all V_z_2 ((V_x_2 != V_z_2 -> hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2)) & (V_z_2 = V_x_2 -> V_y_2 = hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2)))) # label(fact_fun__upd__apply) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2880 (all V_xs_2 all T_b (hAPP(c_List_Orev(T_b),V_xs_2) = c_List_Olist_ONil(T_b) <-> V_xs_2 = c_List_Olist_ONil(T_b))) # label(fact_Nil__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2881 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2882 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2))) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_d_2))))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2883 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> (c_Relation_Oantisym(T_b,V_s_2) -> c_Relation_Oantisym(T_b,V_r_2)))) # label(fact_antisym__subset) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2884 (all V_xs_2 all T_b hAPP(c_List_Oset(T_b),c_List_Oconcat(T_b,V_xs_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xs_2)),c_List_Oset(T_b))) # label(fact_set__concat) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2885 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))))) # label(fact_finite__subset) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2886 (all V_s_2 all V_B_2 all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (c_Relation_Orefl__on(T_b,V_B_2,V_s_2) -> c_Relation_Orefl__on(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_refl__on__Int) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2887 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Un__assoc) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2888 (all V_r_2 all V_x_2 all T_b -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),c_List_Olist_ONil(T_b))),c_List_Olexord(T_b,V_r_2)))) # label(fact_lexord__Nil__right) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2889 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2))))) # label(fact_folding__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2890 (all V_A_2 all V_xs_2 all T_b hAPP(hAPP(c_List_Omap(tc_prod(T_b,tc_Nat_Onat),T_b),c_Product__Type_Ofst(T_b,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_b,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_prod(T_b,tc_Nat_Onat)),c_member(tc_Nat_Onat)),c_Product__Type_Osnd(T_b,tc_Nat_Onat))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_b,tc_Nat_Onat),V_xs_2),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))))) = c_List_Osublist(T_b,V_xs_2,V_A_2)) # label(fact_sublist__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2891 (all V_y all V_x (hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y) -> V_y = V_x)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2892 (all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OPls))))) # label(fact_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2893 (all V_r_2 all T_b (c_Wellfounded_Oacyclic(T_b,V_r_2) <-> (all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_acyclic__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2894 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_List_Olistsp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_x_2),c_List_Olists(T_b,V_A_2))))) # label(fact_listsp__lists__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2895 (all V_b all V_a all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))) -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)))))) # label(fact_min__max_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2896 (all V_B_2 all V_g_2 all T_d all V_A_2 all V_f_2 all T_c all T_b (class_Rings_Osemiring__0(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_b),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_b),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_b)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_b),tc_fun(T_d,T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,T_b),tc_fun(tc_fun(T_d,T_b),tc_fun(T_d,T_b)),T_c),c_COMBB(T_b,T_b,T_d)),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Otimes__class_Otimes(T_b)),V_f_2))),V_g_2))),V_B_2)),V_A_2))) # label(fact_setsum__product) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2897 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_b),T_c),c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_b)),V_f_2)),V_B_2)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_prod(T_c,T_d),T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2))))) # label(fact_setprod__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2898 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))))) # label(fact_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2899 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))))) # label(fact_inf__sup__ord_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2900 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) <-> c_Nat__Transfer_Ois__nat(V_x_2))) # label(fact_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2901 (all V_A_2 all T_b hAPP(c_Set_Ovimage(T_b,T_b,c_Fun_Oid(T_b)),V_A_2) = V_A_2) # label(fact_vimage__id) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2902 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))))) # label(fact_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2903 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2))))))) # label(fact_Max_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2904 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)))) # label(fact_inf2D2) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2905 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> V_xs_2 = c_List_Oinsert(T_b,V_x_2,V_xs_2))) # label(fact_in__set__insert) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2906 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__Un) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2907 (all V_y_2 all V_x_2 all V_r_2 all T_b ((all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_a),B_b)),V_r_2)) | B_a = B_b | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_b),B_a)),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olexord(T_b,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_y_2),V_x_2)),c_List_Olexord(T_b,V_r_2))) | V_y_2 = V_x_2)) # label(fact_lexord__linear) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2908 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2))))))))) # label(fact_folding__one_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2909 (all V_xs_2 all V_g_2 all T_d all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_b),hAPP(c_Fun_Ocomp(T_c,T_b,T_d,V_f_2),V_g_2)),V_xs_2)) # label(fact_map__map) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2910 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) -> -(hBOOL(hAPP(V_A_2,V_x_2)) -> -hBOOL(hAPP(V_B_2,V_x_2))))) # label(fact_inf1E) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2911 (all V_a_2 all V_f_2 all V_A_2 all T_c all T_b (class_Orderings_Olinorder(T_b) & class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),V_A_2),V_f_2)),V_a_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(V_f_2,B_x)),V_a_2))))))) # label(fact_INF__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2912 (all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2913 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_w),V_z)))) # label(fact_of__int__diff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2914 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_sup__top__right) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2915 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2916 (all T_b all V_P_2 ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) -> hBOOL(hAPP(V_P_2,c_Hilbert__Choice_OEps(T_b,V_P_2))))) # label(fact_someI__ex) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2917 (all V_y_2 all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) = hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_x_2),V_y_2)))) # label(fact_image__uminus__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2918 (all V_P_2 all V_n_2 ((exists B_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2)) & hBOOL(hAPP(V_P_2,B_m)))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))) & hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2919 (all V_v_2 all V_i_2 all V_xs_2 all V_x_2 all T_b hAPP(hAPP(c_List_Olist__update(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_i_2),V_v_2) = c_Nat_Onat_Onat__case(tc_List_Olist(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_v_2),V_xs_2),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_b),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,T_b,tc_List_Olist(T_b)),c_List_Olist__update(T_b,V_xs_2)),V_v_2)),V_i_2)) # label(fact_list__update_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2920 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2921 (all V_x_2 hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(c_Int_ORep__Integ,V_x_2)),c_Int_OInteg))) # label(fact_Rep__Integ) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2922 (all V_n_2 all V_a_2 all T_b (class_Rings_Omult__zero(T_b) & class_Rings_Ono__zero__divisors(T_b) & class_Rings_Ozero__neq__one(T_b) & class_Power_Opower(T_b) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_b) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_b) & V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2923 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2))))) # label(fact_in__mono) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2924 (all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olist_ONil(T_b) = c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,c_List_Olist_ONil(T_b)))) # label(fact_sort__key__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2925 (all V_j_2 all V_i_2 all V_P_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_Oall__interval__int(V_P_2,V_i_2,V_j_2))) # label(fact_all__interval__int__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2926 (all T_b c_Relation_Otrans(tc_fun(T_b,tc_HOL_Obool),c_Wellfounded_Ofinite__psubset(T_b))) # label(fact_trans__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2927 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),hAPP(c_List_OremoveAll(T_b,V_x_2),V_xs_2)) = hAPP(c_List_OremoveAll(T_c,hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)))) # label(fact_map__removeAll__inj) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2928 (all V_A_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) | V_b_2 = V_a_2)) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2929 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hAPP(V_f_2,V_x_2) = V_y_2 -> V_x_2 = hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),V_y_2)))) # label(fact_inv__f__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2930 (all V_f_2 all V_S_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(V_f_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_S_2)))))) # label(fact_setprod__pos__nat__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2931 (all V_p_2 all V_c_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_c_2),V_p_2) = hAPP(hAPP(V_c_2,hAPP(c_Product__Type_Ofst(T_c,T_d),V_p_2)),hAPP(c_Product__Type_Osnd(T_c,T_d),V_p_2))) # label(fact_split__def) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2932 (all V_f_2 all V_y_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),hAPP(V_f_2,V_x_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),c_List_Olist_Olist__size(T_b,V_f_2,V_xs_2)))))) # label(fact_list__size__estimation_H) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2933 (all V_qr all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)),V_qr)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(c_Divides_OnegateSnd,V_qr))))) # label(fact_divmod__int__rel__neg) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2934 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Osucc(V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k)))) # label(fact_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2935 (all V_c_2 all V_b_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_c_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_b_2),V_c_2))))))) # label(fact_min__max_Ofold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2936 (all V_A_2 all V_f_2 all T_b hAPP(c_Int_Onat,hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_b),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2)) # label(fact_transfer__nat__int__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2937 (all V_x_2 all V_r_2 all T_b (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_x_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))))) # label(fact_irrefl__tranclI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2938 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a))) # label(fact_diff__0) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2939 (all V_b all V_a ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Divides_OnegDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))) -> c_Divides_OnegDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OnegDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))))) # label(fact_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2940 (all V_k_2 all V_l_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord_OgreaterThanLessThan(tc_Nat_Onat,hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2))) # label(fact_dvd_OgreaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2941 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))))) # label(fact_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2942 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2943 (all V_C_2 all V_A_2 all T_b ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_X),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),B_X),V_C_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2)),V_C_2)))) # label(fact_Union__least) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2944 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)),V_a)))) # label(fact_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2945 (all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_subsetI) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2946 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2))))) # label(fact_Sup__fin__Sup) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2947 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)))) # label(fact_sup__compl__top) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2948 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2)) = hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Pow__Int__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2949 (all V_B_H_2 all V_A_H_2 all V_A_2 all V_f_2 all T_b all T_c (hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = V_A_H_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_H_2),V_A_H_2)) -> V_B_H_2 = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Oimage(T_b,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_b,V_A_2,V_f_2)),V_B_H_2))))) # label(fact_image__inv__into__cancel) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2950 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2951 (all V_d_2 all V_b_2 all V_c_2 all V_a_2 all T_b (class_Divides_Osemiring__div(T_b) -> (V_a_2 != c_Groups_Ozero__class_Ozero(T_b) -> (V_c_2 != c_Groups_Ozero__class_Ozero(T_b) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_c_2),V_d_2)) -> (hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_b),V_d_2),V_c_2) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_b),V_b_2),V_a_2) <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_c_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_d_2)))))))) # label(fact_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2952 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Sup__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2953 (all V_r all V_q hAPP(c_Divides_OnegateSnd,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_r))) # label(fact_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2954 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (V_y_2 = V_x_2 <-> hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_y_2) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_x_2)))) # label(fact_atMost__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2955 (all V_D_2 all V_C_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_C_2,V_D_2) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_D_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Fun_Obij__betw(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_D_2)))))) # label(fact_bij__betw__combine) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2956 (all V_x all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> V_x = hAPP(c_List_Ohd(T_a),c_List_Oreplicate(T_a,V_n,V_x)))) # label(fact_hd__replicate) # label(axiom) # label(non_clause). [assumption]. 8.88/8.90 2957 (all V_v all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) # label(fact_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2958 (all V_b_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> ((exists B_x (V_b_2 = hAPP(V_f_2,B_x) & (all B_y (hAPP(V_f_2,B_y) = V_b_2 -> B_x = B_y)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2959 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel(T_b,V_r_2)),c_List_Olistrel(T_b,V_s_2))))) # label(fact_listrel__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2960 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Un__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2961 (all V_y all V_ys all V_x all V_xs all T_a hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_y) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys))) # label(fact_list__update__length) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2962 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> c_Relation_Otrans(tc_List_Olist(T_b),c_List_Olistrel(T_b,V_r_2)))) # label(fact_listrel__trans) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2963 (all V_y_2 all V_f_2 all T_c all T_b hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),V_y_2) = c_Hilbert__Choice_OEps(T_b,hAPP(hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),V_y_2))) # label(fact_inv__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2964 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (V_y_2 != V_x_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> hAPP(V_f_2,V_x_2) != hAPP(V_f_2,V_y_2)))))) # label(fact_inj__on__contraD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2965 (all V_x all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> V_x = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x))) # label(fact_mult__idem) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 2966 (all V_n all V_m ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> c_List_Oupto(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)) = c_List_Olist_ONil(tc_Int_Oint)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> c_List_Oupto(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)) = hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)))))) # label(fact_upto__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2967 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__sup__aci_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2968 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_y)))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> V_x_2 = c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)))))) # label(fact_Min__eqI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2969 (all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_R_2),V_S_2)))) # label(fact_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2970 (all V_R_2 all T_b c_Transitive__Closure_Otrancl(T_b,c_Transitive__Closure_Ortrancl(T_b,V_R_2)) = c_Transitive__Closure_Ortrancl(T_b,V_R_2)) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2971 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),V_x)))))) # label(fact_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2972 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))))) # label(fact_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2973 (all V_m_2 c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2974 (all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_UN__empty2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2975 (all V_n V_n = hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_n))) # label(fact_nat__of__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2976 (all V_A_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))))) # label(fact_trancl__subset__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2977 (all V_l all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_l)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_l)))))) # label(fact_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2978 (all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2979 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_b,V_B_2))))))) # label(fact_Max_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2980 (all V_xs all V_n all V_m all T_a hAPP(c_List_Orotate(T_a,V_m),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_List_Orotate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_xs)) # label(fact_rotate__rotate) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2981 (all T_b c_Nitpick_Odenom(T_b) = hAPP(c_Fun_Ocomp(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,T_b,c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint)),c_Nitpick_ORep__Frac(T_b))) # label(fact_Nitpick_Odenom__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2982 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2))))))) # label(fact_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2983 (all V_x all V_i all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_i,V_x)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Oreplicate(T_a,V_i,V_x))) # label(fact_replicate__append__same) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2984 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w)) # label(fact_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2985 (all V_x_2 all V_xs_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_set__subset__Cons) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2986 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_inj__on__diff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2987 (all V_xs_2 all T_b (hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) -> hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)))) # label(fact_card__distinct) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2988 (all V_b_2 all V_a_2 all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_a_2)),V_r_2))))) # label(fact_symD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2989 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2990 (all V_zs_2 all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_ys_2),V_zs_2)),c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_zs_2)),c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2))))))) # label(fact_listrel__rtrancl__trans) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2991 (all V_k_2 all V_n_2 all V_P_2 ((V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hBOOL(hAPP(V_P_2,V_n_2))) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_P_2,B_j)))))) <-> hBOOL(hAPP(V_P_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n_2,V_k_2))))) # label(fact_split__mod) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2992 (all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> V_a = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_a))) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2993 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2994 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_div__by__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2995 (all V_v all V_u all T_b (class_Int_Onumber(T_b) & class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)))) # label(fact_max__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2996 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_min__max_OInf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2997 (all V_xs all V_m all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Otake(T_a),V_m),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_drop__take) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2998 (all V_t_2 all V_B_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_t_2))))))) # label(fact_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 2999 (all V_S_2 all V_R_2 all T_b (c_Wellfounded_Owf(T_b,V_R_2) -> (c_Wellfounded_Owf(T_b,V_S_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_S_2)),V_R_2)) -> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2)))))) # label(fact_wf__union__compatible) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3000 (all V_d_2 all T_b all V_P_2 ((-(exists B_x ((all B_y (hBOOL(hAPP(V_P_2,B_y)) -> B_y = B_x)) & hBOOL(hAPP(V_P_2,B_x)))) -> c_FunDef_OTHE__default(T_b,V_d_2,V_P_2) = V_d_2) & ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & (all B_y (hBOOL(hAPP(V_P_2,B_y)) -> B_y = B_x)))) -> c_HOL_OThe(T_b,V_P_2) = c_FunDef_OTHE__default(T_b,V_d_2,V_P_2)))) # label(fact_THE__default__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3001 (all V_x all T_a c_List_Oreplicate(T_a,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_x) = c_List_Olist_ONil(T_a)) # label(fact_replicate__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3002 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I18_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3003 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3004 (all V_R_2 all T_b (c_Wellfounded_Owf(T_b,V_R_2) -> c_Wellfounded_OwfP(T_b,c_FunDef_Oin__rel(T_b,T_b,V_R_2)))) # label(fact_wf__in__rel) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3005 (all V_c all V_b all V_a all T_a (class_Groups_Oordered__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))))) # label(fact_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3006 (all V_xss_2 all T_c all V_xs_2 all T_b hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),c_List_Ofoldr(tc_List_Olist(T_c),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_c)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_c)),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_List_Ofilter(tc_List_Olist(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_c)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_c)))),V_xss_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))),c_List_Ofoldr(tc_List_Olist(T_c),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_c)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xss_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_transpose__aux__max) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3007 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3008 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)))) # label(fact_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3009 (all V_B_2 all V_f_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_c),c_Fun_Oid(T_c)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))))) # label(fact_setprod__reindex__id) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3010 (all V_z_H all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)))) # label(fact_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3011 (all V_f_2 all V_R_2 all T_c all T_b hAPP(c_Relation_Oconverse(T_b,T_b),hAPP(hAPP(c_Relation_Oinv__image(T_c,T_b),V_R_2),V_f_2)) = hAPP(hAPP(c_Relation_Oinv__image(T_c,T_b),hAPP(c_Relation_Oconverse(T_c,T_c),V_R_2)),V_f_2)) # label(fact_converse__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3012 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_b))) # label(fact_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3013 (all V_g_2 all V_f_2 all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_List_Olist_Olist__size(T_b,V_f_2,V_xs_2)),c_List_Olist_Olist__size(T_b,V_g_2,V_xs_2))))) # label(fact_list__size__pointwise) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3014 (all V_n all V_m (V_m = V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3015 (all V_n all T_a (class_Rings_Olinordered__idom(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_n)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_abs__power__minus__one) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3016 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zle__int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3017 (all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> V_y_2 = hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_y_2)))) # label(fact_of__nat__inverse) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3018 (all V_n all V_m (c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n)) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_m))) # label(fact_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3019 (all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z)) # label(fact_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3020 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3021 (all V_A_2 all T_b c_Relation_Osingle__valued(T_b,T_b,c_Relation_OId__on(T_b,V_A_2))) # label(fact_single__valued__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3022 (all V_k all V_v ((V_k = V_v -> c_Random_Oinc__shift(V_v,V_k) = c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) & (V_v != V_k -> c_Random_Oinc__shift(V_v,V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_k),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))))) # label(fact_inc__shift__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3023 (all V_w_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w_2)))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))))) # label(fact_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3024 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2))))) # label(fact_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3025 (all T_c all T_b all V_x_2 all V_f_2 all V_b_2 (V_b_2 = hAPP(V_f_2,V_x_2) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_range__eqI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3026 (all V_f_2 all V_S_2 all V_T_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_c))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_S_2)))))) # label(fact_setprod__mono__one__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3027 (all V_xs all V_x all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_n)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) # label(fact_drop__Suc__Cons) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3028 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x))) # label(fact_compl__inf__bot) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3029 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zless__int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3030 (all V_x all V_m all V_n all T_a c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m),V_x) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),c_List_Oreplicate(T_a,V_m,V_x))) # label(fact_replicate__add) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3031 (all V_f_2 all V_n_2 all V_m_2 all T_b hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)),V_f_2) = hAPP(c_Fun_Ocomp(T_b,T_b,T_b,hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_m_2),V_f_2)),hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_n_2),V_f_2))) # label(fact_funpow__add) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3032 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_l_2))))) # label(fact_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3033 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2)),V_n_2))))) # label(fact_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3034 (all V_y_2 all V_x_2 all V_b_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_y_2))))))) # label(fact_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3035 (all V_k_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_k_2)))) # label(fact_Compl__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3036 (all V_p_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_p_2),V_r_2)) -> c_Wellfounded_Owf(T_b,V_p_2)))) # label(fact_wf__subset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3037 (all V_f_2 all V_A_2 all T_b (class_Groups_Ocomm__monoid__mult(T_b) & class_Groups_Ocomm__monoid__add(T_b) -> (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_b),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_b,tc_Nat_Onat),V_f_2),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))))) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3038 (all V_n all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)))),V_n) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m)),V_n)) # label(fact_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3039 (all V_y_2 all V_xs_2 all V_x_2 all T_b (V_x_2 = V_y_2 | hBOOL(hAPP(hAPP(c_List_Omember(T_b),V_xs_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_List_Omember(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_y_2)))) # label(fact_member__rec_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3040 (all T_1 class_Nat_Osize(tc_List_Olist(T_1))) # label(arity_List__Olist__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3041 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2))) # label(fact_subset__refl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3042 (all V_s_2 all V_r_2 all V_I_2 all T_e all T_c all T_d all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool)),V_I_2),V_r_2)),V_s_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBC(T_e,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),T_e),c_Relation_Orel__comp(T_b,T_d,T_c)),V_r_2)),V_s_2))) # label(fact_rel__comp__UNION__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3043 (all V_A_2 all T_b -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_not__psubset__empty) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3044 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))))) # label(fact_not__zle__0__negative) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3045 (all V_z_H_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H_2)) -> (V_z_H_2 = V_z_2 <-> hAPP(c_Int_Onat,V_z_2) = hAPP(c_Int_Onat,V_z_H_2))))) # label(fact_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3046 (all V_n all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) # label(fact_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3047 (all V_f_2 all V_r_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),B_x),V_r_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_c),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_c),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_c,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2))),V_f_2)),B_x)))) <-> c_Equiv__Relations_Ocongruent(T_b,T_c,V_r_2,V_f_2))) # label(fact_congruent__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3048 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Big__Operators_Ocomm__monoid__big(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_F_2,V_g_2),V_A_2) = V_z_2) & (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_F_2,V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),V_A_2)))) # label(fact_comm__monoid__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3049 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_b all T_c hAPP(hAPP(hAPP(c_New__Random__Sequence_Oneg__bind(T_c,T_b,V_R_2,V_f_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Oneg__bind(T_c,T_b,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b)))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b))))),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b)))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_neg__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3050 (all V_list_H all V_a_H all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H) != c_List_Olist_ONil(T_a)) # label(fact_list_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3051 (all V_c_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))))) # label(fact_subsetD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3052 (all V_A_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_A_2))) # label(fact_image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3053 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (hAPP(hAPP(c_Set_Oinsert(T_b),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2) <-> V_b_2 = V_a_2 & V_b_2 = V_c_2))) # label(fact_atLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3054 (all V_r_2 all V_f_2 all T_b ((all B_i hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(V_f_2,hAPP(c_Nat_OSuc,B_i))),hAPP(V_f_2,B_i))),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) -> (c_Wellfounded_Owf(T_b,c_Transitive__Closure_Otrancl(T_b,V_r_2)) -> (exists B_i all B_k hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),B_k)) = hAPP(V_f_2,B_i))))) # label(fact_wf__weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3055 (all V_B_2 all V_b_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)))))) # label(fact_SigmaI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3056 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a)))))) # label(fact_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3057 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (V_y_2 = V_x_2 <-> hAPP(c_Int_Onat,V_x_2) = hAPP(c_Int_Onat,V_y_2))))) # label(fact_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3058 (all V_y_2 all V_x_2 all V_f_2 all V_r_2 all T_c all T_b (c_Predicate_Oinv__imagep(T_b,T_c,V_r_2,V_f_2,V_x_2,V_y_2) <-> hBOOL(hAPP(hAPP(V_r_2,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))))) # label(fact_in__inv__imagep) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3059 (all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) # label(fact_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3060 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2) # label(fact_Int__absorb) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3061 (all V_b_2 all V_a_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),V_r_2))))) # label(fact_equiv__class__nondisjoint) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3062 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_b (c_Relation_Osingle__valued(T_b,T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_z_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_z_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))))) # label(fact_single__valued__confluent) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3063 (all V_f_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2)))) # label(fact_eq__card__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3064 (all V_m all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m)),V_m) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__minus__add__max) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3065 (all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_card__ge__0__finite) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3066 (all V_g_2 all T_d all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_B_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_A_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_A_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_F_2,V_B_2)),V_g_2)))))) # label(fact_folding_Ocommute__left__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3067 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_b)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b))) # label(fact_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3068 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_B_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Sigma__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3069 (all V_x_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) -> c_List_Olinorder__class_Osorted(T_b,c_List_Olinorder__class_Oinsort__insert__key(T_b,T_b,c_COMBI(T_b),V_x_2,V_xs_2))))) # label(fact_sorted__insort__insert) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3070 (all V_B_2 all V_A_2 all T_d all V_r_2 all T_b all T_c hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Relation_OImage(T_c,T_b,V_r_2)),V_B_2))) # label(fact_Image__UN) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3071 (all V_f_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) <-> hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3072 (all V_f_2 all V_X_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),hAPP(V_f_2,V_X_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_X_2),c_Inductive_Ocomplete__lattice__class_Ogfp(T_b,V_f_2)))))) # label(fact_gfp__upperbound) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3073 (all V_g_2 all V_f_2 all T_c all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_g_2)))))) # label(fact_INT__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3074 (all V_a_2 all V_b_2 all V_A_2 all T_b (class_Lattices_Osemilattice__sup(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_b_2),V_A_2))))) # label(fact_fold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3075 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg(V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3076 (all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),V_B_2)) -> (c_Fun_Oinj__on(T_c,T_b,V_g_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_g_2),V_B_2)),V_A_2)) -> (exists B_h c_Fun_Obij__betw(T_b,T_c,B_h,V_A_2,V_B_2))))))) # label(fact_Cantor__Bernstein) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3077 (all V_j_2 all V_i_2 all V_P_2 (c_List_Oall__interval__int(V_P_2,V_i_2,V_j_2) <-> c_List_Olist__all(tc_Int_Oint,V_P_2,c_List_Oupto(V_i_2,V_j_2)))) # label(fact_list__all__iff__all__interval__int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3078 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> V_y_2 = V_x_2))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3079 (all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3080 (all V_F_2 all V_h_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_Ovimage(T_b,T_c,V_h_2),V_F_2))) -> (hAPP(c_Set_Oimage(T_b,T_c,V_h_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2))))) # label(fact_finite__vimageD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3081 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) <-> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3082 (all V_xq_2 all V_s_2 all T_b c_Option_Ooption_Ooption__case(tc_Nat_Onat,tc_prod(T_b,tc_Lazy__Sequence_Olazy__sequence(T_b)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),hAPP(hAPP(c_COMBC(T_b,tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat)),T_b),c_COMBC(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),T_b),hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Lazy__Sequence_Olazy__sequence(T_b)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat)),T_b),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Lazy__Sequence_Olazy__sequence(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),T_b),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_s_2))),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_b,V_s_2))))),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_Lazy__Sequence_Oyield(T_b),V_xq_2)) = hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_b,V_s_2),V_xq_2)) # label(fact_lazy__sequence__size__code) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3083 (all V_v all V_u all T_b (class_Orderings_Oord(T_b) & class_Int_Onumber(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))))) # label(fact_min__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3084 (all V_xs_2 all T_b all V_x_2 all V_P_2 ((hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) & (-hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2) = hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_filter_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3085 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3086 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 3087 (all V_b all V_a all V_c all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_mult__divide__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3088 (all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_abs__one) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3089 (all V_w all V_z all V_x all V_y all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_z),V_w)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_x),V_y)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_w),V_z)))))) # label(fact_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3090 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_not__less0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3091 (all V_A_2 all V_l_H_2 all V_l_2 all T_b c_List_Osublist(T_b,hAPP(hAPP(c_List_Oappend(T_b),V_l_2),V_l_H_2),V_A_2) = hAPP(hAPP(c_List_Oappend(T_b),c_List_Osublist(T_b,V_l_2,V_A_2)),c_List_Osublist(T_b,V_l_H_2,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_l_2)))),V_A_2))))) # label(fact_sublist__append) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3092 (all V_x_2 all V_f_2 all T_b all T_c all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k_2 -> hAPP(c_Random_Oiterate(T_c,T_b,V_k_2,V_f_2),V_x_2) = hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2)) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_k_2 -> hAPP(c_Random_Oiterate(T_c,T_b,V_k_2,V_f_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_c,T_b,tc_prod(T_c,T_b)),hAPP(V_f_2,V_x_2)),c_Random_Oiterate(T_c,T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_k_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),V_f_2))))) # label(fact_iterate_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3093 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3094 (all V_P_2 all V_n_2 ((all B_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2)) -> hBOOL(hAPP(V_P_2,B_m)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3095 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z)) # label(fact_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3096 (all V_u_2 all V_l_2 hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2))) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_2)) # label(fact_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3097 (all V_ys_2 all V_xs_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),V_ys_2))) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_ys_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_ys_2))) -> (hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(c_List_Oset(T_b),V_ys_2) -> V_xs_2 = V_ys_2)))))))) # label(fact_map__sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3098 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> V_a_2 = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_Osup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3099 (all V_s_2 all V_r_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Otrancl(T_b,V_s_2)))))) # label(fact_trancl__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3100 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)))) # label(fact_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3101 (all V_r_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_r_2) = c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))) # label(fact_rtrancl__r__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3102 (all V_r_2 all V_A_2 all T_c all V_S_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) -> c_Relation_Orefl__on(T_c,hAPP(V_A_2,B_x),hAPP(V_r_2,B_x)))) -> c_Relation_Orefl__on(T_c,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_S_2),V_r_2)))) # label(fact_refl__on__INTER) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3103 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3104 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3105 (all V_f_2 all V_m_2 all V_n_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))),hAPP(V_f_2,hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_setsum__cl__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3106 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3107 (all T_2 all T_1 (class_Orderings_Obot(T_1) -> class_Orderings_Obot(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Obot) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3108 (all V_s_2 all V_r_2 all T_c all T_d all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_r_2),V_s_2) = hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_b),c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_b),c_COMBC(T_d,tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_b,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_d),tc_fun(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_b,T_d)))),c_Product__Type_OPair(T_b,T_d)))),V_r_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_d,T_c)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_d,T_c),tc_fun(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_d,T_c)))),hAPP(c_COMBC(T_d,T_c,tc_prod(T_d,T_c)),c_Product__Type_OPair(T_d,T_c))))),V_s_2)))))) # label(fact_rel__comp__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3109 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l))) # label(fact_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3110 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3111 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b c_List_OtakeWhile(tc_prod(T_b,T_c),hAPP(c_Fun_Ocomp(T_b,tc_HOL_Obool,tc_prod(T_b,T_c),V_P_2),c_Product__Type_Ofst(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Ozip(T_b,T_c),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)),V_ys_2)) # label(fact_zip__takeWhile__fst) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3112 (all V_r_2 all V_q_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_b),c_Nitpick_Oplus__frac(T_b,V_q_2,c_Nitpick_Ouminus__frac(T_b,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> c_Nitpick_Oless__eq__frac(T_b,V_q_2,V_r_2))) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3113 (all V_xs_2 all V_P_2 all T_b hAPP(c_List_Oset(T_b),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),hAPP(c_List_Oset(T_b),V_xs_2)))),V_P_2))) # label(fact_set__filter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3114 (all V_j all V_i c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Oupt(V_i,V_j))) # label(fact_sorted__upt) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3115 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2))))))) # label(fact_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3116 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2) = hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2))))) # label(fact_lfp__unfold) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3117 (all V_Y_2 all T_b V_Y_2 = hAPP(c_Set_Oimage(T_b,T_b,c_COMBI(T_b)),V_Y_2)) # label(fact_image__ident) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3118 (all V_n all V_m hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odivmod__nat(V_m,V_n)) # label(fact_divmod__nat__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3119 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> V_A_2 = hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_inj__vimage__image__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3120 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = V_x_2 & c_Groups_Ozero__class_Ozero(T_b) = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_y_2),V_y_2))),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3121 (all V_B_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2))))) # label(fact_Un__least) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3122 (all V_w hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),V_w) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) # label(fact_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3123 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_u_2)))) # label(fact_ivl__disj__int_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3124 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)),V_c)))) # label(fact_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3125 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_div__by__1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3126 (all T_b c_Wellfounded_Owf(tc_fun(T_b,tc_HOL_Obool),c_Wellfounded_Ofinite__psubset(T_b))) # label(fact_wf__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3127 (all V_xs_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oremdups(T_b,V_xs_2)) <-> V_xs_2 = c_List_Oremdups(T_b,V_xs_2))) # label(fact_length__remdups__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3128 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool)),c_List_Oset(T_b)),c_Enum_Osublists(T_b,V_xs_2)))))) # label(fact_distinct__set__sublists) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3129 (all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) <-> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_card__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3130 (all V_xs all T_a c_List_Osplice(T_a,V_xs,c_List_Olist_ONil(T_a)) = V_xs) # label(fact_splice__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3131 (all V_c_2 all V_b_2 all V_A_2 all T_b (class_Lattices_Osemilattice__sup(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_c_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_b_2),V_c_2))))))) # label(fact_fold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3132 (all V_r_2 all T_b (c_Nitpick_Owf_H(T_b,V_r_2) <-> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) | c_Nitpick_Ounknown(tc_HOL_Obool)) & c_Wellfounded_Oacyclic(T_b,V_r_2))) # label(fact_Nitpick_Owf_H__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3133 (all V_k_2 all V_N_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),B_x)))) -> c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_k_2),V_N_2))) # label(fact_inj__on__diff__nat) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3134 (all V_c all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_c),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_b),V_c)))))) # label(fact_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3135 (all V_A_2 all V_f_2 all T_b hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),V_A_2) = c_HOL_OThe(T_b,c_Finite__Set_Ofold1Set(T_b,V_f_2,V_A_2))) # label(fact_fold1__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3136 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3137 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Osgn__class_Osgn(T_a,V_a) = c_Groups_Osgn__class_Osgn(T_a,c_Groups_Osgn__class_Osgn(T_a,V_a)))) # label(fact_sgn__sgn) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3138 (all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3139 (all V_s_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(V_F_2,V_A_2),V_s_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_s_2),V_A_2)))) # label(fact_folding_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3140 (all V_r_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel1(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_listrel1__rtrancl__subset__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3141 (all V_x_1 all T_a hAPP(c_Fun_Oid(T_a),V_x_1) = V_x_1) # label(fact_id__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3142 (all V_A_2 all V_f_2 all T_b hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_b),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_A_2))) # label(fact_int__setsum) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3143 (all V_r_2 all V_q_2 all V_b_2 all V_a_2 (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a_2,V_b_2),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_2),V_r_2))) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),V_b_2),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(hAPP(c_If(tc_Int_Oint),hAPP(hAPP(c_fequal,V_r_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_q_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_q_2)),c_Groups_Oone__class_Oone(tc_Int_Oint)))),hAPP(hAPP(hAPP(c_If(tc_Int_Oint),hAPP(hAPP(c_fequal,V_r_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b_2),V_r_2))))))) # label(fact_zminus1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3144 (all T_b c_Relation_Osingle__valued(T_b,T_b,c_Relation_OId(T_b))) # label(fact_single__valued__Id) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3145 (all V_y_2 all V_r_2 all T_b ((all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_y_2),V_y_2)),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__irreflexive) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3146 (all V_n all V_m all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),V_l)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_l) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3147 (all V_p_2 all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_p_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_p_2)),hAPP(hAPP(c_List_Onth(T_c),V_ys_2),V_p_2)))))) # label(fact_list__all2__nthD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3148 (all V_r_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),c_List_Olist_ONil(T_b))),c_List_Olistrel(T_b,V_r_2))) -> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_listrel__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3149 (all V_n_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_b),V_f_2),c_List_Oupt(V_m_2,V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_m_2,V_n_2))))) # label(fact_interv__listsum__conv__setsum__set__nat) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3150 (all V_f_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_F_2)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),V_f_2),V_F_2)))) # label(fact_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3151 (all V_A_2 all V_C_2 all T_b ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_X),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),B_X)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_C_2))))) # label(fact_InterI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3152 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> V_a = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3153 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))))) # label(fact_rtrancl__listrel1__if__listrel) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3154 (all V_B_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(hAPP(V_f_2,hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__image__simple_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3155 (all V_prod_2 all V_i_2 all V_h_2 all T_g all T_f all V_g_2 all V_f_2 all T_c all T_e all T_b all T_d hAPP(c_Product__Type_Omap__pair(T_d,T_b,T_e,T_c,V_f_2,V_g_2),hAPP(c_Product__Type_Omap__pair(T_f,T_d,T_g,T_e,V_h_2,V_i_2),V_prod_2)) = hAPP(c_Product__Type_Omap__pair(T_f,T_b,T_g,T_c,hAPP(c_Fun_Ocomp(T_d,T_b,T_f,V_f_2),V_h_2),hAPP(c_Fun_Ocomp(T_e,T_c,T_g,V_g_2),V_i_2)),V_prod_2)) # label(fact_map__pair_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3156 (all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3157 (all V_d all V_b all V_c all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_d))))))) # label(fact_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3158 (all V_a_2 all T_b c_FunDef_Ois__measure(T_b,V_a_2)) # label(fact_is__measure_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3159 (all V_B_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_B_2),c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2))))) # label(fact_sup__Sup__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3160 (all V_w all V_z hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3161 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2) = V_n_2))) # label(fact_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3162 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> c_Nat__Transfer_Onat__set(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3163 (all V_k all V_x all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),c_List_Oreplicate(T_a,V_k,V_x)) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_k,V_x)),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_append__replicate__commute) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3164 (all V_w all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3165 (all V_k all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_k)),V_j) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3166 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Nat__Transfer_Otsub(V_x,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3167 (all V_n all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)))) # label(fact_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3168 (all V_z_2 all V_w_2 all T_b (class_Int_Oring__char__0(T_b) -> (V_z_2 = V_w_2 <-> hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_w_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)))) # label(fact_of__int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3169 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_m))))) # label(fact_diff__less) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3170 (all T_d all V_f_2 all V_r2_2 all T_c all V_r1_2 all T_b ((all B_y1 all B_z1 (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y1),B_z1)),V_r1_2)) -> (all B_y2 all B_z2 (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y2),B_z2)),V_r2_2)) -> hAPP(hAPP(V_f_2,B_z1),B_z2) = hAPP(hAPP(V_f_2,B_y1),B_y2))))) -> c_Equiv__Relations_Ocongruent2(T_b,T_c,T_d,V_r1_2,V_r2_2,V_f_2))) # label(fact_congruent2I_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3171 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) # label(fact_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3172 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_ORange(T_b,T_c),V_r_2))))) # label(fact_RangeI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3173 (all V_y all V_x hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_x),V_y))) # label(fact_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3174 (all V_w all V_z2 all V_z1 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_w)) # label(fact_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3175 (all V_y all V_x (c_Nat__Transfer_Ois__nat(V_x) -> (c_Nat__Transfer_Ois__nat(V_y) -> c_Nat__Transfer_Ois__nat(hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y))))) # label(fact_Divides_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3176 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = V_a_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3177 (all V_A_2 all V_f_2 all T_c all V_r_2 all T_b (class_Rings_Osemiring__0(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_r_2),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),hAPP(c_Groups_Otimes__class_Otimes(T_b),V_r_2)),V_f_2)),V_A_2))) # label(fact_setsum__right__distrib) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3178 (all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_c) & class_Enum_Oenum(T_b) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_b),hAPP(hAPP(c_COMBS(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_Orderings_Oord__class_Oless__eq(T_c)),V_f_2)),V_g_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))))) # label(fact_order__fun_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3179 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_compl__sup) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3180 (all V_S_2 all V_R_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_R_2)),c_Transitive__Closure_Ortrancl(T_b,V_S_2))),c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_rtrancl__Un__subset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3181 (all V_b_2 all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),V_b_2) = c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_Sup__binary) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3182 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),c_Groups_Oone__class_Oone(tc_Int_Oint)) = V_z) # label(fact_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3183 (all V_Q_2 all V_P_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_P_2))) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_Q_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2)),V_Q_2)))))) # label(fact_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.92 3184 (all V_a all V_r all V_q all V_c all V_b (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_c -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r),V_c))))))) # label(fact_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3185 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),V_j))) # label(fact_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3186 (all V_r_2 all V_x_2 all V_y_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b))),hAPP(hAPP(c_COMBC(T_b,T_b,tc_prod(T_b,T_b)),c_Product__Type_OPair(T_b,T_b)),V_y_2))),c_Transitive__Closure_Ortrancl(T_b,V_r_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b))),hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2))),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))))) = c_Transitive__Closure_Otrancl(T_b,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_x_2)),V_r_2))) # label(fact_trancl__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3187 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin))) # label(fact_number__of__Min) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3188 (all V_xs_2 all V_x_2 all V_Q_2 all T_b hAPP(c_List_Ofilter(T_b,V_Q_2),c_List_Oremove1(T_b,V_x_2,V_xs_2)) = c_List_Oremove1(T_b,V_x_2,hAPP(c_List_Ofilter(T_b,V_Q_2),V_xs_2))) # label(fact_filter__remove1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3189 (all V_X_2 all V_a_2 all V_P_2 all T_b all V_A_2 (c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_OCollect(T_b)),V_P_2)) = V_A_2 -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_OCollect(T_b)),V_P_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> ((all B_z (hBOOL(hAPP(hAPP(c_member(T_b),B_z),V_X_2)) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_A_2)),B_z)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))))) # label(fact_def__Collect__coinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3190 (all V_n all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))) # label(fact_power__minus__even) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3191 (all V_A_2 all T_b hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),c_List_Olist_ONil(T_b)))) # label(fact_listsp_ONil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 3192 (all T_b all T_c all V_y_2 all V_x_2 all V_f_2 (V_y_2 = hAPP(V_f_2,V_x_2) -> c_Fun_Ofun__upd(T_c,T_b,V_f_2,V_x_2,V_y_2) = V_f_2)) # label(fact_fun__upd__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3193 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_image__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3194 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)))) # label(fact_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3195 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) -> ((all B_M (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_M),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_M)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2)))))) # label(fact_finite__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3196 (all V_s_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> (c_Relation_Osingle__valued(T_b,T_c,V_s_2) -> c_Relation_Osingle__valued(T_b,T_c,V_r_2)))) # label(fact_single__valued__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3197 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> (hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2) <-> V_B_2 = V_A_2)))) # label(fact_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3198 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2))))))))) # label(fact_Min_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3199 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3200 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)))))))) # label(fact_Sup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3201 (all V_c all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b)))))))) # label(fact_divide__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3202 (all V_r_2 all V_zb_2 all V_za_2 all V_xb_2 all V_xa_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_xa_2),V_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2))) -> (hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_xa_2),V_xb_2) != hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_za_2),V_zb_2) -> -(all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_xa_2),V_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b))),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2)))))))) # label(fact_converse__rtranclE2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3203 (all V_a_2 all V_b_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(V_B_2,V_a_2),V_b_2))))) # label(fact_INF1__D) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3204 (all V_P_2 all V_Q_2 (-hBOOL(V_Q_2) | hBOOL(hAPP(hAPP(c_fdisj,V_P_2),V_Q_2)))) # label(help_c__fdisj__2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3205 (all V_xs all V_n all T_a c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs))) # label(fact_butlast__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3206 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2)),V_A_2))))) # label(fact_vimage__subsetI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3207 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_ys))) # label(fact_take__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3208 (all V_x all V_k all V_i all T_a hAPP(hAPP(c_List_Odrop(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_k),V_i),V_x)) # label(fact_drop__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3209 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_Olinorder__class_Osorted__list__of__set(T_b,V_A_2))) & c_List_Olinorder__class_Osorted(T_b,c_List_Olinorder__class_Osorted__list__of__set(T_b,V_A_2)) & V_A_2 = hAPP(c_List_Oset(T_b),c_List_Olinorder__class_Osorted__list__of__set(T_b,V_A_2))))) # label(fact_sorted__list__of__set) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3210 (all V_r_2 all T_b c_Transitive__Closure_Otrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) = hAPP(c_Relation_Oconverse(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2))) # label(fact_trancl__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3211 (all V_xs_2 all V_a_2 all T_b (class_Groups_Omonoid__add(T_b) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),c_List_Ofoldr(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_xs_2,c_Groups_Ozero__class_Ozero(T_b))) = hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_xs_2))) # label(fact_foldl__foldr1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3212 (all T_c all T_b c_Sum__Type_OPlus(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_b,T_c),tc_HOL_Obool))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3213 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3214 (all V_l_2 (c_Int_OBit1(V_l_2) = c_Int_OMin <-> V_l_2 = c_Int_OMin)) # label(fact_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3215 (all V_P_2 all V_Q_2 (hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)) | -hBOOL(V_Q_2))) # label(help_c__fimplies__2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3216 (all T_d all V_f_2 all V_Q_2 all V_p_2 all T_b all T_c all V_P_2 (hBOOL(hAPP(hAPP(V_P_2,hAPP(c_Product__Type_Osnd(T_c,T_b),V_p_2)),hAPP(c_Product__Type_Ofst(T_c,T_b),V_p_2))) -> ((all B_x all B_y (hBOOL(hAPP(hAPP(V_P_2,B_y),B_x)) -> hBOOL(hAPP(V_Q_2,hAPP(hAPP(V_f_2,B_x),B_y))))) -> hBOOL(hAPP(V_Q_2,hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_b,T_d),V_f_2),V_p_2)))))) # label(fact_exE__realizer) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3217 (all V_a all V_b all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_c),V_b)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c)))) # label(fact_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3218 (all V_I_2 all V_xs_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Osublist(T_b,V_xs_2,V_I_2)))))) # label(fact_notin__set__sublistI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3219 (all V_N_2 ((exists B_m all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_x),B_m)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2)))) # label(fact_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3220 (all T_b (class_Orderings_Olinorder(T_b) -> c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)))) # label(fact_min__max_Ofun__left__comm__idem__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3221 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_bij__betw__imp__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3222 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)))) # label(fact_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3223 (all V_m_2 all V_x_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_x_2 | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3224 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))))) # label(fact_le__maxI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3225 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3226 (all T_b (class_Orderings_Oord(T_b) -> c_Orderings_Oord__class_Omax(T_b) = c_Orderings_Oord_Omax(T_b,c_Orderings_Oord__class_Oless__eq(T_b)))) # label(fact_max__ord__max) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3227 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3228 (all V_a all V_y all V_x all T_a all T_b (V_a = hAPP(c_Product__Type_Osnd(T_b,T_a),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_x),V_y)) -> V_y = V_a)) # label(fact_snd__eqD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3229 (all V_n_2 all T_b (class_Rings_Osemiring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)),c_Nat_Osemiring__1__class_ONats(T_b))))) # label(fact_of__nat__in__Nats) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3230 (all V_b_2 all V_B_2 all V_A_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> -hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)))))) # label(fact_SUP1__E) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3231 (all V_c all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c))))))) # label(fact_divide__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3232 (all V_k c_Int_Osucc(V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OMin)) # label(fact_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3233 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Otake(T_a),V_n),c_List_Obutlast(T_a,V_xs)) = hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))) # label(fact_take__butlast) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3234 (all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> c_Relation_Osym(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) # label(fact_sym__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3235 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_List_Olinorder__class_Osorted__list__of__set(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),c_List_Olinorder__class_Osorted__list__of__set(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_sorted__list__of__set__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3236 (all T_b c_List_Olists(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)))) # label(fact_lists__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3237 (all V_d_2 c_Wellfounded_Owf(tc_Int_Oint,c_Int_Oint__ge__less__than2(V_d_2))) # label(fact_wf__int__ge__less__than2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3238 (all V_p_2 all V_g_2 all T_f all T_e all V_f_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,tc_prod(T_c,T_d)),V_g_2),V_p_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,T_b),hAPP(hAPP(c_COMBB(tc_fun(T_f,tc_prod(T_c,T_d)),tc_fun(T_f,T_b),T_e),hAPP(c_COMBB(tc_prod(T_c,T_d),T_b,T_f),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_f_2))),V_g_2)),V_p_2)) # label(fact_split__twice) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3239 (all V_N_2 all V_h_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMax(T_b,V_N_2))))))) # label(fact_Max_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3240 (all V_y all V_x all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_y)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__sum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3241 (all T_b hAPP(c_Set_OCollect(T_b),hAPP(c_COMBK(tc_HOL_Obool,T_b),c_fFalse)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_empty__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3242 (all V_f_2 all T_b all T_c hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_image__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3243 (all V_k_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_k_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_k_2,V_l_2)))) # label(fact_greaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3244 (all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Compl__UNIV__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3245 (all V_a_2 all V_b_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),V_A_2))))) # label(fact_min__max_Ofold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3246 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_one__power2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3247 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)) & (c_Groups_Ozero__class_Ozero(T_a) = V_c -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3248 (all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs))) = V_xs)) # label(fact_id__take__nth__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3249 (all V_k_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_k_2))))) # label(fact_finite__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3250 (all V_xq_2 all T_b c_Option_Ooption_Ooption__case(tc_Nat_Onat,tc_prod(T_b,tc_Lazy__Sequence_Olazy__sequence(T_b)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),hAPP(c_COMBK(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat),T_b),hAPP(hAPP(c_COMBC(tc_Lazy__Sequence_Olazy__sequence(T_b),tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Lazy__Sequence_Olazy__sequence(T_b)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_b)))),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(c_Lazy__Sequence_Oyield(T_b),V_xq_2)) = hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_b)),V_xq_2)) # label(fact_size__code) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3251 (all V_m_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_m_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_m_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_atLeastLessThan__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3252 (all V_A_2 all T_c all V_I_2 all V_k_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_INT__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3253 (all V_a_2 all V_b_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2)))))) # label(fact_pos__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3254 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3255 (all V_x hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_x))) # label(fact_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3256 (all V_xs all V_x all V_n all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_take__Suc__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3257 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Groups_Ocancel__semigroup__add(T_b) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_c_2) <-> V_b_2 = V_c_2))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3258 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3259 (all V_A_2 all V_a_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3260 (all V_list_2 all V_a_2 all V_fa_2 all T_b c_List_Olist_Olist__size(T_b,V_fa_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_2),V_list_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_fa_2,V_a_2)),c_List_Olist_Olist__size(T_b,V_fa_2,V_list_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_list_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3261 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3262 (all V_u_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_u_2)) = V_u_2) # label(fact_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3263 (all V_zs_2 all V_y_2 all V_x_2 all T_b (class_Groups_Osemigroup__add(T_b) -> hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_x_2),V_y_2)),V_zs_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_x_2),hAPP(c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),V_y_2),V_zs_2)))) # label(fact_foldl__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3264 (all V_T_2 all V_S_2 all V_R_2 all T_e all T_c all T_d all T_b hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_e,T_d),V_R_2),V_S_2)),V_T_2) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_e,T_c),V_R_2),hAPP(hAPP(c_Relation_Orel__comp(T_e,T_d,T_c),V_S_2),V_T_2))) # label(fact_O__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3265 (all V_M_2 all V_A_2 all V_i_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),V_M_2)),hAPP(V_M_2,V_i_2)))))) # label(fact_INF__leI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3266 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> (c_Groups_Ozero__class_Ozero(T_a) != V_z -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_w),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)))))) # label(fact_diff__frac__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3267 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3268 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))))) # label(fact_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3269 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_m = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3270 (all V_c_2 all V_b_2 all V_a_2 (V_b_2 = V_a_2 & V_b_2 = V_c_2 <-> c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_b_2) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_dvd_OatLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3271 (all V_m_2 all V_n_2 (V_n_2 = V_m_2 <-> hAPP(c_Code__Numeral_Oof__nat,V_n_2) = hAPP(c_Code__Numeral_Oof__nat,V_m_2))) # label(fact_Code__Numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3272 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield__inverse__zero(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),V_x)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_x))) # label(fact_minus1__divide) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3273 (all V_ys_2 all V_y_2 all V_xs_2 all V_P_2 all T_b (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2)) & -hBOOL(hAPP(V_P_2,V_y_2)) <-> hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2) = c_List_OdropWhile(T_b,V_P_2,V_xs_2))) # label(fact_dropWhile__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3274 (all V_k all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_n),V_k))))) # label(fact_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3275 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3276 (all V_j all V_i ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> c_List_Oupto(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),V_i),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> c_List_Oupto(V_i,V_j) = c_List_Olist_ONil(tc_Int_Oint)))) # label(fact_upto_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3277 (all V_a_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_2)),c_Relation_OId(T_b)))) # label(fact_IdI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3278 (all V_P_2 all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),V_P_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),V_P_2))) # label(fact_Collect__neg__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3279 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3280 (all V_ys all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),c_List_Olist_ONil(T_a)),V_ys) = c_List_Olist_ONil(tc_prod(T_a,T_b))) # label(fact_zip__Nil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3281 (all V_a_2 all V_r_2 all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2)) <-> (all B_x (hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),B_x)))))) # label(fact_accp_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3282 (all V_y_2 all V_f_2 all T_c all T_b all V_x_2 all V_z_2 (V_x_2 != V_z_2 -> hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2))) # label(fact_fun__upd__other) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3283 (all V_m all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_k)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),V_k) = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3284 (all V_b_2 all V_a_2 all V_P_2 (hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2))) <-> (all B_d (V_a_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d) -> hBOOL(hAPP(V_P_2,B_d)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2)) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3285 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> (hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2))))))) # label(fact_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3286 (all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_prod(T_a,T_b))),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))) # label(fact_length__zip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3287 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))))) # label(fact_diff__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3288 (all V_d all T_b all T_a (class_Lazy__Sequence_Osmall__lazy(T_b) & class_Lazy__Sequence_Osmall__lazy(T_a) -> c_Lazy__Sequence_Oproduct(T_a,T_b,c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_a,V_d),c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_b,V_d)) = c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_prod(T_a,T_b),V_d))) # label(fact_small__lazy__prod__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3289 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))) # label(fact_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3290 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_power__minus1__odd) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3291 (all V_qr all V_n all V_m (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),V_qr)) -> V_qr = c_Divides_Odivmod__nat(V_m,V_n))) # label(fact_divmod__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3292 (all V_C_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) = V_A_2))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3293 (all V_x all V_n all T_a V_n = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_length__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3294 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2))) -> -(hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2)))))) # label(fact_SigmaE2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3295 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b))))) # label(fact_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3296 (all V_r_2 all V_A_2 all T_b (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_quotient__is__empty2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3297 (all V_z all V_y all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_y)),V_z) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_y),V_z))) # label(fact_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3298 (all V_g_2 all V_x_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3299 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))))) # label(fact_le__imp__neg__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3300 (all V_D_2 all V_B_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Int__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3301 (all V_A_2 all T_b hAPP(c_Relation_OImage(T_b,T_b,c_Relation_OId(T_b)),V_A_2) = V_A_2) # label(fact_Image__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3302 (all V_x all V_y all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_x = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)))) # label(fact_min__max_Osup__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3303 (all V_w_2 all V_c_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))))) # label(fact_divide__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3304 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) -> hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs) = V_xs)) # label(fact_take__all) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3305 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_a_2))))) # label(fact_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3306 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> V_r_2 = c_Transitive__Closure_Otrancl(T_b,V_r_2))) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3307 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3308 (all V_P_2 hAPP(c_Set_OCollect(tc_Nat_Onat),V_P_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Int_Oint),V_P_2),c_Int_Onat))))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3309 (all T_c all T_b all V_b_2 all V_a_2 all V_r_2 (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_b,T_c,V_r_2),V_b_2),V_a_2)))) # label(fact_conversep_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3310 (all V_h_2 all V_g_2 all V_f_2 all T_g all T_f all T_c all T_e all T_d all T_b hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_d,T_e,T_c),hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_f,T_g,tc_prod(T_d,T_e)),V_f_2),V_g_2)),V_h_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_f,T_g,T_c),V_f_2),hAPP(hAPP(c_COMBC(T_f,tc_fun(T_d,tc_fun(T_e,T_c)),tc_fun(T_g,T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_g,tc_prod(T_d,T_e)),tc_fun(tc_fun(T_d,tc_fun(T_e,T_c)),tc_fun(T_g,T_c)),T_f),c_Product__Type_Oscomp(T_g,T_d,T_e,T_c)),V_g_2)),V_h_2))) # label(fact_scomp__scomp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3311 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2))) # label(fact_vimage__Diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3312 (all V_B_2 all V_A_2 all T_c all V_C_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_C_2)) -> (hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_B_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_C_2)) <-> V_B_2 = V_A_2))) # label(fact_Times__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3313 (all T_a c_List_Otranspose(T_a,c_List_Olist_ONil(tc_List_Olist(T_a))) = c_List_Olist_ONil(tc_List_Olist(T_a))) # label(fact_transpose_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3314 (all V_br all V_bq all V_b all V_ar all V_aq all V_c all V_a (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_aq),V_ar))) -> (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_bq),V_br))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__nat__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_ar),V_br),V_c)))))))) # label(fact_divmod__nat__rel__add1__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3315 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2) = c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OAll(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fimplies),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),c_Orderings_Oord__class_Oless__eq(T_b))))))) # label(fact_Inf__Sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3316 (all V_a_2 all V_b_2 all T_b (V_a_2 = V_b_2 <-> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3317 (all V_w_2 all V_z_2 (V_z_2 != V_w_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),V_w_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),V_w_2)))) # label(fact_zless__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3318 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2))))))) # label(fact_Max__ge) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3319 (all V_m_2 all V_w_2 (hAPP(c_Int_Onat,V_w_2) = V_m_2 <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> V_w_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3320 (all V_g_2 all T_d all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_c,T_d,V_g_2,hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)) -> c_Fun_Oinj__on(T_b,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_g_2),V_f_2),V_A_2)))) # label(fact_comp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3321 (all V_S_2 all V_R_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_x_2),c_Transitive__Closure_Ortrancl(T_b,V_R_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_x_2),c_Transitive__Closure_Ortrancl(T_b,V_S_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_x_2),c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2)))))) # label(fact_in__rtrancl__UnI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3322 (all V_R_2 all T_c all T_b hAPP(c_Relation_ODomain(T_b,T_c),V_R_2) = hAPP(c_Set_Oimage(tc_prod(T_b,T_c),T_b,c_Product__Type_Ofst(T_b,T_c)),V_R_2)) # label(fact_fst__eq__Domain) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3323 (all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3324 (all V_f_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> c_Nat__Transfer_Ois__nat(hAPP(V_f_2,B_x)))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,T_b),c_Int_Onat),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Int_Oint),V_f_2),V_A_2))) # label(fact_transfer__int__nat__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3325 (all V_r_2 all V_ys_2 all V_xs_2 all T_b ((exists B_y exists B_n (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_n)),B_y)),V_r_2)) & V_ys_2 = hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),B_n),B_y) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_listrel1__iff__update) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 3326 (all V_A_2 all T_b (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> (all B_x -hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))))) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3327 (all T_b all V_xs_2 (c_List_Olist_ONil(T_b) != V_xs_2 -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> c_List_OtakeWhile(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),c_List_Olast(T_b,V_xs_2))),V_xs_2) = c_List_Obutlast(T_b,V_xs_2)))) # label(fact_takeWhile__not__last) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3328 (all V_A_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))) -> -c_Fun_Oinj__on(T_c,T_b,V_f_2,V_A_2))) # label(fact_pigeonhole) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3329 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_lessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3330 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Olattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Olattice) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3331 (all T_1 class_Nat_Osize(tc_Hoare__Mirabelle_Otriple(T_1))) # label(arity_Hoare____Mirabelle__Otriple__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3332 (all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> V_x_2 = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_min__max_OInf__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3333 (all V_y all V_x hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)),c_Nitpick_Onat__gcd(V_x,V_y)) = c_Nitpick_Onat__lcm(V_x,V_y)) # label(fact_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3334 (all V_A_2 all V_B_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3335 (all V_n all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin))) # label(fact_power__m1__odd) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3336 (all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xs_2))) -> c_List_Olist_ONil(T_b) = B_x)) <-> c_List_Olist_ONil(tc_List_Olist(T_b)) = c_List_Otranspose(T_b,V_xs_2))) # label(fact_transpose__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3337 (all V_m_2 all V_n_2 c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_n_2,V_m_2) = hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(hAPP(c_Nat_OSuc,V_n_2),V_m_2))) # label(fact_greaterThanLessThan__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3338 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3339 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_List_Olistsp(T_b,hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_A_2)),V_B_2)),V_x_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_b,V_A_2),V_x_2)) & hBOOL(hAPP(c_List_Olistsp(T_b,V_B_2),V_x_2)))) # label(fact_listsp__conj__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3340 (all V_xs all V_x all T_a all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x))) -> hAPP(c_Random_Opick(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_prod(tc_Code__Numeral_Ocode__numeral,T_a)),V_x),V_xs)),V_i) = hAPP(c_Random_Opick(T_a,V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x))) -> hAPP(c_Random_Opick(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_prod(tc_Code__Numeral_Ocode__numeral,T_a)),V_x),V_xs)),V_i) = hAPP(c_Product__Type_Osnd(tc_Code__Numeral_Ocode__numeral,T_a),V_x)))) # label(fact_pick_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3341 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3342 (all V_B_2 all V_a_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Set_Oinsert(T_b),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2))) # label(fact_image__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3343 (all V_ys_2 all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2))))))) # label(fact_rtrancl__listrel1__ConsI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3344 (all V_a3_2 all V_a2_2 all V_a1_2 all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_b,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a1_2),V_a2_2)),V_a3_2)) <-> (exists B_a exists B_A (hAPP(hAPP(c_Set_Oinsert(T_b),V_a1_2),V_a2_2) = hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_A) & -hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_A)) & hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,V_f_2,B_a,B_A),V_a3_2)))))) # label(fact_fold1Set_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3345 (all V_xs_2 all V_P_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_OtakeWhile(T_b,V_P_2,V_xs_2) = V_xs_2)) # label(fact_takeWhile__eq__all__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3346 (all V_a all V_x all V_b all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x))))) # label(fact_min__max_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3347 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_a) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))) # label(fact_min__max_Oinf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3348 (all V_xs_2 all V_f_2 all T_c all T_b hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_length__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3349 (all V_p_2 all V_g_2 all V_f_2 all T_c all T_b c_Product__Type_Oprod_Oprod__size(T_b,T_c,V_f_2,V_g_2,V_p_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_b,T_c),V_p_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_b,T_c),V_p_2)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_prod__size__simp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3350 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_y)),V_x)))) # label(fact_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3351 (all V_a all V_b all V_c all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3352 (all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OMin),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)) # label(fact_mult__Min) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3353 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3354 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3355 (all V_x_2 all V_g_2 all T_e all V_f_2 all T_c all T_b all T_d hAPP(c_Product__Type_Oapfst(T_e,T_b,T_c,hAPP(c_Fun_Ocomp(T_d,T_b,T_e,V_f_2),V_g_2)),V_x_2) = hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_f_2),hAPP(c_Product__Type_Oapfst(T_e,T_d,T_c,V_g_2),V_x_2))) # label(fact_apfst__compose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3356 (all V_list_2 all V_a_2 all V_f2_2 all V_f1_2 all T_c all T_b hAPP(c_List_Olist_Olist__case(T_b,T_c,V_f1_2,V_f2_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_list_2)) = hAPP(hAPP(V_f2_2,V_a_2),V_list_2)) # label(fact_list_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3357 (all V_B_2 all V_A_2 all T_b all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(V_B_2,B_x))) <-> c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_INTER__UNIV__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3358 (all V_xs all T_a all V_ys ((c_List_Olist_ONil(T_a) != V_ys -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),c_List_Obutlast(T_a,V_ys))) & (c_List_Olist_ONil(T_a) = V_ys -> c_List_Obutlast(T_a,V_xs) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))))) # label(fact_butlast__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3359 (all V_f_2 all T_c all T_b c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_inj__on__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3360 (all V_G_2 all V_tsa_2 all V_G_Ha_2 all T_b (c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_Ha_2,V_tsa_2) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_G_Ha_2) -> c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_tsa_2)))) # label(fact_cut) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3361 (all V_f_2 all T_c all V_xs_2 all T_b all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)),V_n_2) = hAPP(V_f_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_n_2)))) # label(fact_nth__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3362 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),T_b,T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_b),V_B_2)),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_b),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_b),T_b,T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),V_A_2)),hAPP(c_COMBC(T_c,T_d,T_b),V_f_2))))) # label(fact_INF__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3363 (all V_x_2 all T_b (class_Lattices_Olattice(T_b) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2)) # label(fact_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3364 (all V_a all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_a),hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs))) = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_a))) # label(fact_upd__conv__take__nth__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3365 (all V_B_2 all V_A_2 all T_b (class_Lattices_Odistrib__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_B_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBS(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal))),c_Lattices_Osemilattice__sup__class_Osup(T_b)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))))))))))))) # label(fact_sup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3366 (all V_P_2 all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Ofilter(T_c,V_P_2),V_xs_2)))))) # label(fact_sorted__filter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3367 (all V_A_2 all V_f_2 all T_c all T_b (class_Rings_Osemiring__1(T_b) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,T_c),c_Nat_Osemiring__1__class_Oof__nat(T_b)),V_f_2)),V_A_2))) # label(fact_of__nat__setsum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3368 (all V_xs_2 all T_b all V_x_2 all V_P_2 ((-hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_OdropWhile(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) & (hBOOL(hAPP(V_P_2,V_x_2)) -> c_List_OdropWhile(T_b,V_P_2,V_xs_2) = c_List_OdropWhile(T_b,V_P_2,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_dropWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3369 (all V_l_2 (c_Int_OPls = V_l_2 <-> c_Int_OPls = c_Int_OBit0(V_l_2))) # label(fact_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3370 (all V_r_2 all T_b (c_Wellfounded_Oacyclic(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))) <-> c_Wellfounded_Oacyclic(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),c_Predicate_Oconversep(T_b,T_b,V_r_2)))))) # label(fact_acyclicP__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3371 (all V_r_2 all T_b ((all B_x1 hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),B_x1))) <-> c_Wellfounded_OwfP(T_b,V_r_2))) # label(fact_wfP__accp__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3372 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_i_2,V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) # label(fact_atLeastLessThan__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3373 (all V_y_2 all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> c_SetInterval_Oord__class_OatLeastAtMost(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) = hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_x_2,V_y_2)))) # label(fact_image__uminus__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3374 (all V_g_2 all V_f_2 all T_c all T_b c_Fun_Ooverride__on(T_b,T_c,V_f_2,V_g_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = V_f_2) # label(fact_override__on__emptyset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3375 (all V_R_2 all T_b V_R_2 = c_Predicate_Opred__comp(T_b,T_b,T_b,c_fequal,c_Predicate_Opred__comp(T_b,T_b,T_b,V_R_2,c_fequal))) # label(fact_eq__comp__r) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3376 (all V_x c_Int_Osucc(c_Int_Opred(V_x)) = V_x) # label(fact_succ__pred) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3377 (all V_b_2 all V_a_2 (V_b_2 = V_a_2 -> c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_b_2) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_dvd_OatLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3378 (all V_xs_2 all V_f_2 all T_c all T_b c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(T_c,tc_List_Olist(T_b)),V_f_2),V_xs_2)) = c_List_Omaps(T_c,T_b,V_f_2,V_xs_2)) # label(fact_concat__map__maps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3379 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = V_x_2)))) # label(fact_Max_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3380 (all V_k hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Nat_OSuc,hAPP(c_Code__Numeral_Onat__of,V_k))) = c_Code__Numeral_OSuc__code__numeral(V_k)) # label(fact_Suc__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3381 (all V_xa_2 all V_x_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_x_2),V_xa_2)),c_Wellfounded_Omax__ext(T_b,V_R_2))) <-> c_Wellfounded_Omax__extp(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_R_2),V_x_2,V_xa_2))) # label(fact_max__extp__max__ext__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3382 (all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_c) & class_Enum_Oenum(T_b) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBS(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),c_fequal),V_f_2)),V_g_2)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,T_c)),V_f_2),V_g_2))))) # label(fact_order__fun_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3383 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)) -> c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2))))) # label(fact_wf__converse__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3384 (all V_A_2 all V_l_2 all V_x_2 all T_b hAPP(hAPP(c_List_Oappend(T_b),hAPP(hAPP(hAPP(c_If(tc_List_Olist(T_b)),hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b))),c_List_Olist_ONil(T_b))),c_List_Osublist(T_b,V_l_2,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_A_2)))) = c_List_Osublist(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_l_2),V_A_2)) # label(fact_sublist__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3385 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = V_x_2))))) # label(fact_Min_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3386 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)))))) # label(fact_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3387 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x),V_y)) = c_Nat__Transfer_Otsub(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3388 (all V_a_2 all V_f_2 all V_xs_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_a_2)),hAPP(V_f_2,B_x))))) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_2),V_xs_2) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_a_2),V_xs_2)))) # label(fact_insort__is__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3389 (all V_P_2 (-hBOOL(V_P_2) | -hBOOL(hAPP(c_fNot,V_P_2)))) # label(help_c__fNot__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3390 (all T_b (class_Rings_Osemiring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Oone__class_Oone(T_b)),c_Nat_Osemiring__1__class_ONats(T_b))))) # label(fact_Nats__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3391 (all V_a_2 all V_r_2 all T_b all T_c hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_prod(T_c,T_b),tc_fun(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_c,T_b))),hAPP(c_Product__Type_OPair(T_c,T_b),V_a_2))),V_r_2))) # label(fact_Image__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3392 (all V_h_2 all V_B_2 all V_f_2 all T_c all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_h_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_h_2),V_f_2)),V_B_2)))) # label(fact_setprod__reindex) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3393 (all V_n V_n = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_max__0R) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3394 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),V_n_2)),c_Transitive__Closure_Ortrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_pred__nat__trancl__eq__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3395 (all V_b_2 all V_a_2 all T_b (class_Rings_Osemiring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Nat_Osemiring__1__class_ONats(T_b))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Nat_Osemiring__1__class_ONats(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_b))))))) # label(fact_Nats__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3396 (all V_g_2 all V_x_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2))))) # label(fact_setprod_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3397 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3398 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3399 (all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_w_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3400 (all V_x all T_a V_x = hAPP(c_Fun_Oid(T_a),V_x)) # label(fact_id__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3401 (all V_vs_2 all V_us_2 all V_P_2 all V_ys_2 all T_c all V_xs_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) -> (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) & c_List_Olist__all2(T_b,T_c,V_P_2,V_us_2,V_vs_2) <-> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_us_2),hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_vs_2))))) # label(fact_list__all2__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3402 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2))))) # label(fact_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3403 (all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (class_Groups_Ominus(T_b) -> hAPP(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,T_b)),V_A_2),V_B_2),V_x_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(V_A_2,V_x_2)),hAPP(V_B_2,V_x_2)))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3404 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_b) = V_a)) # label(fact_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3405 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = V_m -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m | c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n)) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3406 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Finite__Set_Ocard(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_card__Pow) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3407 (all T_d all T_c all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)))) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)))) # label(fact_INF2__I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3408 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))))))) # label(fact_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3409 (all V_B_2 all V_A_2 all V_C_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3410 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)))))) # label(fact_inj__on__strict__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3411 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x)),V_x))))))) # label(fact_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3412 (all V_x_2 all T_b c_List_Ofilter(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(c_fequal,V_x_2))) = c_List_OremoveAll(T_b,V_x_2)) # label(fact_removeAll__filter__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3413 (all V_i all V_xs all T_a hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)) = V_xs) # label(fact_list__update__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3414 (all V_r_2 all T_b c_List_Olistrelp(T_b,V_r_2,c_List_Olist_ONil(T_b),c_List_Olist_ONil(T_b))) # label(fact_listrelp_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3415 (all V_a_2 all T_b (class_Int_Oring__char__0(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),c_Groups_Oone__class_Oone(T_b)),V_a_2)),V_a_2) != c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_Ints__odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3416 (all V_n_2 hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_lessThan__Suc__eq__insert__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3417 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_m))))) # label(fact_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3418 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3419 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(V_f_2,V_x_2))))) # label(fact_folding_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3420 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3421 (all V_ys all V_xs all T_a all V_n ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n) = hAPP(hAPP(c_List_Onth(T_a),V_ys),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n)))) # label(fact_nth__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3422 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_card_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3423 (all V_r_2 all T_b c_List_Olistrelp(T_b,V_r_2,c_List_Olist_ONil(T_b),c_List_Olist_ONil(T_b))) # label(fact_listrelp_ONil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3424 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3425 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_e all T_d all T_c hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_e,T_b),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_b),V_g_2),hAPP(V_f_2,V_x_2))) # label(fact_scomp__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3426 (all V_u_2 all V_l_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_2)))) # label(fact_finite__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3427 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)))) # label(fact_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3428 (all V_xs_2 all V_g_2 all T_c all V_f_2 all T_b c_List_Olist_Olist__size(T_b,V_f_2,hAPP(hAPP(c_List_Omap(T_c,T_b),V_g_2),V_xs_2)) = c_List_Olist_Olist__size(T_c,hAPP(c_Fun_Ocomp(T_b,tc_Nat_Onat,T_c,V_f_2),V_g_2),V_xs_2)) # label(fact_list__size__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3429 (all V_z_2 all V_x_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> hAPP(hAPP(V_f_2,V_x_2),V_z_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)))) # label(fact_fun__left__comm__idem_Ofun__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3430 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)))) # label(fact_Int__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3431 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))) -> c_Divides_OnegDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OnegDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))) -> c_Divides_OnegDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))))) # label(fact_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3432 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Oordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3433 (all V_xs_2 all T_b hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_b),hAPP(c_List_Onth(T_b),V_xs_2)),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) = V_xs_2) # label(fact_map__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3434 (all V_a_2 all V_xs_2 all V_f_2 all T_b all T_c hAPP(c_List_Ofoldl(T_b,T_c,hAPP(c_COMBC(T_c,T_b,T_b),V_f_2),V_a_2),hAPP(c_List_Orev(T_c),V_xs_2)) = c_List_Ofoldr(T_c,T_b,V_f_2,V_xs_2,V_a_2)) # label(fact_foldr__foldl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3435 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Oidom(T_b) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))) <-> c_Groups_Ozero__class_Ozero(T_b) = V_c_2 | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_a_2),V_b_2))))) # label(fact_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3436 (all V_x_2 all V_g_2 all T_e all V_f_2 all T_b all T_c all T_d hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2))),hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_e,T_d),V_x_2))) = hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),hAPP(c_Product__Type_Oapfst(T_e,T_b,T_d,V_g_2),V_x_2))) # label(fact_apsnd__apfst) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3437 (all V_N_2 c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc,V_N_2)) # label(fact_inj__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3438 (all V_xs_2 all V_a_2 all V_f_2 all T_c all T_b c_List_Ofoldr(T_c,T_b,hAPP(c_COMBC(T_b,T_c,T_b),V_f_2),hAPP(c_List_Orev(T_c),V_xs_2),V_a_2) = hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,V_a_2),V_xs_2)) # label(fact_foldl__foldr) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3439 (all V_c_2 all T_c all T_b all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_c_2))) & (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> V_c_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_c_2))))) # label(fact_UN__constant) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3440 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)))))) # label(fact_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3441 (all V_xs_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_b),c_COMBI(T_b)),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),V_xs_2)))) # label(fact_distinct__listsum__conv__Setsum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 3442 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3443 (all V_b all V_a ((c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b)))) # label(fact_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3444 (all V_P_2 all T_b c_List_OdropWhile(T_b,V_P_2,c_List_Olist_ONil(T_b)) = c_List_Olist_ONil(T_b)) # label(fact_dropWhile_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3445 (all V_A_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__mult(T_b) & class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,T_b),V_f_2),V_A_2))) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3446 (all V_z_2 all T_c all T_b V_z_2 = hAPP(c_Product__Type_Omap__pair(T_b,T_b,T_c,T_c,c_COMBI(T_b),c_COMBI(T_c)),V_z_2)) # label(fact_map__pair__ident) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3447 (all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_x))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3448 (all V_m all V_n hAPP(hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_n,V_m)) = c_Code__Numeral_Odiv__mod__code__numeral(V_n,V_m)) # label(fact_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3449 (all V_xs_2 all T_b hAPP(c_Set_OPow(T_b),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_Set_Oimage(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),c_List_Oset(T_b)),hAPP(c_List_Oset(tc_List_Olist(T_b)),c_Enum_Osublists(T_b,V_xs_2)))) # label(fact_sublists__powset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3450 (all V_x_2 all V_z_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_x_2)) -> V_x_2 = V_z_2)) # label(fact_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3451 (all V_l_2 c_List_Oreturn__list(V_l_2) = hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),c_Int_Onat),V_l_2)) # label(fact_return__list__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3452 (all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a)) = c_List_Oinsert(T_a,V_x,c_List_Olist_ONil(T_a))) # label(fact_insert__Nil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3453 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2)))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3454 (all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_C_2))) -> -(all B_X (hBOOL(hAPP(hAPP(c_member(T_b),V_A_2),B_X)) -> -hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_X),V_C_2)))))) # label(fact_UnionE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3455 (all V_xs_2 all V_n_2 all T_b (hAPP(c_List_Orotate(T_b,V_n_2),V_xs_2) = c_List_Olist_ONil(T_b) <-> c_List_Olist_ONil(T_b) = V_xs_2)) # label(fact_rotate__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3456 (all V_A_2 all V_f_2 all T_c all T_b (class_Groups_Oab__group__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),c_Groups_Ouminus__class_Ouminus(T_b)),V_f_2)),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)))) # label(fact_setsum__negf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3457 (all V_B_2 all V_C_2 all T_c all V_A_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_UN__extend__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3458 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_folding__one_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3459 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a) -> V_b = V_c))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3460 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)))) # label(fact_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3461 (all T_b (class_Lattices_Osemilattice__inf(T_b) -> c_Finite__Set_Ofun__left__comm__idem(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)))) # label(fact_fun__left__comm__idem__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3462 (all V_xs_2 all V_P_2 all T_b c_List_OdropWhile(T_b,V_P_2,V_xs_2) = hAPP(hAPP(c_List_Odrop(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))),V_xs_2)) # label(fact_dropWhile__eq__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3463 (all V_a_2 all V_b_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))))))) # label(fact_pos__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3464 (all T_c all V_f_2 all T_b (c_FunDef_Ois__measure(T_b,V_f_2) -> c_FunDef_Ois__measure(tc_prod(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,tc_Nat_Onat,tc_prod(T_c,T_b)),V_f_2),c_Product__Type_Osnd(T_c,T_b))))) # label(fact_measure__snd) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3465 (all V_f_2 all V_A_2 all V_B_2 all T_b all T_c (class_Groups_Oordered__comm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_b (hBOOL(hAPP(hAPP(c_member(T_b),B_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_b))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2)))))))) # label(fact_setsum__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3466 (all V_r_2 all T_b (c_Relation_Oirrefl(T_b,V_r_2) <-> (all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_x)),V_r_2))))) # label(fact_irrefl__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3467 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (V_x_2 = V_y_2 <-> hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_y_2) = hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_x_2)))) # label(fact_lessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3468 (all V_R_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Quotient_ORespects(T_b,V_R_2))) <-> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)))) # label(fact_in__respects) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3469 (all V_xs_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(c_List_OremoveAll(T_b,V_x_2),V_xs_2) = V_xs_2)) # label(fact_removeAll__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3470 (all V_D_2 all V_B_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_D_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Un__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3471 (all V_x all V_z all V_y all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_mult__imp__le__div__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3472 (all V_xs_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Oinsert(T_b,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) # label(fact_not__in__set__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3473 (all V_y all V_i all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Nat_OSuc,V_i)),V_y) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_y))) # label(fact_list__update__code_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3474 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit0(V_k)) = c_Int_OBit0(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))) # label(fact_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3475 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> (exists B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),B_y)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))))))) # label(fact_rel__pow__Suc__D2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3476 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (V_m_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)))) # label(fact_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3477 (all V_m all T_a (class_Rings_Olinordered__semidom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3478 (all V_ys_2 all V_xs_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),V_ys_2)) = hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) # label(fact_set__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3479 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = V_x_2))))) # label(fact_min__max_OSup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3480 (all V_B_2 all V_A_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Set_Oimage(T_b,T_c,V_f_2)),V_B_2)))) # label(fact_bij__image__INT) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3481 (all V_x all V_n all T_a hAPP(c_List_Orev(T_a),c_List_Oreplicate(T_a,V_n,V_x)) = c_List_Oreplicate(T_a,V_n,V_x)) # label(fact_rev__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3482 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3483 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)))))) # label(fact_Inf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3484 (all V_P_2 all V_k_2 all T_b (class_Orderings_Owellorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_k_2),c_Orderings_Oord__class_OLeast(T_b,V_P_2))) -> -hBOOL(hAPP(V_P_2,V_k_2))))) # label(fact_not__less__Least) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3485 (all V_f_2 all V_a_2 all V_A_2 all T_b all T_c (class_Rings_Oring(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)))))) # label(fact_setsum__diff1__ring) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3486 (all V_P_2 all V_A_2 all T_c all T_b hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OAll(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fimplies),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),V_P_2))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Set_OCollect(T_b)),hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),V_P_2)))) # label(fact_Collect__ball__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3487 (all V_b_2 all V_B_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))))) # label(fact_insertI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3488 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2)))))) # label(fact_less__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3489 (all V_Q_2 all V_P_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_Q_2))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),V_P_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fdisj),V_P_2)),V_Q_2)))))) # label(fact_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3490 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> V_m = V_n))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3491 (all V_r all V_q all V_n all V_m (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r))) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_r)) # label(fact_mod__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3492 (all V_v_H hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) # label(fact_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3493 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))))) # label(fact_less__SucI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3494 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_INT__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3495 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2))))))) # label(fact_min__max_OInf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3496 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_n) = V_m)) # label(fact_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3497 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)))))) # label(fact_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3498 (all V_P_2 all V_a_2 all V_n_2 all T_b (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(V_P_2,V_a_2)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,V_n_2,V_a_2)))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_Ball__set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3499 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_i),c_List_Oupt(hAPP(c_Nat_OSuc,V_i),V_j)))) # label(fact_upt__conv__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3500 (all V_y_2 all V_x_2 all T_b (class_Lattices_Obounded__lattice__top(T_b) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),V_y_2) = c_Orderings_Otop__class_Otop(T_b) <-> V_y_2 = c_Orderings_Otop__class_Otop(T_b) & c_Orderings_Otop__class_Otop(T_b) = V_x_2))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3501 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_B_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Inf__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3502 (all V_i all T_a c_List_Oconcat(T_a,c_List_Oreplicate(tc_List_Olist(T_a),V_i,c_List_Olist_ONil(T_a))) = c_List_Olist_ONil(T_a)) # label(fact_concat__replicate__trivial) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3503 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)),V_k)) # label(fact_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3504 (all V_u_2 all V_l_2 hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2))) # label(fact_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3505 (all V_k_2 all V_P_2 all V_d_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2)) -> ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_d_2))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),V_d_2)))))))))) # label(fact_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3506 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_mult__minus1__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3507 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),V_xs_2)))) # label(fact_set__insort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3508 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (V_A_2 = V_B_2 <-> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)))) # label(fact_inj__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3509 (all V_a_2 all V_f_2 all T_c all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_b,tc_fun(T_c,tc_HOL_Obool),V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_f_2))))) # label(fact_UN__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3510 (all V_r all V_q all V_n all V_m (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r))) -> V_q = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n))) # label(fact_div__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3511 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_OSuc,V_n))) = hAPP(hAPP(c_List_Onth(T_a),hAPP(c_List_Orev(T_a),V_xs)),V_n))) # label(fact_rev__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3512 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OPls) = V_k) # label(fact_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3513 (all V_b_2 all V_B_2 all V_a_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2)) -> V_a_2 = V_b_2) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_B_2))))) # label(fact_insertCI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3514 (all V_n_2 all V_m_2 (V_n_2 != V_m_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3515 (all V_y all V_x all T_a (class_Rings_Olinordered__ring(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_y))),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3516 (all V_f2_2 all V_f1_2 all T_b hAPP(c_Nat_Onat_Onat__rec(T_b,V_f1_2,V_f2_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_f1_2) # label(fact_nat__rec__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3517 (all V_y_2 all V_x_2 all V_z_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_y_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)))))) # label(fact_less__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3518 (all V_m V_m = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_div__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3519 (all V_v_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2)))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3520 (all T_b all V_R_2 all V_y_2 all V_y_H_2 all T_c all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_x_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_H_2),V_y_2)),hAPP(V_R_2,V_x_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_H_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2))),c_Recdef_Osame__fst(T_b,T_c,V_P_2,V_R_2)))))) # label(fact_same__fstI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3521 (all V_n all V_xs all V_x all T_a hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Nat_OSuc,V_n))) # label(fact_nth__Cons__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3522 (all V_y all V_x all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> ((all B_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),B_z)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),B_z),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),B_z),V_x)),V_y))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))))) # label(fact_field__le__mult__one__interval) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3523 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Fields_Ofield__inverse__zero(T_b) -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = V_a_2 <-> (V_c_2 = c_Groups_Ozero__class_Ozero(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = V_a_2) & (V_c_2 != c_Groups_Ozero__class_Ozero(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2) = V_b_2)))) # label(fact_eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3524 (all V_v_H all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))))) # label(fact_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3525 (all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n))) # label(fact_power__one) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3526 (all V_f_2 all T_d all T_c all T_b c_Product__Type_Ocurry(T_b,T_c,T_d,hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_f_2)) = V_f_2) # label(fact_curry__split) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3527 (all V_d all V_c all V_b all V_a all T_a (class_Groups_Oordered__cancel__ab__semigroup__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d))))))) # label(fact_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3528 (all V_y_2 all V_x_2 all V_b_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Oone__class_Oone(T_b)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_b_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)))))) # label(fact_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3529 (all V_A_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_List_Olists(T_b,V_A_2))) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_ys_2),c_List_Olists(T_b,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)),c_List_Olists(T_b,V_A_2))))) # label(fact_append__in__lists__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3530 (all V_R_2 all T_c all T_b V_R_2 = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_c,T_c),V_R_2),c_Relation_OId(T_c))) # label(fact_R__O__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3531 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),V_ys_2))))) # label(fact_set__zip__rightD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3532 (all V_q all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3533 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3534 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Odivision__ring(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_c_2 -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) <-> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))))) # label(fact_nonzero__eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3535 (all V_xs_2 all T_c all V_x_2 all V_f_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_List_Oset(T_c),V_xs_2)))) -> V_xs_2 = c_List_Olinorder__class_Oinsort__insert__key(T_c,T_b,V_f_2,V_x_2,V_xs_2)))) # label(fact_insort__insert__key__triv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3536 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2))))) # label(fact_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3537 (all V_x all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_last__ConsR) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3538 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),V_x_2)))))) # label(fact_Min__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3539 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)))) # label(fact_Int__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3540 (all V_A_2 all V_f_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) -> -(all B_x (V_b_2 = hAPP(V_f_2,B_x) -> -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_imageE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3541 (all V_x_2 all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3542 (all V_d all V_b all V_c all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_c),V_d))))))) # label(fact_min__max_Osup__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3543 (all V_z_2 all V_y_2 all V_A_2 all V_b_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_b_2,V_A_2),V_y_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_z_2),V_y_2))))))) # label(fact_fold__graph__insert__swap) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3544 (all V_f_2 all V_a_2 all V_A_2 all T_b all T_c (class_Groups_Oab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_setsum__diff1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3545 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3546 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> c_Finite__Set_Ofun__left__comm__idem(T_d,T_c,hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,T_c),T_d),V_f_2),V_g_2)))) # label(fact_fun__left__comm__idem_Ofun__left__comm__idem__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3547 (all V_M_2 all T_b all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),hAPP(c_COMBK(T_c,T_b),V_M_2)) = V_M_2))) # label(fact_INF__const) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3548 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3549 (all V_x all T_a (class_Orderings_Opreorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_x)))) # label(fact_order__refl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3550 (all V_N_2 all V_h_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMin(T_b,V_N_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2))))))) # label(fact_Min_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3551 (all V_P_2 all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2)),V_P_2))),V_P_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2)),V_P_2)))))) # label(fact_lfp__induct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3552 (all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3553 (all V_B_2 all V_A_2 all T_c all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool))),V_B_2))) # label(fact_Compl__INT) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3554 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> V_y_2 = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)))) # label(fact_min__max_Ole__iff__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3555 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c)) # label(fact_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3556 (all T_1 all T_2 class_Nat_Osize(tc_sum(T_2,T_1))) # label(arity_sum__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3557 (all V_A_2 all V_I_2 all T_c all T_b exists B_f (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,tc_prod(T_c,T_b),B_f),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2))),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_I_2),V_A_2))) & c_Fun_Oinj__on(T_b,tc_prod(T_c,T_b),B_f,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_Ex__inj__on__UNION__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3558 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_i)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_j))))) # label(fact_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 3559 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)))) # label(fact_diff__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3560 (all V_xs_2 all V_P_2 all T_b all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OdropWhile(T_b,V_P_2,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_b),c_List_OdropWhile(T_b,V_P_2,V_xs_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_b),V_xs_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)))))) # label(fact_dropWhile__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3561 (all V_B_2 all V_A_2 all T_c all T_b hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool))),V_B_2))) # label(fact_Compl__UN) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3562 (all V_z_2 all V_x_2 all V_y_2 all V_w_2 all T_b (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b) -> (V_z_2 = V_y_2 | V_x_2 = V_w_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_y_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_w_2),V_y_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_x_2),V_z_2))))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3563 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) # label(fact_INT__extend__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3564 (all V_n all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(tc_List_Olist(T_a)),c_Enum_On__lists(T_a,V_n,V_xs))))) # label(fact_distinct__n__lists) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3565 (all V_b all V_b_H all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b_H))))))) # label(fact_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3566 (all V_y all V_i all T_a hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_y) = c_List_Olist_ONil(T_a)) # label(fact_list__update__code_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3567 (all V_r_2 all V_ys_2 all V_xs_2 all T_b ((all B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_n)),hAPP(hAPP(c_List_Onth(T_b),V_ys_2),B_n))),V_r_2)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,V_r_2))))) # label(fact_listrel__iff__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3568 (all V_r_2 all T_b c_Transitive__Closure_Ortrancl(T_b,V_r_2) = c_Transitive__Closure_Otrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3569 (all V_list_H_2 all V_a_H_2 all V_list_2 all V_a_2 all T_b (hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_H_2),V_list_H_2) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_a_2),V_list_2) <-> V_a_2 = V_a_H_2 & V_list_H_2 = V_list_2)) # label(fact_list_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3570 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3571 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a))) # label(fact_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3572 (all V_c all V_a all V_b all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_c)))) # label(fact_min__max_Oinf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3573 (all V_z_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_b,V_f_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = V_z_2) # label(fact_fold__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3574 (all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_ComplD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3575 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))),c_List_Olist_ONil(T_a))))) # label(fact_take__hd__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3576 (all V_u_2 all V_M_2 all V_A_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_M_2,B_x)),V_u_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),V_A_2),V_M_2)),V_u_2))))) # label(fact_SUP__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3577 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__all(T_b,V_P_2,V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_list__all__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3578 (all V_r_2 all T_b (c_Wellfounded_OwfP(T_b,V_r_2) -> c_Wellfounded_Oacyclic(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))))) # label(fact_wfP__acyclicP) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3579 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Transitive__Closure_Ortrancl(T_b,V_s_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),c_Transitive__Closure_Ortrancl(T_b,V_s_2))))) # label(fact_rtrancl__subset__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3580 (all V_A_2 all V_a_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) & -hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) <-> c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)))) # label(fact_inj__on__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3581 (all V_B_2 all V_A_2 all V_C_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_C_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_inj__on__image__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3582 (all V_r_2 all V_x_2 all V_y_2 all T_b (c_Wellfounded_Oacyclic(T_b,V_r_2) & -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) <-> c_Wellfounded_Oacyclic(T_b,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_x_2)),V_r_2)))) # label(fact_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3583 (all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Union__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3584 (all V_b all V_x all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x))))) # label(fact_le__infI1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3585 (all V_g_2 all V_h_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Fun_Oinj__on(T_b,T_b,V_h_2,V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(c_Fun_Ocomp(T_b,T_c,T_b,V_g_2),V_h_2)),V_A_2))))) # label(fact_setprod_Oreindex) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3586 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3587 (all V_s_2 all V_B_2 all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> (c_Relation_Orefl__on(T_b,V_B_2,V_s_2) -> c_Relation_Orefl__on(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_refl__on__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3588 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = V_B_2 -> (hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_h_2),V_f_2) = V_g_2 -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_d),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_h_2),V_B_2)))))) # label(fact_setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3589 (all V_c_2 all V_b_2 all V_A_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_b_2),V_c_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_c_2),V_A_2))))))) # label(fact_inf__le__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3590 (all V_a_H all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_b) -> c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a_H),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b)))) # label(fact_mod__minus__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3591 (all T_2 all T_1 (class_Orderings_Oorder(T_1) -> class_Orderings_Oorder(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3592 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k))) # label(fact_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3593 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = V_a_2)))) # label(fact_min__max_Oinf__Sup__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3594 (all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3595 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) & hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),V_xs_2)))))) # label(fact_distinct__insort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3596 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3597 (all V_k_2 all V_less_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThan(T_b,V_less_2,V_k_2))) <-> hBOOL(hAPP(hAPP(V_less_2,V_k_2),V_i_2)))) # label(fact_ord_OgreaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3598 (all V_ns_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_List_Oset(tc_Nat_Onat),V_ns_2))) -> B_x = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),V_ns_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_listsum__eq__0__nat__iff__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3599 (all V_a_2 all V_D_2 all V_C_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_C_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_D_2))))) # label(fact_insert__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3600 (all V_xs all V_x all T_a V_xs = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_tl_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3601 (all V_f_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) -> -(all B_x hAPP(V_f_2,B_x) != V_b_2))) # label(fact_rangeE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3602 (all V_p_2 all T_b all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_b,T_b),hAPP(c_COMBK(tc_fun(T_b,T_b),T_c),c_COMBI(T_b))),V_p_2) = hAPP(c_Product__Type_Osnd(T_c,T_b),V_p_2)) # label(fact_snd__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3603 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)))) # label(fact_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3604 (all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ly))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3605 (all V_n all V_m (V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) -> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3606 (all V_P_2 all V_ys_2 all T_c all V_xs_2 all T_b ((hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Otake(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Otake(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))),V_ys_2)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Ozip(T_b,T_c),hAPP(hAPP(c_List_Otake(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))),V_xs_2)),hAPP(hAPP(c_List_Otake(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))),V_ys_2))))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2))))) # label(fact_zip__obtain__same__length) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3607 (all V_z_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_z_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_rtrancl__trancl__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3608 (all T_a (class_Orderings_Olinorder(T_a) -> c_List_Olinorder__class_Osorted(T_a,c_List_Olist_ONil(T_a)))) # label(fact_sorted_ONil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3609 (all T_b c_Transitive__Closure_Otrancl(T_b,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) # label(fact_trancl__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3610 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3611 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3612 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Orev(T_a),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_rev_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3613 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3614 (all T_c all T_b c_Fun_Oinj__on(tc_fun(T_b,T_c),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_b,T_c)),c_Set_OCollect(tc_prod(T_b,T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_b,T_c)),c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool)),hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal)))),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,T_c),tc_HOL_Obool)))) # label(fact_inj__graph) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3615 (all T_2 all T_1 (class_Orderings_Oord(T_1) -> class_Orderings_Oord(tc_fun(T_2,T_1)))) # label(arity_fun__Orderings_Oord) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3616 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k)))))) # label(fact_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3617 (all V_Z_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,V_Z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_Z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_neg__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3618 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_c)))))) # label(fact_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3619 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_OCollect(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c)),c_fequal)),V_B_2)))))) # label(fact_UN__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3620 (all V_n all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n))) # label(fact_power__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3621 (all V_nat hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Nat_Onat_Onat__size(V_nat)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Nat_Onat_Onat__size(hAPP(c_Nat_OSuc,V_nat))) # label(fact_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3622 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__insert2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3623 (all V_xs all T_a all V_ys (V_ys = c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_last__appendL) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3624 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) != V_xs) # label(fact_not__Cons__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3625 (all V_y all V_j all V_x all V_i all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),c_List_Oreplicate(T_a,V_i,V_x)),c_List_Oreplicate(T_b,V_j,V_y)) = c_List_Oreplicate(tc_prod(T_a,T_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_i),V_j),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y))) # label(fact_zip__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3626 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3627 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_n)))) # label(fact_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3628 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_I_2),V_B_2))) # label(fact_Sigma__Diff__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3629 (all V_a_2 all V_S_2 all T_b (class_Orderings_Olinorder(T_b) & class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Complete__Lattice_OInf__class_OInf(T_b,V_S_2)),V_a_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),B_x),V_a_2))))))) # label(fact_Inf__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3630 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2) = c_Groups_Oone__class_Oone(T_c)))) # label(fact_setprod_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3631 (all V_i c_List_Oupt(V_i,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_List_Olist_ONil(tc_Nat_Onat)) # label(fact_upt__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3632 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> (exists B_ys exists B_zs (hAPP(hAPP(c_List_Oappend(T_b),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_zs)) = V_xs_2 & -hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),B_zs))))))) # label(fact_in__set__conv__decomp__last) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3633 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_c all T_b (V_b_H_2 = V_b_2 & V_a_2 = V_a_H_2 <-> hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_H_2),V_b_H_2))) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3634 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3635 (all V_zs all V_ys all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_zs) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Oappend(T_a),V_ys),V_zs))) # label(fact_append__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3636 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_IntI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3637 (all V_P_2 c_Nat__Transfer_Onat__set(hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),V_P_2)))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3638 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x))))) # label(fact_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3639 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))))) # label(fact_rtrancl__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3640 (all V_Q_2 all V_P_2 (hBOOL(V_Q_2) | -hBOOL(V_P_2) | -hBOOL(hAPP(hAPP(c_fimplies,V_P_2),V_Q_2)))) # label(help_c__fimplies__3) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3641 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_length__rev) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3642 (all V_j_2 all V_i_2 all V_P_2 (c_List_Olist__ex(tc_Nat_Onat,V_P_2,c_List_Oupt(V_i_2,V_j_2)) <-> -c_List_Oall__interval__nat(hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat,c_fNot),V_P_2),V_i_2,V_j_2))) # label(fact_list__ex__iff__not__all__inverval__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3643 (all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_vimage__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3644 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_x)))))))) # label(fact_finite__UN) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3645 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2))))))) # label(fact_folding__one_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3646 (all V_Y_2 all V_X_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_X_2),V_Y_2) | V_Y_2 = V_X_2)))) # label(fact_quotient__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3647 (all V_x_2 all T_b (class_Lattices_Olattice(T_b) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2)) # label(fact_Inf__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3648 (all V_B_2 all V_A_2 hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2)))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3649 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2))),c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2)))))) # label(fact_lfp__lemma2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3650 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_DiffI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3651 (all V_g_2 all V_x_2 all V_y_2 all T_d all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_y_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,hAPP(V_f_2,V_y_2)),V_g_2)))) # label(fact_folding_Ocommute__left__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3652 (all V_A_2 all T_b V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3653 (all V_R_2 all T_b ((all B_x all B_y (hAPP(V_R_2,B_y) = hAPP(V_R_2,B_x) <-> hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)))) <-> c_Equiv__Relations_Oequivp(T_b,V_R_2))) # label(fact_equivp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3654 (all V_x all V_z all V_d (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_z))),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_d)))))) # label(fact_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3655 (all V_d_2 all T_b (class_Lazy__Sequence_Osmall__lazy(T_b) -> c_Lazy__Sequence_Oappend(tc_List_Olist(T_b),hAPP(c_Lazy__Sequence_Osingle(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),hAPP(hAPP(hAPP(c_If(tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_b))),hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_d_2)),hAPP(c_Lazy__Sequence_Obind(tc_prod(T_b,tc_List_Olist(T_b)),tc_List_Olist(T_b),c_Lazy__Sequence_Oproduct(T_b,tc_List_Olist(T_b),c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_d_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_List_Olist(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_d_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,tc_List_Olist(T_b),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_List_Olist(T_b),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_b))),T_b),hAPP(c_COMBB(tc_List_Olist(T_b),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_b)),tc_List_Olist(T_b)),c_Lazy__Sequence_Osingle(tc_List_Olist(T_b)))),c_List_Olist_OCons(T_b))))),c_Lazy__Sequence_Oempty(tc_List_Olist(T_b)))) = c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_List_Olist(T_b),V_d_2))) # label(fact_small__lazy__list_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3656 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3657 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)))))) # label(fact_folding_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3658 (all V_s_2 all V_r_2 all T_b all T_c c_Predicate_Oconversep(T_c,T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool))),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),c_Predicate_Oconversep(T_c,T_b,V_r_2)),c_Predicate_Oconversep(T_c,T_b,V_s_2))) # label(fact_converse__join) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3659 (all T_b c_Finite__Set_Ofun__left__comm__idem(T_b,tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_fun__left__comm__idem__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3660 (all V_y_2 all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(c_List_Orev(T_b),V_xs_2)) -> (c_List_Olist_ONil(T_b) != V_xs_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_y_2) = c_List_Ofoldr(T_b,T_b,c_Orderings_Oord__class_Omax(T_b),V_xs_2,V_y_2)) & (V_xs_2 = c_List_Olist_ONil(T_b) -> V_y_2 = c_List_Ofoldr(T_b,T_b,c_Orderings_Oord__class_Omax(T_b),V_xs_2,V_y_2))))) # label(fact_foldr__max__sorted) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3661 (all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_List_Ofoldl(tc_List_Olist(T_b),T_b,hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b))),c_List_Olist_ONil(T_b)),V_xs_2) = c_List_Olinorder__class_Osort__key(T_b,T_b,c_COMBI(T_b),V_xs_2))) # label(fact_sort__foldl__insort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3662 (all V_B_2 all V_C_2 all T_c all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_UN__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3663 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3664 (all V_A_2 all V_y_2 all V_x_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3665 (all V_w_2 all V_b_2 all V_a_2 all T_b (class_Int_Onumber(T_b) & class_Fields_Olinordered__field__inverse__zero(T_b) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2))))))) # label(fact_le__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3666 (all V_l_2 all V_k_2 (hAPP(c_Code__Numeral_Onat__of,V_l_2) = hAPP(c_Code__Numeral_Onat__of,V_k_2) <-> V_l_2 = V_k_2)) # label(fact_Code__Numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3667 (all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> -(all B_x V_y_2 != hAPP(c_Code__Numeral_Onat__of,B_x)))) # label(fact_nat__of__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3668 (all V_x_2 all V_a_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_x_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_a_2)),V_r_2))))) # label(fact_wf__asym) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3669 (all V_f_2 all V_X_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_f_2)),V_X_2))),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)))))) -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_f_2)),V_X_2))),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)))),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_f_2)),V_X_2))),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))))))))) # label(fact_coinduct3__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3670 (all V_P_2 all V_xs_2 all T_b all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),V_n_2)))))) # label(fact_list__ball__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3671 (all V_xs_2 all V_x_2 all T_b (hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)) = hAPP(c_List_Orev(T_b),V_xs_2) <-> V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)))) # label(fact_singleton__rev__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 3672 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_b))) # label(fact_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3673 (all V_k hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_k)) = V_k) # label(fact_of__nat__nat__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3674 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3675 (all V_t_2 all V_D_2 all V_d_2 all T_b (class_Rings_Odvd(T_b) & class_Rings_Ocomm__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),V_D_2)) -> (all B_x all B_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),B_x),V_t_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_b),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),B_k),V_D_2))),V_t_2)))))))) # label(fact_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3676 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_B_2)),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)))))))) # label(fact_fold1__antimono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3677 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oring(T_b) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2) = V_c_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)))) # label(fact_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3678 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (exists B_n hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n))) = V_x))) # label(fact_negD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3679 (all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3680 (all V_b_2 all V_a_2 all T_b (class_Rings_Oring__1(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Int_Oring__1__class_OInts(T_b))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),c_Int_Oring__1__class_OInts(T_b))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_b))))))) # label(fact_Ints__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3681 (all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Range__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3682 (all V_P_2 all V_b_2 all T_c all V_a_2 all T_b (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_b_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_a_2) -> ((all B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_a_2))) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_a_2),B_n)),hAPP(hAPP(c_List_Onth(T_c),V_b_2),B_n))))) -> c_List_Olist__all2(T_b,T_c,V_P_2,V_a_2,V_b_2)))) # label(fact_list__all2__all__nthI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3683 (all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_c all T_b (c_Typedef_Otype__definition(T_b,T_c,V_Rep_2,V_Abs_2,V_A_2) -> V_x_2 = hAPP(V_Abs_2,hAPP(V_Rep_2,V_x_2)))) # label(fact_type__definition_ORep__inverse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3684 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Orderings_Oorder(T_c) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)))))) # label(fact_strict__mono__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3685 (all V_B_2 all V_A_2 all T_b ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_X),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),B_X),V_B_2)))) -> (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2)),V_B_2))))) # label(fact_Inter__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3686 (all V_n hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))))) # label(fact_neg__zminus__int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3687 (all V_f_2 all T_c all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Relation_Oinv__image(T_b,T_c),V_r_2),V_f_2)))) # label(fact_wf__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3688 (all V_C_2 all V_B_2 all V_A_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_B_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_c,V_r_2),V_C_2)),V_B_2)))) # label(fact_Image__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3689 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_a all B_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b))) -> ((-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_a),B_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B_b)))) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3690 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OgreaterThanAtMost(T_b,V_less__eq_2,V_less_2,V_l_2,V_u_2))) <-> hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)))) # label(fact_ord_OgreaterThanAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3691 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a))) # label(fact_power__commutes) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3692 (all V_x_2 all T_b (c_List_Olist_ONil(T_b) = c_List_Oremdups(T_b,V_x_2) <-> V_x_2 = c_List_Olist_ONil(T_b))) # label(fact_remdups__eq__nil__right__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3693 (all V_r_2 all V_q_2 all T_b c_Nitpick_Oplus__frac(T_b,V_q_2,V_r_2) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_b),c_Nitpick_Oint__lcm(hAPP(c_Nitpick_Odenom(T_b),V_q_2),hAPP(c_Nitpick_Odenom(T_b),V_r_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,T_b),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,T_b),tc_Int_Oint),c_Nitpick_Ofrac(T_b)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_b),V_q_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint)),hAPP(c_Nitpick_Odenom(T_b),V_q_2))))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_b),V_r_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint)),hAPP(c_Nitpick_Odenom(T_b),V_r_2)))))),c_COMBI(tc_Int_Oint)))) # label(fact_Nitpick_Oplus__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3694 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = V_n)) # label(fact_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3695 (all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) & hBOOL(hAPP(c_List_Odistinct(T_b),V_ys_2)) & c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),V_ys_2)))) # label(fact_distinct__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3696 (all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_less__add__one) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3697 (all V_B_2 all V_f_2 all T_c all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_a_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2))))) # label(fact_vimage__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3698 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) <-> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b)),V_x_2),V_xs_2))))) # label(fact_sorted__insort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3699 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))))))) # label(fact_min__max_OSup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3700 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))) <-> (exists B_f (c_Fun_Oinj__on(T_b,T_c,B_f,V_A_2) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,B_f),V_A_2)),V_B_2)))))))) # label(fact_inj__on__iff__card__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3701 (all V_z_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),V_z_2)))) # label(fact_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3702 (all V_N_2 all V_h_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_N_2))))))) # label(fact_min__max_OInf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3703 (all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),V_xs),c_List_Olist_ONil(T_a)) = V_xs) # label(fact_append__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3704 (all V_A_2 all V_x_2 (c_Nat__Transfer_Ois__nat(V_x_2) -> (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))))))) # label(fact_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3705 (all T_a hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a))) = c_Enum_Osublists(T_a,c_List_Olist_ONil(T_a))) # label(fact_sublists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3706 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) <-> c_Groups_Ozero__class_Ozero(T_b) != V_a_2))) # label(fact_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3707 (all T_a hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_a)),c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_lazy__sequence_Osize_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3708 (all V_z_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_z_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2)),V_z_2))))) # label(fact_Sup__least) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3709 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_B_2),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))))) # label(fact_Un__eq__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3710 (all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))) # label(fact_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3711 (all V_xs_2 all V_f_2 all V_P_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofilter(T_b,V_P_2),c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,V_xs_2)) = c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)))) # label(fact_filter__sort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3712 (all V_x all T_a (class_Fields_Ofield(T_a) & class_Int_Onumber__ring(T_a) -> V_x = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_divide__Numeral1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3713 (all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> c_Fun_Oinj__on(tc_List_Olist(T_b),tc_List_Olist(T_c),hAPP(c_List_Omap(T_b,T_c),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) # label(fact_inj__mapI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3714 (all V_A_2 all V_r_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),V_A_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),B_x),V_b_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_ImageE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3715 (all V_f_2 all V_X_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,V_X_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)))))) # label(fact_weak__coinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3716 (all V_B_2 all V_C_2 all T_c all V_a_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_b),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__extend__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3717 (all V_B_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2))))))) # label(fact_Sup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3718 (all V_P_2 hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Nat_Onat),V_P_2),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)))) = hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),V_P_2))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3719 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = V_A_2))) # label(fact_INT__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3720 (all V_r_2 all V_A_2 all T_b (c_Relation_Orefl__on(T_b,V_A_2,V_r_2) -> c_Relation_Orefl__on(tc_List_Olist(T_b),c_List_Olists(T_b,V_A_2),c_List_Olistrel(T_b,V_r_2)))) # label(fact_listrel__refl__on) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3721 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_trancl__rtrancl__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3722 (all V_k c_Int_OBit1(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_k)),V_k)) # label(fact_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3723 (all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_A_2)))) # label(fact_Inf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3724 (all V_n all T_a c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_n),c_Groups_Oone__class_Oone(tc_Int_Oint))) = c_Nitpick_Onumber__of__frac(T_a,V_n)) # label(fact_Nitpick_Onumber__of__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3725 (all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_a_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_B_2)) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_b,tc_HOL_Obool)),V_a_2),V_B_2))) # label(fact_Inter__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3726 (all V_r_2 all T_b c_Relation_Osym(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) # label(fact_sym__Int__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3727 (all T_b all V_r_2 all T_c ((all B_i c_Wellfounded_OwfP(T_c,hAPP(V_r_2,B_i))) -> ((all B_i all B_j (hAPP(V_r_2,B_j) != hAPP(V_r_2,B_i) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Predicate_ODomainP(T_c,T_c,hAPP(V_r_2,B_i))),c_Predicate_ORangeP(T_c,T_c,hAPP(V_r_2,B_j))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> c_Wellfounded_OwfP(T_c,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),V_r_2))))) # label(fact_wfP__SUP) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3728 (all V_xs_2 all V_f_2 all T_b all T_c c_List_Obutlast(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Obutlast(T_c,V_xs_2))) # label(fact_map__butlast) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3729 (all V_n all V_x hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x),V_n))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3730 (all V_R_2 all T_b ((all B_x all B_y (hBOOL(hAPP(hAPP(V_R_2,B_y),B_y)) & hAPP(V_R_2,B_x) = hAPP(V_R_2,B_y) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_x)) <-> hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)))) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) <-> c_Equiv__Relations_Opart__equivp(T_b,V_R_2))) # label(fact_part__equivp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3731 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_c all T_b (c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_H_2,V_b_H_2) = c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_2,V_b_2) <-> V_b_2 = V_b_H_2 & V_a_2 = V_a_H_2)) # label(fact_pair__box_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3732 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))))) # label(fact_le__maxI1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3733 (all V_k all V_l ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_SMT_Oz3mod(V_k,V_l) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) = c_SMT_Oz3mod(V_k,V_l)))) # label(fact_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3734 (all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)))) # label(fact_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3735 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_b) & class_Orderings_Oorder(T_c) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))))))) # label(fact_strict__monoD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3736 (all V_code__numeral c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3737 (all V_y_2 all V_x_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_x_2)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_atMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3738 (all V_xs_2 all T_b all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Odrop(T_b),V_m_2),V_xs_2))),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Odrop(T_b),V_n_2),V_xs_2)))))) # label(fact_set__drop__subset__set__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3739 (all V_xs_2 all V_x_2 all T_b hAPP(c_List_Oset(T_b),hAPP(c_List_OremoveAll(T_b,V_x_2),V_xs_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_set__removeAll) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3740 (all V_f_2 all T_c all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))))) # label(fact_image__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3741 (all V_xs_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> V_xs_2 = c_List_Oremove1(T_b,V_x_2,V_xs_2))) # label(fact_remove1__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3742 (all V_X_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),V_X_2)),V_X_2)) -> V_X_2 = hAPP(c_Relation_OImage(T_b,T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)),V_X_2))) # label(fact_Image__closed__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3743 (all T_b c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) = hAPP(c_Set_OPow(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_Pow__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3744 (all V_S_2 all T_c all T_b hAPP(c_Relation_ODomain(T_b,T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_S_2),c_Relation_ODomain(T_b,T_c))) # label(fact_Domain__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3745 (all T_b (class_Groups_Omonoid__add(T_b) -> c_List_Ofoldl(T_b,T_b,c_Groups_Oplus__class_Oplus(T_b),c_Groups_Ozero__class_Ozero(T_b)) = c_List_Omonoid__add__class_Olistsum(T_b))) # label(fact_listsum__foldl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3746 (all V_n_2 all V_m_2 (c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) <-> V_m_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) & c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n_2)) # label(fact_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3747 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_tl__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3748 (all V_xs_2 all V_f_2 all V_c_2 all T_c all T_b (class_Rings_Osemiring__0(T_b) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),hAPP(hAPP(c_COMBB(T_b,T_b,T_c),hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2)),V_f_2)),V_xs_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))))) # label(fact_listsum__const__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3749 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),c_List_Olists(T_b,V_A_2)),c_List_Olists(T_b,V_B_2))))) # label(fact_lists__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3750 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_right__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3751 (all V_b all V_a ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_OnegDivAlg(V_a,V_b) = c_Divides_Odivmod__int(V_a,V_b)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_Odivmod__int(V_a,V_b) = hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = c_Divides_Odivmod__int(V_a,V_b)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a -> c_Divides_Odivmod__int(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) = c_Divides_Odivmod__int(V_a,V_b)))))) # label(fact_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3752 (all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_x_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)))) # label(fact_image__uminus__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3753 (all V_R_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Otrancl(T_b,V_R_2))) <-> (exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),B_n),V_R_2))))))) # label(fact_trancl__power) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3754 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3755 (all V_y_2 all V_x_2 all T_b (class_Groups_Oordered__comm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_y_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_x_2),V_y_2) = c_Groups_Ozero__class_Ozero(T_b) <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_b) & c_Groups_Ozero__class_Ozero(T_b) = V_y_2))))) # label(fact_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3756 (all V_n all V_m all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3757 (all V_f_2 all V_A_2 all T_b all T_c (class_Rings_Ocomm__semiring__1(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((exists B_x (hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c))))) # label(fact_setprod__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3758 (all V_m_2 all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))))) # label(fact_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3759 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a)))) # label(fact_divide__self) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3760 (all V_n all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3761 (all V_b_2 all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omin(T_b)),V_b_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_a_2),V_b_2))))))) # label(fact_min__max_Ofold__inf__le__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3762 (all V_A_2 all V_P_2 all T_b all T_c hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(c_COMBC(T_c,T_b,tc_HOL_Obool),V_P_2)))) = hAPP(c_Relation_OImage(T_c,T_b,hAPP(c_Set_OCollect(tc_prod(T_c,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_b,tc_HOL_Obool),V_P_2))),V_A_2)) # label(fact_Image__Collect__split) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3763 (all V_n_2 all V_m_2 all T_b c_List_Orotate(T_b,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)) = hAPP(c_Fun_Ocomp(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Orotate(T_b,V_m_2)),c_List_Orotate(T_b,V_n_2))) # label(fact_rotate__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3764 (all V_x all V_ys all V_xs all T_a all V_n ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n),V_x) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_n),V_x)),V_ys)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n),V_x) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist__update(T_a,V_ys),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_x))))) # label(fact_list__update__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3765 (all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_a_2) <-> V_a_2 = c_List_Olist_ONil(T_b) | (exists B_xs exists B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),hAPP(c_List_Oset(T_b),B_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),B_xa)))) & c_List_Olinorder__class_Osorted(T_b,B_xs) & hAPP(hAPP(c_List_Olist_OCons(T_b),B_x),B_xs) = V_a_2))))) # label(fact_sorted_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3766 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_b),V_A_2)),hAPP(c_Set_OPow(T_b),V_B_2))),hAPP(c_Set_OPow(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Un__Pow__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3767 (all V_k all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k)) # label(fact_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3768 (all V_n_2 all V_m_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | (exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_n_2)) & hAPP(c_Nat_OSuc,B_j) = V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3769 (all V_l_2 all V_k_2 (c_Int_OBit1(V_l_2) = c_Int_OBit1(V_k_2) <-> V_l_2 = V_k_2)) # label(fact_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3770 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(c_List_Oset(T_b),V_ys_2))) -> (hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2) <-> V_ys_2 = V_xs_2))) # label(fact_inj__on__map__eq__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3771 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2) -> V_x_2 = V_y_2))) # label(fact_injD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3772 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__left__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3773 (all V_g_2 all V_f_2 all T_b all T_c (class_Orderings_Oord(T_c) -> ((all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3774 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n) = V_m)) # label(fact_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3775 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x))))) # label(fact_leI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3776 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y) = V_x))) # label(fact_min__max_Oinf__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3777 (all V_ns_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & (all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_List_Oset(tc_Nat_Onat),V_ns_2))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B_x)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_ns_2))) # label(fact_sum__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3778 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))))) # label(fact_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3779 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_d)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))))) # label(fact_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3780 (all T_b c_Set_OBall(T_b) = c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_HOL_Obool)) # label(fact_INFI__bool__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3781 (all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_c) & class_Orderings_Oorder(T_b) -> (c_Orderings_Oorder__class_Ostrict__mono(T_b,T_c,V_f_2) -> c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2)))) # label(fact_strict__mono__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3782 (all V_B_2 all T_c all T_b all V_y_2 all V_x_2 all V_A_2 (hBOOL(hAPP(hAPP(V_A_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2I1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3783 (all V_x_2 all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_ys_2))),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_listrel1I2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3784 (all V_u_2 hAPP(c_Nat_OSuc,V_u_2) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_u_2))) # label(fact_card__atMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3785 (all V_x_2 all V_y_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) -> (V_x_2 = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)))))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3786 (all V_y all V_x c_Nitpick_Oint__lcm(V_x,V_y) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Nitpick_Onat__lcm(hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_y))))) # label(fact_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 3787 (all V_B_2 all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2)))))) # label(fact_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3788 (all V_a_2 c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_a_2) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OatLeastAtMost__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3789 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)))) # label(fact_sup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3790 (all V_z_2 all V_R_2 all V_n_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_z_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2)))))) # label(fact_rel__pow__Suc__I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3791 (all V_n all V_m all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3792 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2)) = V_A_2)) # label(fact_Union__quotient) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3793 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) = c_Divides_OposDivAlg(V_a,V_b)))) # label(fact_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3794 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b)) # label(fact_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3795 (all V_c_2 all V_b_2 all V_w_2 all T_b (class_Int_Onumber(T_b) & class_Fields_Olinordered__field__inverse__zero(T_b) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)))))) # label(fact_less__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3796 (all V_A_2 all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) # label(fact_Compl__partition) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3797 (all V_p all T_b all T_a hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),hAPP(c_Product__Type_Ofst(T_a,T_b),V_p)),hAPP(c_Product__Type_Osnd(T_a,T_b),V_p)) = V_p) # label(fact_pair__collapse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3798 (all V_B_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))) # label(fact_Sigma__empty1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3799 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_j))))) # label(fact_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3800 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) -> V_xs = hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__length01) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3801 (all V_r_2 all V_A_2 all T_b (c_Relation_Ototal__on(T_b,V_A_2,V_r_2) <-> c_Relation_Ototal__on(T_b,V_A_2,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))) # label(fact_total__on__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3802 (all V_q all V_y all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_q))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3803 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Min__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3804 (all V_r_2 all T_b hAPP(c_Relation_ODomain(T_b,T_b),V_r_2) = hAPP(c_Relation_ODomain(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2))) # label(fact_trancl__domain) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3805 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) <-> c_Wellfounded_OwfP(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_r_2)))) # label(fact_wfP__wf__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3806 (all V_z_2 (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_z_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_z_2)),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3807 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2))))))) # label(fact_Min_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3808 (all V_y all V_n all V_x (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_y,V_n) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y),V_x)) -> (exists B_q hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),B_q)) = V_x)))) # label(fact_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3809 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c all T_d hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_d),V_x_2),V_y_2)) = hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),hAPP(V_f_2,V_y_2))) # label(fact_apsnd__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3810 (all V_n_2 all V_m_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_n_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))))) # label(fact_atLeastLessThanSuc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3811 (all V_P_2 all T_b (class_Orderings_Owellorder(T_b) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) -> hBOOL(hAPP(V_P_2,c_Orderings_Oord__class_OLeast(T_b,V_P_2)))))) # label(fact_LeastI__ex) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3812 (all V_w_2 all V_z_2 hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),c_COMBI(tc_Int_Oint)))) # label(fact_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3813 (all V_x_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2))) <-> hBOOL(hAPP(c_Predicate_ODomainP(T_b,T_c,hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_r_2)),V_x_2)))) # label(fact_DomainP__Domain__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3814 (all T_a hAPP(c_List_Otl(T_a),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_tl_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3815 (all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,V_xs_2) <-> (all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i)),hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_j)))))))))) # label(fact_sorted__equals__nth__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3816 (all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),c_Groups_Ozero__class_Ozero(T_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_divide__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3817 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__all(T_b,V_P_2,V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_list__all__iff__raw) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3818 (all V_xs_2 all V_x_2 all V_r_2 all T_b c_List_Oset__Cons(T_b,hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),hAPP(c_Relation_OImage(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Olistrel(T_b,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),V_xs_2),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) = hAPP(c_Relation_OImage(tc_List_Olist(T_b),tc_List_Olist(T_b),c_List_Olistrel(T_b,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) # label(fact_listrel__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3819 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3820 (all V_R_2 all V_f_2 all V_T_2 all T_c all V_S_2 all T_b all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d),hAPP(hAPP(c_COMBS(T_b,tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_d),tc_fun(tc_fun(T_c,tc_HOL_Obool),T_d),T_b),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d)),V_f_2)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),c_Set_OCollect(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_T_2)))),V_R_2)))),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_b,tc_HOL_Obool),T_d),hAPP(hAPP(c_COMBB(tc_fun(T_b,T_d),tc_fun(tc_fun(T_b,tc_HOL_Obool),T_d),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_d)),hAPP(c_COMBC(T_b,T_c,T_d),V_f_2))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),c_Set_OCollect(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2)))),hAPP(c_COMBC(T_b,T_c,tc_HOL_Obool),V_R_2))))),V_T_2))))) # label(fact_setsum__setsum__restrict) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3821 (all V_xss_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xss_2))) -> c_List_Olist_ONil(T_b) = B_x)) <-> c_List_Oconcat(T_b,V_xss_2) = c_List_Olist_ONil(T_b))) # label(fact_Nil__eq__concat__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3822 (all V_x_2 all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Oremove1(T_c,V_x_2,V_xs_2)))))) # label(fact_sorted__map__remove1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3823 (all V_f_2 all T_b hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_f_2) = c_Fun_Oid(T_b)) # label(fact_funpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3824 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_c))) # label(fact_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3825 (all V_u_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2))) # label(fact_ivl__disj__un_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3826 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Orderings_Oord__class_Omin(T_b),V_x_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))))))))) # label(fact_min__max_Oinf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3827 (all V_xs_2 all V_P_2 all T_b hAPP(c_Product__Type_Ofst(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Opartition(T_b,V_P_2,V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)) # label(fact_partition__filter1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3828 (all V_a_2 all V_y_2 all V_A_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (exists B_y_H (V_y_2 = hAPP(hAPP(V_f_2,V_a_2),B_y_H) & hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),B_y_H)))))))) # label(fact_fun__left__comm_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3829 (all V_f_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_x),V_S_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_x)))) -> ((all B_T1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_T1),V_S_2)) -> (all B_T2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_T2),V_S_2)) -> (B_T2 != B_T1 -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_T1)) -> (hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_T2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(V_f_2,B_x))))))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2)),V_S_2)))))) # label(fact_setsum__UNION__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3830 (all V_C_2 all V_A_2 all T_b ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_X),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),B_X)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_C_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),V_A_2))))) # label(fact_Inter__greatest) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3831 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3832 (all V_n_2 all V_m_2 all T_b (class_Nat_Osemiring__char__0(T_b) -> (hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2) <-> V_n_2 = V_m_2))) # label(fact_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3833 (all V_A_2 all V_C_2 all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Un__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3834 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))))) # label(fact_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3835 (all V_A_2 all V_z_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_b,V_f_2),V_z_2),V_A_2) = c_HOL_OThe(T_b,c_Finite__Set_Ofold__graph(T_c,T_b,V_f_2,V_z_2,V_A_2))) # label(fact_fold__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3836 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))))) # label(fact_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3837 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (exists B_x (c_List_Olinorder__class_Osorted(T_b,B_x) & hBOOL(hAPP(c_List_Odistinct(T_b),B_x)) & (all B_y (V_A_2 = hAPP(c_List_Oset(T_b),B_y) & hBOOL(hAPP(c_List_Odistinct(T_b),B_y)) & c_List_Olinorder__class_Osorted(T_b,B_y) -> B_x = B_y)) & V_A_2 = hAPP(c_List_Oset(T_b),B_x)))))) # label(fact_finite__sorted__distinct__unique) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3838 (all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_m))))) # label(fact_le__cube) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3839 (all V_n_2 all V_m_2 all T_b (class_Rings_Olinordered__semidom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_n_2)))))) # label(fact_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3840 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_b),V_A_2)))) # label(fact_Pow__top) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3841 (all V_ys all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) != hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys) -> V_xs != V_ys)) # label(fact_neq__if__length__neq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3842 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_drop__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3843 (all V_l_2 hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Int_Oint),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_l_2) = c_List_Oembed__list(V_l_2)) # label(fact_embed__list__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3844 (all V_xs all V_i all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__take) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3845 (all V_k c_Int_OBit1(V_k) = c_Int_Osucc(c_Int_OBit0(V_k))) # label(fact_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3846 (all V_i_2 all V_xs_2 all V_A_2 all T_b hAPP(hAPP(c_List_Omap(tc_prod(T_b,tc_Nat_Onat),T_b),c_Product__Type_Ofst(T_b,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_b,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_prod(T_b,tc_Nat_Onat)),c_member(tc_Nat_Onat)),c_Product__Type_Osnd(T_b,tc_Nat_Onat))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_b,tc_Nat_Onat),V_xs_2),c_List_Oupt(V_i_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))))) = hAPP(hAPP(c_List_Omap(tc_prod(T_b,tc_Nat_Onat),T_b),c_Product__Type_Ofst(T_b,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_b,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_prod(T_b,tc_Nat_Onat)),c_member(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_prod(T_b,tc_Nat_Onat),tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_prod(T_b,tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Product__Type_Osnd(T_b,tc_Nat_Onat))),V_i_2))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_b,tc_Nat_Onat),V_xs_2),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))))) # label(fact_sublist__shift__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3847 (all T_2 all T_1 (class_Lattices_Obounded__lattice(T_1) -> class_Lattices_Obounded__lattice__top(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3848 (all V_B_2 all V_A_2 all T_c all T_b hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2)))) = c_Nitpick_Oprod(T_b,T_c,V_A_2,V_B_2)) # label(fact_Nitpick_Oprod__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3849 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a)) & (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(T_a)))) # label(fact_divide__self__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3850 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oordered__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b_2),V_a_2)),V_e_2)),V_d_2)))))) # label(fact_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3851 (all V_x_2 all V_r_2 all T_b (c_Predicate_Oreflp(T_b,V_r_2) -> hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_x_2)))) # label(fact_reflpE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3852 (all V_z_2 all V_g_2 all V_f_2 all T_c all T_b V_z_2 = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_b,T_c,V_f_2),V_g_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_fold__image__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3853 (all V_x hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_x)) = V_x) # label(fact_nat__of__inverse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3854 (all V_x all V_i all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)) -> V_xs = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x))) # label(fact_list__update__beyond) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3855 (all V_P_2 all V_n_2 all V_m_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_P_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(V_P_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_P_2),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_m_2,V_n_2)))))) # label(fact_setsum__head) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3856 (all V_b_2 all V_a_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_b,T_c,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)) -> hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))))) # label(fact_curryE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3857 (all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) -> hAPP(c_List_Orotate1(T_a),V_xs) = V_xs)) # label(fact_rotate1__length01) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3858 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m)) = hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,hAPP(c_Code__Numeral_Oof__nat,V_n)),hAPP(c_Code__Numeral_Oof__nat,V_m))) # label(fact_subtract__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3859 (all V_xs_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_List_Oset(T_b),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)))) # label(fact_card__length) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3860 (all V_C_2 all V_g_2 all T_d all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,V_B_2) -> (c_Fun_Obij__betw(T_c,T_d,V_g_2,V_B_2,V_C_2) -> c_Fun_Obij__betw(T_b,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_g_2),V_f_2),V_A_2,V_C_2)))) # label(fact_bij__betw__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3861 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)))) # label(fact_times__divide__eq__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3862 (all V_f_2 all V_g_2 all T_c all V_X_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_g_2),V_X_2)),hAPP(V_f_2,hAPP(c_Set_Oimage(T_b,T_c,V_g_2),V_X_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_g_2,V_a_2)),c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_c,tc_HOL_Obool),V_f_2)))))) # label(fact_weak__coinduct__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3863 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a))) # label(fact_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3864 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3865 (all V_B_2 all V_f_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_c),c_Fun_Oid(T_c)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_B_2))))) # label(fact_setsum__reindex__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3866 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)))) # label(fact_diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3867 (all V_n V_n = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_n)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) # label(fact_Suc__code__numeral__minus__one) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3868 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_b_2),V_a_2)),V_r_2)))) # label(fact_converseD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3869 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Oone__class_Oone(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3870 (all V_ys_2 all V_xs_2 all T_b hAPP(c_List_Otl(T_b),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(T_b),T_b,hAPP(c_List_Otl(T_b),V_ys_2),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Oappend(T_b)),V_ys_2))),V_xs_2)) # label(fact_tl__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3871 (all T_b (class_Orderings_Olinorder(T_b) -> c_Finite__Set_Ofun__left__comm(T_b,tc_List_Olist(T_b),c_List_Olinorder__class_Oinsort__key(T_b,T_b,c_COMBI(T_b))))) # label(fact_fun__left__comm__insort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3872 (all V_y_2 all V_x_2 all T_b (class_Lattices_Oboolean__algebra(T_b) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2) <-> V_x_2 = V_y_2))) # label(fact_compl__eq__compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3873 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3874 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3875 (all V_R_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_R_2) -> c_Wellfounded_Owf(T_b,V_R_2))) # label(fact_wf__no__loop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3876 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_A_2))) # label(fact_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3877 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3878 (all V_R_2 all V_g_2 all V_f_2 all T_e all T_d all V_c_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),V_c_2),hAPP(c_Set_Oimage(tc_prod(T_d,T_e),tc_prod(T_b,T_c),c_Product__Type_Omap__pair(T_d,T_b,T_e,T_c,V_f_2,V_g_2)),V_R_2))) -> -(all B_x all B_y (hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_y)) = V_c_2 -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),B_x),B_y)),V_R_2)))))) # label(fact_prod__fun__imageE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3879 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) # label(fact_minus__mult__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3880 (all V_ys_2 all T_c all V_y_2 all V_x_2 all V_f_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),V_ys_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))))) # label(fact_insort__key_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3881 (all V_xs all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_listsum__rev) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3882 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_c)))) # label(fact_inf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3883 (all V_B_2 all T_b all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)) = V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_less__by__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3884 (all V_code__numeral hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3885 (all V_f_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_card__image__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3886 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)))) # label(fact_Min_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3887 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j))))) # label(fact_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3888 (all V_b_2 all V_a_2 all T_b (class_Groups_Ogroup__add(T_b) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2) = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2) = V_b_2))) # label(fact_minus__equation__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3889 (all V_a_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_a_2),V_A_2)))) # label(fact_Sup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3890 (all V_ys_2 all V_xs_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) # label(fact_list__all2__lengthD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3891 (all V_ys all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Osplice(T_a,V_xs,V_ys)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) # label(fact_length__splice) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3892 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) <-> (all B_Q ((exists B_x hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_Q))) -> (exists B_x ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_x)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),B_y),B_Q)))) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),B_Q)))))))) # label(fact_wf__eq__minimal) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3893 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3894 (all V_f_2 all V_P_2 all V_xs_2 all T_b all T_c (class_Groups_Omonoid__add(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> (-hBOOL(hAPP(V_P_2,B_x)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_c)))) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2))))) # label(fact_listsum__map__filter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3895 (all V_b all V_a all T_a (class_Rings_Oordered__cancel__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))))) # label(fact_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3896 (all V_n hAPP(c_Nat_OSuc,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 3897 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setprod_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3898 (all V_M_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_M_2)) -> (exists B_h c_Fun_Obij__betw(tc_Nat_Onat,T_b,B_h,hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),V_M_2)),V_M_2)))) # label(fact_ex__bij__betw__nat__finite) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3899 (all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_minus__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3900 (all V_B_2 all V_k_2 all V_A_2 all T_b ((all B_n hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)),V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_n),V_k_2))),V_B_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_A_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_B_2))) # label(fact_UN__finite2__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3901 (all V_a all T_a (class_Rings_Omult__zero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3902 (all V_ys all V_y all V_xs all V_n all T_a (hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys) -> V_y = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n))) # label(fact_nth__via__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3903 (all V_k c_Int_Opred(c_Int_OBit1(V_k)) = c_Int_OBit0(V_k)) # label(fact_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3904 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3905 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2))))) # label(fact_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3906 (all V_n_2 hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2)) # label(fact_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3907 (all V_h_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_b),c_Groups_Otimes__class_Otimes(T_c)),V_g_2)),V_h_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_h_2),V_A_2))))) # label(fact_setprod_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3908 (all V_x_2 all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))))) # label(fact_setprod_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3909 (all V_i_2 all V_P_2 hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),V_P_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2)))))) # label(fact_finite__M__bounded__by__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3910 (all V_xs_2 all T_b c_Random_Opick(T_b,V_xs_2) = c_Random_Opick(T_b,hAPP(c_List_Ofilter(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Code__Numeral_Ocode__numeral,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_COMBK(tc_HOL_Obool,T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))))),V_xs_2))) # label(fact_pick__drop__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3911 (all V_R_2 all V_p_2 all T_b ((exists B_n hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),B_n),V_R_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Ortrancl(T_b,V_R_2))))) # label(fact_rtrancl__power) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3912 (all V_n all V_m (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m) -> V_n = V_m))) # label(fact_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3913 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_B_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,T_c),V_A_2),V_f_2)))))) # label(fact_INF__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3914 (all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_a) = V_a)) # label(fact_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3915 (all V_P_2 all V_A_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_F_2),V_A_2)) -> (hBOOL(hAPP(V_P_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> ((all B_a all B_F (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),B_F)) -> (hBOOL(hAPP(hAPP(c_member(T_b),B_a),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),B_a),B_F)) -> (hBOOL(hAPP(V_P_2,B_F)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Set_Oinsert(T_b),B_a),B_F)))))))) -> hBOOL(hAPP(V_P_2,V_F_2))))))) # label(fact_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3916 (all V_a all V_b all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)))))) # label(fact_le__supI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3917 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) <-> V_x_2 = V_y_2))))) # label(fact_inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3918 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3919 (all V_P_2 all T_b V_P_2 = hAPP(c_Set_OCollect(T_b),V_P_2)) # label(fact_Collect__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3920 (all V_r_H all V_q_H all V_b_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q_H),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) # label(fact_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3921 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3922 (all V_d_2 all V_c_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_a_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2),V_d_2))) # label(fact_Int__atLeastAtMostR2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3923 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = c_Groups_Ozero__class_Ozero(T_a) -> V_a = c_Groups_Ozero__class_Ozero(T_a) | c_Groups_Ozero__class_Ozero(T_a) = V_b))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3924 (all V_c_2 all V_b_2 all V_a_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)))))) # label(fact_less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3925 (all V_A_2 all V_f_2 all T_d all T_c all T_b all V_I_2 (V_I_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_I_2)) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(V_A_2,B_i)))) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_inj__on__INTER) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3926 (all V_k all V_v_H all V_v ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H))),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k))))) # label(fact_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3927 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_divide__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3928 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> -(hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))))) # label(fact_IntE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3929 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_B_2)),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) -> V_A_2 = V_B_2)))) # label(fact_card__seteq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3930 (all V_b all V_x all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_x)))))) # label(fact_le__supI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3931 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2))))))) # label(fact_folding_Ounion) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3932 (all V_n_2 ((exists B_m V_n_2 = hAPP(c_Nat_OSuc,B_m)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3933 (all V_R_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_R_2)),c_Relation_OId(T_b)) = c_Transitive__Closure_Ortrancl(T_b,V_R_2)) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3934 (all V_d_2 all V_b_2 all V_m_2 all T_c all T_b all V_c_2 all V_a_2 (V_a_2 != V_c_2 -> c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_m_2,V_c_2,V_d_2),V_a_2,V_b_2) = c_Fun_Ofun__upd(T_b,T_c,c_Fun_Ofun__upd(T_b,T_c,V_m_2,V_a_2,V_b_2),V_c_2,V_d_2))) # label(fact_fun__upd__twist) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3935 (all V_r_2 all V_I_2 all T_e all V_s_2 all T_c all T_d all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),T_e),hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_s_2)),V_r_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_s_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_I_2),V_r_2))) # label(fact_rel__comp__UNION__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3936 (all V_lazy__sequence_H all V_a_H all T_a c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a_H,V_lazy__sequence_H) != c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a)) # label(fact_lazy__sequence_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3937 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3938 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y))) # label(fact_inf__sup__aci_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3939 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3940 (all V_R_2 all V_n_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Ortrancl(T_b,V_R_2))))) # label(fact_rel__pow__imp__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3941 (all V_x_2 all V_g_2 all V_f_2 all T_b all T_e all T_d all T_c hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_e,T_b),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_b),V_g_2),hAPP(V_f_2,V_x_2))) # label(fact_scomp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3942 (all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),c_List_Olist_ONil(T_b)) = c_List_Olist_ONil(tc_prod(T_a,T_b))) # label(fact_zip_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3943 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OField(T_b),V_r_2)),hAPP(c_Relation_OField(T_b),V_s_2))))) # label(fact_mono__Field) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3944 (all V_B_2 all V_A_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_subset__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3945 (all V_x_2 all T_b hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_iso__tuple__UNIV__I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3946 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,T_b,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_COMBK(tc_Nat_Onat,T_b),c_Groups_Oone__class_Oone(tc_Nat_Onat))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) # label(fact_card_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3947 (all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_bij__is__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3948 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3949 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3950 (all V_g_2 all V_f_2 all T_c all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_g_2)))))) # label(fact_UN__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3951 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_Opred(V_k_2)),V_l_2)))) # label(fact_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3952 (all V_w_2 all V_v_2 all T_b (class_Int_Onumber(T_b) & class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))))) # label(fact_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3953 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_power__0__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3954 (all V_xs all T_a all V_y all V_x ((V_y != V_x -> hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),hAPP(c_List_OremoveAll(T_a,V_x),V_xs))) & (V_y = V_x -> hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(c_List_OremoveAll(T_a,V_x),V_xs)))) # label(fact_removeAll_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3955 (all V_A_2 all V_C_2 all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_C_2)),V_A_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_x),V_C_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),B_x),V_A_2))))) # label(fact_Union__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3956 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2)))))) # label(fact_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3957 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ry))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3958 (all V_b all V_a hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_Odivmod__int(V_a,V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) # label(fact_div__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3959 (all V_x_2 all V_P_2 all T_b (class_Orderings_Oorder(T_b) -> (hBOOL(hAPP(V_P_2,V_x_2)) -> ((all B_y (hBOOL(hAPP(V_P_2,B_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),B_y)))) -> c_Orderings_Oord__class_OLeast(T_b,V_P_2) = V_x_2)))) # label(fact_Least__equality) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3960 (all V_n_2 all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Nat_OSuc,V_n_2) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_fequal),c_Int_Onat)),V_n_2))))))) # label(fact_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3961 (all V_a_2 all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Relation_OImage(T_b,T_b,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_equiv__class__self) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3962 (all V_k_2 all V_N_2 all T_b (class_Groups_Olinordered__ab__semigroup__add(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(T_b),V_k_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_N_2))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_k_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_N_2)))))) # label(fact_add__Max__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3963 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Un__upper1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3964 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3965 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3966 (all V_k (hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != V_k -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_k)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Code__Numeral_Onat__of,V_k))))) # label(fact_code__numeral__decr) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3967 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)))) # label(fact_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3968 (all V_R_2 all T_b (c_Predicate_Oreflp(T_b,V_R_2) -> (c_Predicate_Osymp(T_b,V_R_2) -> (c_Predicate_Otransp(T_b,V_R_2) -> c_Equiv__Relations_Oequivp(T_b,V_R_2))))) # label(fact_equivpI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3969 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_j)))))) # label(fact_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3970 (all V_a all T_a (class_Rings_Olinordered__ring(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))))) # label(fact_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3971 (all T_c all V_A_2 all V_c_2 all T_b ((-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_c,T_b,hAPP(c_COMBK(T_b,T_c),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) & (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_c,T_b,hAPP(c_COMBK(T_b,T_c),V_c_2)),V_A_2) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_vimage__const) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3972 (all V_b all V_x all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_x)))))) # label(fact_min__max_Ole__supI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3973 (all V_ys all V_xs all V_n all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Odrop(T_b),V_n),V_ys)) = hAPP(hAPP(c_List_Odrop(tc_prod(T_a,T_b)),V_n),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))) # label(fact_drop__zip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3974 (all V_y_2 all V_x_2 all V_S_2 all V_R_2 all T_c all T_b (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_S_2)),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_sup__Un__eq2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3975 (all V_xs all T_a c_List_Oremdups(T_a,c_List_Oremdups(T_a,V_xs)) = c_List_Oremdups(T_a,V_xs)) # label(fact_remdups__remdups) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3976 (all V_x_2 all V_y_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding(T_b,T_c,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_y_2)),hAPP(V_f_2,V_x_2)))) # label(fact_folding_Ocommute__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3977 (all V_a2_2 all V_a1_2 all T_c all T_b c_Nitpick_Ofold__graph_H(T_b,T_c,V_a1_2,V_a2_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_a2_2)) # label(fact_fold__graph_H_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3978 (all V_A_2 all T_b hAPP(c_Set_OPow(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = hAPP(c_Set_OCollect(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),c_HOL_OEx(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBS(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_fconj)),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool))),c_fequal)),c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),c_Set_OPow(T_b)))))) # label(fact_Pow__Compl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3979 (all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_subset__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3980 (all V_ys_2 all V_xs_2 all T_b (V_xs_2 = V_ys_2 <-> hAPP(c_List_Orev(T_b),V_xs_2) = hAPP(c_List_Orev(T_b),V_ys_2))) # label(fact_rev__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3981 (all T_2 all T_1 (class_Complete__Lattice_Ocomplete__lattice(T_1) -> class_Complete__Lattice_Ocomplete__lattice(tc_fun(T_2,T_1)))) # label(arity_fun__Complete__Lattice_Ocomplete__lattice) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3982 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3983 (all V_z_2 all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__max_Ole__sup__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3984 (all V_A_2 all V_I_2 all T_c all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2)),V_A_2))) # label(fact_Un__INT__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3985 (all V_g_2 all V_B_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setsum__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3986 (all V_b_2 all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2))))) # label(fact_atLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3987 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_u_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y_2),V_v_2))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3988 (all V_r_2 all T_b hAPP(c_Relation_OField(T_b),V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_b),V_r_2)),hAPP(c_Relation_ORange(T_b,T_b),V_r_2))) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3989 (all V_a_2 all V_A_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_A_2)),V_a_2)))))) # label(fact_fold1__belowI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3990 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) -> hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs) = c_List_Olist_ONil(T_a))) # label(fact_drop__all) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3991 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3992 (all V_a_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))))) # label(fact_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3993 (all V_xs_2 all T_c all V_y_2 all V_x_2 all V_f_2 all T_b (class_Orderings_Olinorder(T_b) -> (hAPP(V_f_2,V_x_2) != hAPP(V_f_2,V_y_2) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_y_2),V_xs_2))))) # label(fact_insort__key__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3994 (all V_f_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (c_Orderings_Oorder__class_Omono(T_b,T_b,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2)),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(T_b,V_f_2))))))) # label(fact_lfp__lemma3) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3995 (all V_ys_2 all V_x_2 all V_xs_2 all T_b (hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b))) = V_ys_2 <-> V_ys_2 != c_List_Olist_ONil(T_b) & c_List_Olast(T_b,V_ys_2) = V_x_2 & c_List_Obutlast(T_b,V_ys_2) = V_xs_2)) # label(fact_snoc__eq__iff__butlast) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3996 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Rings_Olinordered__ring__strict(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),V_a_2)))))) # label(fact_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3997 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_m_2),V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3998 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_z3)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_z2)),V_z3)) # label(fact_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 3999 (all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_x)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_x)) = V_x)) # label(fact_mult__sgn__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4000 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4001 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_m_2))))) # label(fact_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4002 (all V_y_2 all V_x_2 (hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2) <-> V_y_2 = V_x_2)) # label(fact_transfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4003 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_minus__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4004 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_divide__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4005 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> c_Fun_Obij__betw(T_b,T_c,V_f_2,V_A_2,hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))) # label(fact_inj__on__imp__bij__betw) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4006 (all V_x all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_x)) -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)))))) # label(fact_min__max_Ole__supE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4007 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)))) # label(fact_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4008 (all V_y_2 all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))))) # label(fact_card__Diff2__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4009 (all V_x all T_a all V_xs V_xs != hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) # label(fact_not__Cons__self) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4010 (all V_r_2 all V_S_2 all T_d all T_b all T_c hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_S_2),V_r_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_S_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),T_d),c_Relation_Oconverse(T_c,T_b)),V_r_2))) # label(fact_converse__UNION) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 4011 (all V_z_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))))) # label(fact_of__int__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4012 (all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),V_b))) # label(fact_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4013 (all T_b (class_Orderings_Olinorder(T_b) & class_Lattices_Osemilattice__sup(T_b) -> c_Lattices_Osemilattice__sup__class_Osup(T_b) = c_Orderings_Oord__class_Omax(T_b))) # label(fact_sup__max) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4014 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))))) # label(fact_rtrancl_Ortrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4015 (all V_g_2 all V_x_2 all V_f_2 all T_c all T_b all T_d (hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_d,T_c),V_x_2)) = hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_d,T_c),V_x_2)) <-> hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_f_2),V_x_2) = hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_g_2),V_x_2))) # label(fact_apfst__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4016 (all V_g_2 all V_f_2 all T_e all T_d all V_R_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2))),hAPP(c_Set_Oimage(tc_prod(T_b,T_c),tc_prod(T_d,T_e),c_Product__Type_Omap__pair(T_b,T_d,T_c,T_e,V_f_2,V_g_2)),V_R_2))))) # label(fact_map__pair__imageI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4017 (all V_m all V_l hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),V_l),V_m) = c_List_Oreturn__list(hAPP(hAPP(c_List_Oappend(tc_Int_Oint),c_List_Oembed__list(V_l)),c_List_Oembed__list(V_m)))) # label(fact_transfer__nat__int__list__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4018 (all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x) = V_x)) # label(fact_Sup__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4019 (all V_n all V_m hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),c_Divides_Odivmod__nat(V_m,V_n)))) # label(fact_divmod__nat__rel__divmod__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4020 (all V_B_2 all T_b all V_x_2 all V_A_2 (hBOOL(hAPP(V_A_2,V_x_2)) -> hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1I1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4021 (all V_a c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin))) # label(fact_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4022 (all V_f_2 all V_F_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_F_2)) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,tc_Nat_Onat),V_f_2),V_F_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_setprod__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4023 (all V_x_2 all V_P_2 ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_y),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> hBOOL(hAPP(V_P_2,hAPP(c_Code__Numeral_Oof__nat,B_y))))) -> hBOOL(hAPP(V_P_2,V_x_2)))) # label(fact_of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4024 (all V_A_2 all V_f_2 all T_c all V_y_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_y_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) -> hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_b,V_A_2,V_f_2),V_y_2)) = V_y_2)) # label(fact_f__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4025 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_prod(T_2,T_1)))) # label(arity_prod__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4026 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))) # label(fact_card_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4027 (all T_b all V_P_2 ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> hBOOL(hAPP(V_P_2,c_Hilbert__Choice_OEps(T_b,V_P_2))))) # label(fact_some__eq__ex) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4028 (all V_xs_2 all V_x_2 all V_zs_2 all V_ys_2 all T_b ((exists B_ys_H (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),B_ys_H),V_zs_2) & V_ys_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_ys_H))) | V_zs_2 = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2) & V_ys_2 = c_List_Olist_ONil(T_b) <-> hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),V_zs_2) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) # label(fact_append__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4029 (all V_n all V_m hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) # label(fact_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4030 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(hAPP(c_List_Oappend(T_a),c_List_Obutlast(T_a,V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),c_List_Olast(T_a,V_xs)),c_List_Olist_ONil(T_a))) = V_xs)) # label(fact_append__butlast__last__id) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4031 (all V_z all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z)))) # label(fact_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4032 (all V_l all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_l)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_n)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_l),V_m)))))) # label(fact_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4033 (all V_a_2 all V_B_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)))) # label(fact_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4034 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_divide__minus1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4035 (all V_i_2 all V_k_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)),V_i_2)))) # label(fact_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4036 (all V_xs_2 all V_x_2 all T_b hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) # label(fact_List_Oset_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4037 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c)) # label(fact_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4038 (all V_x all V_z all V_y all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))) # label(fact_add__num__frac) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4039 (all V_ns_2 all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_ns_2))))) # label(fact_start__le__sum) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4040 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) <-> (all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (B_j != B_i -> hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i) != hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_j)))))))) # label(fact_distinct__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4041 (all V_a_2 all V_x_2 all V_A_2 all V_b_2 all T_b (class_Groups_Oab__semigroup__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_b_2,V_A_2),V_x_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_b,c_Groups_Otimes__class_Otimes(T_b),V_a_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),V_x_2))))))) # label(fact_fold__graph__permute__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4042 (all V_R_2 all T_b all T_c c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Relation_OImage(T_c,T_b,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Image__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4043 (all V_A_2 all T_b c_Relation_Osym(T_b,c_Relation_OId__on(T_b,V_A_2))) # label(fact_sym__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4044 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4045 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4046 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_x != V_y))) # label(fact_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4047 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)))))) # label(fact_min__max_Osup__Inf1__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4048 (all V_a_2 all V_c_2 all V_b_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2)) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))))))) # label(fact_divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4049 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_k))))) # label(fact_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4050 (all V_r_2 all V_A_2 all V_f_2 all T_c all T_b (class_Rings_Osemiring__0(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Groups_Otimes__class_Otimes(T_b)),V_f_2)),V_r_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),V_A_2)),V_r_2))) # label(fact_setsum__left__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4051 (all V_x_2 all T_b all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2))))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4052 (all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> c_Equiv__Relations_Oequiv(tc_List_Olist(T_b),c_List_Olists(T_b,V_A_2),c_List_Olistrel(T_b,V_r_2)))) # label(fact_equiv__listrel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4053 (all V_u_2 all V_f_2 ((all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_n),hAPP(V_f_2,B_n)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),V_f_2)),V_u_2)))))) # label(fact_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4054 (all V_f2_2 all V_f1_2 all T_c all T_b V_f1_2 = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_b,T_c,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c))) # label(fact_lazy__sequence_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4055 (all V_z ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4056 (all V_P_2 all V_f_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),V_f_2))) -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),V_f_2)),hAPP(c_Set_OCollect(T_b),V_P_2))))) -> hBOOL(hAPP(V_P_2,B_x)))) -> hBOOL(hAPP(V_P_2,V_a_2)))))) # label(fact_lfp__induct__set) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4057 (all V_xs_2 all V_x_2 all V_n_2 all T_b hAPP(hAPP(c_List_Odrop(T_b),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)) = c_Nat_Onat_Onat__case(tc_List_Olist(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Odrop(T_b)),V_xs_2),V_n_2)) # label(fact_drop__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4058 (all V_r_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),hAPP(c_Relation_OField(T_b),V_r_2)),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_Relation_OField(T_b),V_r_2)))))) # label(fact_trancl__subset__Field2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4059 (all V_ys_2 all V_xs_2 all T_b (c_List_Olist_ONil(T_b) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2) <-> c_List_Olist_ONil(T_b) = V_xs_2 & V_ys_2 = c_List_Olist_ONil(T_b))) # label(fact_Nil__is__append__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4060 (all V_g_2 all V_h_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Fun_Oinj__on(T_b,T_b,V_h_2,V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(c_Fun_Ocomp(T_b,T_c,T_b,V_g_2),V_h_2)),V_A_2))))) # label(fact_setsum_Oreindex) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4061 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4062 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_R2_2 all V_R1_2 all T_b c_New__DSequence_Opos__union(T_b,hAPP(hAPP(hAPP(V_R1_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(hAPP(V_R2_2,V_nrandom_2),V_size_2),V_seed_2)) = c_New__Random__Sequence_Opos__union(T_b,V_R1_2,V_R2_2,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__union__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4063 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) = c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) # label(fact_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4064 (all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4065 (all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Relation_OId(T_b))) <-> V_a_2 = V_b_2)) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4066 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4067 (all V_n_2 all T_b hAPP(hAPP(c_Nat_Ocompow(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_n_2),c_List_Orotate1(T_b)) = c_List_Orotate(T_b,V_n_2)) # label(fact_rotate__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4068 (all V_n_2 all V_P_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(V_P_2,B_i),B_j))))))) # label(fact_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4069 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4070 (all V_i_2 all V_xq_2 all V_f_2 all T_b all T_c c_Lazy__Sequence_Omap(T_c,T_b,V_f_2,hAPP(V_xq_2,V_i_2)) = c_New__DSequence_Opos__map(T_c,T_b,V_f_2,V_xq_2,V_i_2)) # label(fact_New__DSequence_Opos__map__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4071 (all V_x_2 all T_b c_Set_Othe__elem(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = V_x_2) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4072 (all V_x_2 all V_xs_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> (c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Olinorder__class_Oinsort__insert__key(T_c,T_b,V_f_2,V_x_2,V_xs_2)))))) # label(fact_sorted__insort__insert__key) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4073 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))))) # label(fact_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4074 (all V_z_2 all V_m_2 (V_z_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) <-> V_m_2 = hAPP(c_Int_Onat,V_z_2) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)))) # label(fact_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4075 (all V_i_2 all V_n_2 all V_inc_2 all T_b (class_Rings_Osemiring__1(T_b) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_b,V_inc_2,V_n_2,hAPP(V_inc_2,V_i_2)) = c_Nat_Osemiring__1__class_Oof__nat__aux(T_b,V_inc_2,hAPP(c_Nat_OSuc,V_n_2),V_i_2))) # label(fact_of__nat__aux_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4076 (all V_P_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Lazy__Sequence_Osmall__lazy_H__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_d all B_i (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Lazy__Sequence_Osmall__lazy_H__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_d),B_i))) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_d),B_i)) -> hBOOL(hAPP(hAPP(V_P_2,B_d),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_i),c_Groups_Oone__class_Oone(tc_Int_Oint))))) -> hBOOL(hAPP(hAPP(V_P_2,B_d),B_i))))) -> hBOOL(hAPP(hAPP(V_P_2,V_a0_2),V_a1_2))))) # label(fact_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4077 (all V_x_2 all V_P_2 all V_xs_2 all T_b ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i))))) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,V_x_2))))) # label(fact_all__nth__imp__all__set) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4078 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4079 (all V_ys_2 all V_y_2 all V_xs_2 all T_c all T_b hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_b,T_c)),T_b,c_List_Olist_ONil(tc_prod(T_b,T_c)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_prod(T_b,T_c)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_prod(T_b,T_c))),tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_prod(T_b,T_c)))),T_b),c_COMBB(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_List_Olist(tc_prod(T_b,T_c)),tc_List_Olist(tc_prod(T_b,T_c))),T_b),c_List_Olist_OCons(tc_prod(T_b,T_c))),hAPP(hAPP(c_COMBC(T_b,T_c,tc_prod(T_b,T_c)),c_Product__Type_OPair(T_b,T_c)),V_y_2)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_b,T_c))),c_List_Ozip(T_b,T_c)),V_ys_2))),V_xs_2)) # label(fact_zip__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4080 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) <-> -hBOOL(hAPP(c_Nat__Numeral_Oneg,V_x_2)))) # label(fact_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4081 (all V_R_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),V_R_2) = hAPP(c_Set_Oimage(tc_prod(T_c,T_b),T_b,c_Product__Type_Osnd(T_c,T_b)),V_R_2)) # label(fact_snd__eq__Range) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4082 (all V_g_2 all V_f_2 all V_P_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_b),V_P_2)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),V_P_2))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,T_c),hAPP(hAPP(c_COMBS(T_b,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_b),c_If(T_c)),V_P_2)),V_f_2)),V_g_2)),V_A_2)))) # label(fact_setsum__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4083 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_b),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_b),tc_fun(tc_fun(T_c,tc_HOL_Obool),T_b),T_d),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b)),hAPP(c_COMBC(T_c,T_d,T_b),V_f_2))),V_A_2)),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),T_b),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_b),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_b),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_b)),V_f_2)),V_B_2)),V_A_2))) # label(fact_setsum__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4084 (all V_g_2 all T_d all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (c_Fun_Obij__betw(T_c,T_d,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> c_Fun_Obij__betw(T_b,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_g_2),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_bij__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4085 (all V_z_2 all V_f_2 all T_c all T_b hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_z_2))) # label(fact_fold__graph_OemptyI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4086 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hAPP(c_List_Ohd(tc_Nat_Onat),c_List_Oupt(V_i,V_j)) = V_i)) # label(fact_hd__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4087 (all V_f_2 all T_b all V_m_2 all V_n_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_m_2))) -> hAPP(hAPP(c_List_Onth(T_b),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_b),V_f_2),c_List_Oupt(V_m_2,V_n_2))),V_i_2) = hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_i_2)))) # label(fact_nth__map__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4088 (all V_r_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_List_Olist_ONil(T_b))),c_List_Olistrel(T_b,V_r_2)))) # label(fact_listrel_ONil) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4089 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_z)))))) # label(fact_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4090 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool))),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2)) # label(fact_UN__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4091 (all V_x all T_a c_List_Olist_ONil(T_a) = c_List_Oremove1(T_a,V_x,c_List_Olist_ONil(T_a))) # label(fact_remove1_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4092 (all V_a_2 all V_b_2 all T_b (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_b) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b_2),V_a_2) = V_b_2 <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4093 (all V_k_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))))) -> V_A_2 = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))))) # label(fact_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4094 (all V_x_2 all V_f_2 all V_A_2 all T_d all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),V_A_2),hAPP(hAPP(c_COMBC(T_c,T_d,T_b),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,T_b)),V_A_2),V_f_2),V_x_2))) # label(fact_SUPR__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4095 (all T_c all T_b c_Product__Type_Oapfst(T_b,T_b,T_c,c_Fun_Oid(T_b)) = c_Fun_Oid(tc_prod(T_b,T_c))) # label(fact_apfst__id) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4096 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))))) # label(fact_Compl__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4097 (all V_C_2 all V_B_2 all T_d all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_INT__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4098 (all V_r_2 all T_b all T_c hAPP(c_Relation_ORange(T_c,T_b),hAPP(c_Relation_Oconverse(T_b,T_c),V_r_2)) = hAPP(c_Relation_ODomain(T_b,T_c),V_r_2)) # label(fact_Range__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4099 (all V_A_2 all V_R_2 all T_b (c_Wellfounded_Owf(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Relation_OImage(T_b,T_b,V_R_2),V_A_2))) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_A_2))) # label(fact_wfE__pf) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4100 (all V_u_2 all V_l_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2)),V_u_2) = c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4101 (all V_code__numeral_H_2 all V_code__numeral_2 (c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_2) = c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2) <-> V_code__numeral_H_2 = V_code__numeral_2)) # label(fact_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4102 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))),c_List_Olistrel1(T_b,V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) & V_x_2 = V_y_2 | V_ys_2 = V_xs_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)))) # label(fact_Cons__listrel1__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4103 (all T_1 all T_2 (class_Lazy__Sequence_Osmall__lazy(T_2) & class_Lazy__Sequence_Osmall__lazy(T_1) -> class_Lazy__Sequence_Osmall__lazy(tc_prod(T_2,T_1)))) # label(arity_prod__Lazy__Sequence_Osmall__lazy) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4104 (all V_y_2 all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_x_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_y_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) <-> c_Groups_Ozero__class_Ozero(T_b) = V_x_2 & c_Groups_Ozero__class_Ozero(T_b) = V_y_2))) # label(fact_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4105 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_a_2,V_b_2) -> (c_List_Olist__all2(T_b,T_c,V_P_2,V_c_2,V_d_2) -> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Oappend(T_b),V_a_2),V_c_2),hAPP(hAPP(c_List_Oappend(T_c),V_b_2),V_d_2))))) # label(fact_list__all2__appendI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4106 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,tc_fun(T_b,tc_HOL_Obool),c_Set_Oinsert(T_b)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4107 (all V_k_2 all V_less__eq_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatMost(T_b,V_less__eq_2,V_k_2))) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)))) # label(fact_ord_OatMost__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4108 (all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) = V_A_2)) # label(fact_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4109 (all V_n_2 all V_m_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Orderings_Olinorder(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hAPP(V_f_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_m_2),V_n_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),hAPP(V_f_2,V_m_2)),hAPP(V_f_2,V_n_2))))) # label(fact_max__of__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4110 (all V_B_2 all V_A_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,B_x))))))) # label(fact_UN__E) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4111 (all V_y all V_z all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4112 (all V_c all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4113 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_minus__mult__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4114 (all V_A_2 all T_b ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Nitpick_Ocard_H(T_b,V_A_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_Hilbert__Choice_OEps(tc_List_Olist(T_b),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_List_Oset(T_b))),V_A_2))),c_List_Odistinct(T_b))))) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Nitpick_Ocard_H(T_b,V_A_2)))) # label(fact_Nitpick_Ocard_H__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4115 (all V_r_2 all T_b c_List_Olex(T_b,V_r_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),c_List_Olexn(T_b,V_r_2))) # label(fact_lex__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4116 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) # label(fact_min__max_Osup_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4117 (all V_B_2 all V_A_2 all T_b (class_Lattices_Oab__semigroup__idem__mult(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_b,c_Groups_Otimes__class_Otimes(T_b)),V_B_2)))))))) # label(fact_fold1__Un2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4118 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)))) # label(fact_butlast__take) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4119 (all V_A_H_2 all T_c all T_b all V_A_2 (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> ((exists B_f (c_Fun_Oinj__on(T_b,T_c,B_f,V_A_2) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,B_f),V_A_2)),V_A_H_2)))) <-> (exists B_g hAPP(c_Set_Oimage(T_c,T_b,B_g),V_A_H_2) = V_A_2)))) # label(fact_inj__on__iff__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4120 (all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4121 (all T_b c_Fun_Oid(T_b) = c_Hilbert__Choice_Oinv__into(T_b,T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Fun_Oid(T_b))) # label(fact_inv__id) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4122 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b))))) # label(fact_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4123 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> V_xs_2 = c_List_Olinorder__class_Oinsort__insert__key(T_b,T_b,c_COMBI(T_b),V_x_2,V_xs_2)))) # label(fact_insort__insert__triv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4124 (all V_r_2 all V_a_2 all T_b hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)))) # label(fact_rtrancl_Ortrancl__refl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4125 (all V_zs all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (c_List_Olinorder__class_Osorted(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_zs)) -> c_List_Olinorder__class_Osorted(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_zs))))))) # label(fact_sorted__many) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4126 (all V_S_2 all V_f_2 all T_b all T_c hAPP(c_Set_Oimage(T_c,T_b,V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_S_2),c_Set_Oimage(T_c,T_b,V_f_2))) # label(fact_image__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4127 (all V_n all V_m all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))) # label(fact_power__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4128 (all V_v all V_i all T_a hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_v) = c_List_Olist_ONil(T_a)) # label(fact_list__update_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 4129 (all V_z_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),V_z_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_of__int__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4130 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))) # label(fact_inf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4131 (all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),hAPP(c_List_Otl(T_a),V_xs))))) # label(fact_distinct__tl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4132 (all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n) # label(fact_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4133 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = V_x)) # label(fact_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4134 (all V_x_2 all V_Q_2 all V_P_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_P_2),V_Q_2)) -> (hBOOL(hAPP(V_P_2,V_x_2)) -> hBOOL(hAPP(V_Q_2,V_x_2))))) # label(fact_predicate1D) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4135 (all V_q all V_r all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_q)))))) # label(fact_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4136 (all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)),V_x_2)))) # label(fact_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4137 (all V_B_2 all V_A_2 all T_b (c_Fun_Obij__betw(T_b,T_b,c_Fun_Oid(T_b),V_A_2,V_B_2) <-> V_A_2 = V_B_2)) # label(fact_bij__betw__id__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4138 (all V_xs_2 all V_y_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> hAPP(c_List_Ofoldl(T_c,T_b,hAPP(c_COMBC(T_b,T_c,T_c),V_f_2),V_y_2),c_List_Oremdups(T_b,V_xs_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_y_2),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_fun__left__comm_Ofold__set__remdups) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4139 (all V_y_2 all V_xs_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_List_Oset(T_b),hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b),c_fNot),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_y_2))),V_xs_2))) # label(fact_set__minus__filter__out) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4140 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)))))))) # label(fact_strict__below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4141 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) = c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) -> c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))))) # label(fact_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4142 (all V_A_2 all V_S_2 all V_R_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_c,T_b,V_S_2),V_A_2)) = hAPP(c_Relation_OImage(T_c,T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_R_2),V_S_2)),V_A_2)) # label(fact_Un__Image) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4143 (all V_n all V_a all T_a (class_Rings_Oring__1__no__zero__divisors(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4144 (all V_x_2 all V_A_2 all T_b (class_Lattices_Odistrib__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2))))))))) # label(fact_inf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4145 (all V_n all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n))))) # label(fact_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4146 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (((exists B_x hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4147 (all V_z_2 all V_w_2 all T_b (class_Int_Onumber__ring(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> c_Groups_Oone__class_Oone(T_b) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(hAPP(c_HOL_OLet(T_b,T_b),hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_b,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_b),c_Groups_Otimes__class_Otimes(T_b)),hAPP(c_Groups_Otimes__class_Otimes(T_b),V_z_2))),c_COMBI(T_b))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_b),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2)))))) # label(fact_power__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4148 (all V_x_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_x_2)),c_Groups_Oone__class_Oone(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls)))))) # label(fact_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4149 (all V_r_2 all T_b c_Relation_Oirrefl(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))) # label(fact_irrefl__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4150 (all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),V_R_2),V_S_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_b,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_b,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_HOL_Obool)))))),c_Wellfounded_Omax__ext(T_b,V_R_2))))) # label(fact_max__ext__compat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4151 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a),V_b))) # label(fact_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4152 (all V_j_2 all V_i_2 all T_b (class_Rings_Oring__1(T_b) -> hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_i_2),V_j_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_i_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_b),V_j_2)))) # label(fact_of__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4153 (all V_a_2 all V_b_2 all V_c_2 all T_b (class_Rings_Odivision__ring(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) != V_c_2 -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = V_a_2 <-> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))))) # label(fact_nonzero__divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4154 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_card__Diff__subset__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4155 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Obutlast(T_a,V_xs)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_length__butlast) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4156 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_UN__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4157 (all V_r_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_b,V_r_2) -> c_Wellfounded_Owf(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2))))) # label(fact_finite__acyclic__wf__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4158 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)) | hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_Un__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4159 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y))))) # label(fact_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4160 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z))) # label(fact_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4161 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> V_x_2 = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4162 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> ((all B_e (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),B_e)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),B_e))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))))) # label(fact_field__le__epsilon) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4163 (all V_ys all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oconcat(T_a,V_xs)),c_List_Oconcat(T_a,V_ys)) = c_List_Oconcat(T_a,hAPP(hAPP(c_List_Oappend(tc_List_Olist(T_a)),V_xs),V_ys))) # label(fact_concat__append) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4164 (all V_k all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_k)),V_m)))) # label(fact_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4165 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))))) # label(fact_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4166 (all V_k c_Int_Opred(c_Int_OBit0(V_k)) = c_Int_OBit1(c_Int_Opred(V_k))) # label(fact_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4167 (all V_P_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_OdropWhile(T_b,V_P_2,V_xs_2))))) # label(fact_distinct__dropWhile) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4168 (all V_y_2 all V_xs_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_Oremove1(T_b,V_y_2,V_xs_2)))))) # label(fact_notin__set__remove1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4169 (all V_a all V_x all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x))))) # label(fact_le__infI2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4170 (all V_B_2 all V_A_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_b) & class_Lattices_Osemilattice__inf(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2))))))) # label(fact_min__max_Omono__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4171 (all V_R_2 all T_b ((all B_A (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),B_A),hAPP(c_Relation_OImage(T_b,T_b,V_R_2),B_A))) -> B_A = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) -> c_Wellfounded_Owf(T_b,V_R_2))) # label(fact_wfI__pf) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4172 (all V_a_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_2)),V_r_2)))) # label(fact_wf__irrefl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4173 (all V_f_2 all V_m_2 all V_n_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(V_f_2,V_n_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4174 (all V_A_2 all T_b hAPP(c_Relation_ODomain(T_b,T_b),c_Relation_OId__on(T_b,V_A_2)) = V_A_2) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4175 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n))))) # label(fact_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4176 (all V_xs all V_m all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n),V_m)),V_xs) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Otake(T_a),V_m),V_xs))) # label(fact_take__take) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4177 (all V_z all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_z)))))) # label(fact_order__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4178 (all V_r_2 all T_b hAPP(c_Set_OCollect(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fdisj)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBS(tc_List_Olist(T_b),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_b)),c_fconj)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBC(tc_List_Olist(T_b),tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_b),tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),tc_fun(tc_List_Olist(T_b),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_HOL_Obool)),tc_List_Olist(T_b)),hAPP(c_COMBB(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_b)),c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))))),c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b))))),c_List_Olex(T_b,V_r_2)))))) = c_List_Olenlex(T_b,V_r_2)) # label(fact_lenlex__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4179 (all V_i_2 all V_f_2 all V_x_2 all T_b all T_c hAPP(c_New__DSequence_Oneg__bind(T_c,T_b,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Ohb__bind(T_c,T_b,hAPP(V_x_2,V_i_2),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_b))),V_f_2),V_i_2))) # label(fact_New__DSequence_Oneg__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4180 (all V_r_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel(T_b,V_r_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_b),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_listrel__subset__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4181 (all V_n_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_setsum__shift__bounds__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4182 (all V_ns all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),V_ns))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),V_ns),V_k)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),V_ns))))) # label(fact_elem__le__listsum__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4183 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) -> (V_y_2 = V_x_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4184 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_c)))) # label(fact_sup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4185 (all V_xs_2 all V_f_2 all T_b all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2) = c_List_Olist_ONil(T_b))) # label(fact_map__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4186 (all V_r_2 all T_c all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2))) -> -(all B_y -hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),B_y)),V_r_2))))) # label(fact_DomainE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4187 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_b_2)),V_a_2))))) # label(fact_minus__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4188 (all V_xs_2 all V_f_2 all T_b c_List_Opartition(T_b,V_f_2,V_xs_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(c_List_Ofilter(T_b,V_f_2),V_xs_2)),hAPP(c_List_Ofilter(T_b,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,T_b,c_fNot),V_f_2)),V_xs_2))) # label(fact_partition__filter__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4189 (all V_xs_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Oset(T_b),c_List_Olinorder__class_Osort__key(T_b,T_c,V_f_2,V_xs_2)) = hAPP(c_List_Oset(T_b),V_xs_2))) # label(fact_set__sort) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4190 (all V_s_2 all V_r_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),c_Predicate_Oconversep(T_c,T_b,V_r_2)),c_Predicate_Oconversep(T_c,T_b,V_s_2)) = c_Predicate_Oconversep(T_c,T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool))),V_r_2),V_s_2))) # label(fact_converse__meet) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4191 (all T_a (class_Groups_Osgn__if(T_a) -> c_Groups_Osgn__class_Osgn(T_a,c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_sgn0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4192 (all V_n_2 all V_m_2 (V_m_2 = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) & hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = V_n_2 | c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2 & c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4193 (all T_f all T_e all T_d all T_b all T_c all V_x_H_2 all V_g_H_2 all V_f_H_2 all V_x_2 all V_g_2 all V_f_2 (hAPP(V_f_2,hAPP(V_g_2,V_x_2)) = hAPP(V_f_H_2,hAPP(V_g_H_2,V_x_H_2)) -> hAPP(hAPP(c_Fun_Ocomp(T_e,T_b,T_f,V_f_H_2),V_g_H_2),V_x_H_2) = hAPP(hAPP(c_Fun_Ocomp(T_c,T_b,T_d,V_f_2),V_g_2),V_x_2))) # label(fact_comp__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4194 (all V_R_2 all T_b (c_Equiv__Relations_Opart__equivp(T_b,V_R_2) -> (exists B_d hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),B_d),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_b,tc_HOL_Obool)),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBS(T_b,T_b,tc_HOL_Obool),V_R_2),c_COMBI(T_b))))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b)),c_fequal)),V_R_2)))))))) # label(fact_part__equivp__typedef) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4195 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))))) # label(fact_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4196 (all V_r_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_r_2)) # label(fact_quotient__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4197 (all V_m all V_n all T_a (class_Rings_Oring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))))) # label(fact_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4198 (all V_d_2 all V_c_2 all V_b_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_c_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)))) # label(fact_Int__atLeastAtMostR1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4199 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)),V_n))) # label(fact_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4200 (all V_A_2 all V_B_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2))))) # label(fact_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4201 (all V_Q_2 all V_P_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_P_2),V_Q_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),B_x)),c_Transitive__Closure_Ortrancl(T_b,V_P_2))) -> (all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_Q_2)) -> B_x = B_y)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_P_2)))))) # label(fact_rtrancl__Un__separatorE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4202 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = V_A_2) # label(fact_Un__Diff__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4203 (all V_x_2 all T_d all T_c all T_b hAPP(hAPP(c_Product__Type_Oscomp(T_b,T_c,T_d,tc_prod(T_c,T_d)),V_x_2),c_Product__Type_OPair(T_c,T_d)) = V_x_2) # label(fact_scomp__Pair) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4204 (all V_x_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hAPP(c_Fun_Othe__inv__into(T_b,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2)) # label(fact_the__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4205 (all V_p_2 all V_c_2 all T_d all T_c all V_z_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_fun(T_b,tc_HOL_Obool)),V_c_2),V_p_2))) -> -(all B_x all B_y (V_p_2 = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_y) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(hAPP(V_c_2,B_x),B_y))))))) # label(fact_mem__splitE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4206 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y))) # label(fact_min__max_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4207 (all V_b_2 all V_a_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_b,T_c,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)) -> hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))))) # label(fact_curryD) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4208 (all V_xs_2 all T_b (V_xs_2 = c_List_Olist_ONil(T_b) <-> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_List_Oset(T_b),V_xs_2))) # label(fact_set__empty2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4209 (all V_v_H_2 all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_H_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))))) # label(fact_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4210 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2))),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_mono__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4211 (all V_M_2 all V_A_2 all V_i_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_M_2,V_i_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),V_M_2)))))) # label(fact_le__SUPI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4212 (all V_A_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b)),tc_HOL_Obool)),c_List_Olistrel(T_b,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olists(T_b,V_A_2)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool),tc_List_Olist(T_b)),c_List_Olists(T_b,V_A_2))))))) # label(fact_listrel__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4213 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)))))) # label(fact_min__max_OSup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4214 (all V_m all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_n),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n))) # label(fact_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4215 (all V_f_2 all V_g_2 all T_d all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_g_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Fun_Ocomp(T_d,tc_fun(T_b,tc_HOL_Obool),T_c,V_g_2),V_f_2))) # label(fact_UN__o) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4216 (all V_g_2 all V_f_2 all T_b all T_c all T_d hAPP(c_Fun_Ocomp(tc_List_Olist(T_d),tc_List_Olist(T_c),tc_List_Olist(T_b),hAPP(c_List_Omap(T_d,T_c),V_f_2)),hAPP(c_List_Omap(T_b,T_d),V_g_2)) = hAPP(c_List_Omap(T_b,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_b,V_f_2),V_g_2))) # label(fact_List_Omap_Ocomp) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4217 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_a))) # label(fact_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4218 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeastLessThan(T_b,V_less__eq_2,V_less_2,V_l_2,V_u_2))))) # label(fact_ord_OatLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4219 (all V_xs_2 all V_f_2 all T_c all V_P_2 all T_b hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_OtakeWhile(T_c,hAPP(c_Fun_Ocomp(T_b,tc_HOL_Obool,T_c,V_P_2),V_f_2),V_xs_2)) = c_List_OtakeWhile(T_b,V_P_2,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) # label(fact_takeWhile__map) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4220 (all V_a_2 all T_b (class_Orderings_Oorder(T_b) -> c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_a_2,V_a_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_atLeastAtMost__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4221 (all V_xs_2 all V_x_2 all V_a_2 all V_f_2 all T_c all T_b hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,V_a_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,hAPP(hAPP(V_f_2,V_a_2),V_x_2)),V_xs_2)) # label(fact_foldl__Cons) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4222 (all V_i_2 all V_inc_2 all T_b (class_Rings_Osemiring__1(T_b) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_b,V_inc_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_i_2) = V_i_2)) # label(fact_of__nat__aux_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4223 (all V_x_2 all V_A_2 all T_c all T_b (class_Groups_Ouminus(T_b) -> hAPP(hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,T_b)),V_A_2),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),hAPP(V_A_2,V_x_2)))) # label(fact_fun__Compl__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4224 (all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x))) # label(fact_sup__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4225 (all V_A_2 all V_A_H_2 all V_r_2 all V_r_H_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_r_H_2),V_r_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_H_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_b,T_c,V_r_H_2),V_A_H_2)),hAPP(c_Relation_OImage(T_b,T_c,V_r_2),V_A_2)))))) # label(fact_Image__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4226 (all V_A_2 all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_OPow(T_b),V_A_2)) = V_A_2) # label(fact_Union__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4227 (all V_R_2 all V_Q_2 all V_P_2 all T_d all T_b all T_c hAPP(hAPP(hAPP(c_COMBB(T_c,T_b,T_d),V_P_2),V_Q_2),V_R_2) = hAPP(V_P_2,hAPP(V_Q_2,V_R_2))) # label(help_c__COMBB__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4228 (all V_d all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_c),V_d)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_d))))))) # label(fact_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4229 (all V_z all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)),V_z))) # label(fact_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4230 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))) # label(fact_min__max_Osup__left__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4231 (all V_A_2 all T_b hAPP(c_Set_Oimage(T_b,tc_List_Olist(T_b),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b))),V_A_2) = c_List_Oset__Cons(T_b,V_A_2,hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool))))) # label(fact_set__Cons__sing__Nil) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4232 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_psubset__imp__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4233 (all V_n_2 all V_bs_2 all V_as_2 all V_P_2 all T_c all T_b (c_List_Olist__all2(T_b,T_c,V_P_2,V_as_2,V_bs_2) -> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Odrop(T_b),V_n_2),V_as_2),hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_bs_2)))) # label(fact_list__all2__dropI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4234 (all V_P_2 all V_xs_2 all T_b (c_List_Olist__all(T_b,V_P_2,V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_Ball__set__list__all) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4235 (all V_l c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4236 (all V_A_2 all V_f_2 all T_c all T_b (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 <-> hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4237 (all V_B_2 all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_IntD1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4238 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4239 (all V_P_2 all V_b_2 all V_a_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),B_x),hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_a_2),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2),B_x)))) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_a_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_b_2) -> c_List_Olist__all2(T_b,T_c,V_P_2,V_a_2,V_b_2)))) # label(fact_list__all2I) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4240 (all T_2 all T_1 (class_Lattices_Oboolean__algebra(T_1) -> class_Lattices_Oboolean__algebra(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Oboolean__algebra) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4241 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4242 (all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j),V_i)) -> c_List_Oupto(V_i,V_j) = c_List_Olist_ONil(tc_Int_Oint))) # label(fact_upto__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4243 (all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool))),V_A_2)))))) # label(fact_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4244 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)))) # label(fact_min__max_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4245 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)))) # label(fact_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4246 (all V_As_2 all V_A_2 all T_b c_List_Oset__Cons(T_b,V_A_2,c_List_Olistset(T_b,V_As_2)) = c_List_Olistset(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_As_2))) # label(fact_listset_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4247 (all V_r_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),c_Wellfounded_Oacc(T_b,V_r_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),V_a_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Wellfounded_Oacc(T_b,V_r_2))))))) # label(fact_acc_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4248 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> (c_Relation_Oantisym(T_b,V_r_2) -> c_Relation_Otrans(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b)))))) # label(fact_trans__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/9.01 4249 (all V_xss_2 all T_b hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),T_b),c_List_Ohd(T_b)),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_b)))),V_xss_2)) = c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_Olist__case(tc_List_Olist(T_b),T_b,c_List_Olist_ONil(T_b),hAPP(hAPP(c_COMBB(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_b),c_COMBK(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_OCons(T_b)),c_List_Olist_ONil(T_b))))),V_xss_2))) # label(fact_transpose__aux__filter__head) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4250 (all T_c all T_b c_Fun_Oid(tc_prod(T_b,T_c)) = c_Product__Type_Omap__pair(T_b,T_b,T_c,T_c,c_COMBI(T_b),c_COMBI(T_c))) # label(fact_map__pair_Oidentity) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4251 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)))) # label(fact_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4252 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_left__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4253 (all V_z c_Nat__Transfer_Ois__nat(hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_z))) # label(fact_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4254 (all T_a (class_Rings_Olinordered__semidom(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4255 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w)) # label(fact_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4256 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4257 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),c_Groups_Oone__class_Oone(T_a))))))) # label(fact_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4258 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))))))) # label(fact_one__less__power) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4259 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b))) # label(fact_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 4260 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))) # label(fact_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4261 (all V_m all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_m))))) # label(fact_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4262 (all V_l_2 all V_k_2 (-(hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2))) <-> c_SetInterval_Oord_OgreaterThanAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4263 (all V_r_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Fun_Oinj__on(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),T_b),c_Equiv__Relations_Oquotient(T_b)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))),V_r_2),V_A_2) -> hAPP(c_Finite__Set_Ocard(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2)) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2)))) # label(fact_card__quotient__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4264 (all V_y_2 all V_x_2 all T_b (class_Orderings_Oorder(T_b) -> (V_y_2 != V_x_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_x_2),V_y_2))))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4265 (all V_A_2 all V_c_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))))) # label(fact_ComplI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4266 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))))))) # label(fact_card__inj__on__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4267 (all V_B_2 all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),V_x_2)),V_B_2)))))) # label(fact_the__inv__into__into) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4268 (all V_A_2 all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__mult(T_b) & class_Rings_Oring__1(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_b),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_b,T_c),c_Int_Oring__1__class_Oof__int(T_b)),V_f_2)),V_A_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,tc_Int_Oint),V_f_2),V_A_2)))) # label(fact_of__int__setprod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4269 (all V_n all V_m all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_i)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_i),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4270 (all V_b all V_x all V_a all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),V_x))))) # label(fact_min__max_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4271 (all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_x) = V_x)) # label(fact_Inf__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4272 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),V_y_2)))))) # label(fact_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4273 (all V_a_2 all V_b_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2)))))) # label(fact_neg__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4274 (all V_P_2 all T_c all T_b c_HOL_OThe(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_HOL_Obool,T_c),V_P_2)),c_Product__Type_OPair(T_b,T_c)))) = c_HOL_OThe(tc_prod(T_b,T_c),V_P_2)) # label(fact_split__paired__The) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4275 (all V_x_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> V_x_2 = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4276 (all V_n_2 all V_m_2 all V_P_2 (hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | (exists B_q (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,B_q)))) & hBOOL(hAPP(V_P_2,B_q)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),B_q)),V_m_2)))))) # label(fact_split__div_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4277 (all V_P_2 all T_b -(exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) & hBOOL(hAPP(V_P_2,B_x))))) # label(fact_bex__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4278 (all V_x_2 hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_x_2)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_nat__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4279 (all V_y_2 all V_x_2 all T_b (class_Orderings_Oorder(T_b) -> (V_y_2 = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_y_2),V_x_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2))))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4280 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_divide__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4281 (all V_k_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),V_k_2)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_greaterThan__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4282 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_b),c_List_Olex(T_b,V_r_2)))) # label(fact_wf__lex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4283 (all T_a (class_Rings_Oring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_of__int__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4284 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> c_Relation_Otrans(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b))))) # label(fact_trans__reflclI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4285 (all V_c all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)))))))) # label(fact_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4286 (all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),c_List_Olist_ONil(T_b)))) # label(fact_insort__key_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4287 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(c_List_Orotate1(T_b),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)))) # label(fact_distinct1__rotate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4288 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4289 (all V_y_2 all T_c all V_g_2 all V_f_2 all V_S_2 all T_b ((all B_w (hBOOL(hAPP(hAPP(c_member(T_b),B_w),V_S_2)) -> hAPP(V_f_2,B_w) = hAPP(V_g_2,B_w))) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_y_2)),V_S_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_g_2),V_y_2)),V_S_2))) # label(fact_vimage__inter__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4290 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_zero__power2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4291 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_set__zip__leftD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4292 (all V_xs_2 all V_f_2 all T_b all V_x_2 all V_P_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(c_List_Ofilter(T_b,V_P_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_b,T_c,V_f_2),V_x_2),V_xs_2)) = hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)))) # label(fact_filter__insort__triv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4293 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2))))) # label(fact_Union__Int__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4294 (all V_P_2 all T_b (class_Enum_Oenum(T_b) -> ((exists B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_b),V_P_2))))) # label(fact_enum__ex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4295 (all V_r_2 all T_c all T_b ((all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_x),B_y)),V_r_2)) -> (all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_x),B_z)),V_r_2)) -> B_y = B_z)))) <-> c_Relation_Osingle__valued(T_b,T_c,V_r_2))) # label(fact_single__valued__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4296 (all V_f_2 all V_n_2 all V_m_2 all T_b (class_Groups_Oab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(V_f_2,V_m_2)),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(T_b)),V_f_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(T_b)),V_f_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = c_Groups_Ozero__class_Ozero(T_b)))) # label(fact_setsum__natinterval__difff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4297 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | V_m_2 = V_n_2)) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4298 (all V_n_2 all V_f_2 all T_b all V_A_2 (hAPP(c_Set_Oimage(tc_Nat_Onat,T_b,V_f_2),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_n_2))) = V_A_2 -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_nat__seg__image__imp__finite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4299 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n_2)),hAPP(c_Code__Numeral_Oof__nat,V_m_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_less__eq__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4300 (all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Un__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4301 (all V_R_2 all T_b (c_Equiv__Relations_Oequivp(T_b,V_R_2) -> c_Equiv__Relations_Opart__equivp(T_b,V_R_2))) # label(fact_equivp__implies__part__equivp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4302 (all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> c_List_Olinorder__class_Osorted(T_a,hAPP(c_List_Otl(T_a),V_xs))))) # label(fact_sorted__tl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4303 (all V_A_2 all V_x_2 all T_c all T_b hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_A_2)))) # label(fact_card__cartesian__product__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4304 (all V_nat_H_1 c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H_1)) # label(fact_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4305 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4306 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_c_2,V_d_2)))) # label(fact_Int__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4307 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_k)))) # label(fact_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4308 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> (exists B_ys exists B_zs V_xs_2 = hAPP(hAPP(c_List_Oappend(T_b),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),B_zs))))) # label(fact_in__set__conv__decomp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4309 (all V_i_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_i_2))))) # label(fact_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4310 (all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_b,V_f_2),V_A_2)))) # label(fact_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4311 (all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4312 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),V_B_2)) = c_Big__Operators_Olattice_OInf__fin(T_b,c_Orderings_Oord__class_Omin(T_b),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBS(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal))),c_Orderings_Oord__class_Omax(T_b)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))))))))))))) # label(fact_min__max_Osup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4313 (all V_B_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_B_2) = V_B_2) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4314 (all V_b_2 all V_a_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2))) -> hBOOL(hAPP(hAPP(V_f_2,V_a_2),V_b_2)))) # label(fact_splitD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4315 (all V_g_2 all V_f_2 all T_e all T_d all T_c all T_b hAPP(hAPP(c_COMBS(tc_prod(T_b,T_c),T_c,T_d),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_c,T_d),tc_prod(T_b,T_c)),V_f_2),hAPP(hAPP(c_COMBB(T_b,T_e,tc_prod(T_b,T_c)),V_g_2),c_Product__Type_Ofst(T_b,T_c)))),c_Product__Type_Osnd(T_b,T_c)) = hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_c,T_d),T_b),V_f_2),V_g_2))) # label(fact_split__comp__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4316 (all V_A_2 all T_c all T_b all V_y_2 all V_x_2 all V_B_2 (hBOOL(hAPP(hAPP(V_B_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2I2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4317 (all V_f_2 all V_S_2 all V_T_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_T_2),V_S_2))) -> c_Groups_Oone__class_Oone(T_c) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_T_2)))))) # label(fact_setprod__mono__one__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4318 (all V_A_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) & V_y_2 = V_x_2 <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Relation_OId__on(T_b,V_A_2))))) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4319 (all V_f_2 all V_y_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_y_2),hAPP(V_f_2,V_x_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_y_2),c_List_Olist_Olist__size(T_b,V_f_2,V_xs_2)))))) # label(fact_list__size__estimation) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4320 (all V_xs all V_x all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_n)) & (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_n)))) # label(fact_nth__Cons_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4321 (all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) <-> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_Compl__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4322 (all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs))) # label(fact_nth__drop_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4323 (all V_B_2 all V_a_2 all T_c all T_b all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_Oinsert(T_c),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_UN__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4324 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2)) # label(fact_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4325 (all T_b c_Fun_Oinj__on(T_b,tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_inj__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4326 (all V_c_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))))) # label(fact_psubsetD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4327 (all V_B_2 all V_A_2 all T_b c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2))) # label(fact_Union__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4328 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),V_a_2))))) # label(fact_minus__le__self__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4329 (all T_2 all T_1 (class_Lattices_Odistrib__lattice(T_1) -> class_Lattices_Odistrib__lattice(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Odistrib__lattice) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4330 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),c_List_Oinsert(T_a,V_x,V_xs))))) # label(fact_distinct__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4331 (all V_x_2 all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)) -> hBOOL(hAPP(V_B_2,V_x_2)))) # label(fact_inf1D2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4332 (all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__minus) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4333 (all V_s_2 all V_r_2 all T_d all T_b all T_c hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),hAPP(c_Relation_Oconverse(T_d,T_b),V_s_2)),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2)) = hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_b),V_r_2),V_s_2))) # label(fact_converse__rel__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4334 (all V_b all V_a all T_a c_Nitpick_OAbs__Frac(T_a,c_Nitpick_Onorm__frac(V_a,V_b)) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),V_a),V_b)) # label(fact_Nitpick_Ofrac__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4335 (all V_B_2 all V_A_2 all V_C_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_UN__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4336 (all V_n_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)) # label(fact_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4337 (all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_B_2,B_x))))) -> hAPP(c_Finite__Set_Ocard(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_Nat_Onat,T_b),c_Finite__Set_Ocard(T_c)),V_B_2)),V_A_2)))) # label(fact_card__SigmaI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4338 (all V_z all V_x all V_y all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)))))) # label(fact_mult__imp__div__pos__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4339 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olex(T_b,V_r_2))) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_x_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_y_2) & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_x_2),V_y_2)),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__lex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4340 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_b))) # label(fact_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4341 (all V_n_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x_2)),V_n_2))) <-> c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_x_2 | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4342 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_diff__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4343 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)))) # label(fact_INF2__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4344 (all T_a (class_Rings_Olinordered__semidom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4345 (all V_y_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> (V_y_2 = V_x_2 <-> hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_y_2) = hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_x_2)))) # label(fact_atLeast__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4346 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,V_f_2),V_x_2),V_A_2) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4347 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))))))) # label(fact_neg__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4348 (all V_r all V_l all V_k ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l)) -> c_Random_Ominus__shift(V_r,V_k,V_l) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_k),V_l)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_k),V_l)) -> c_Random_Ominus__shift(V_r,V_k,V_l) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_r),V_k)),V_l)))) # label(fact_minus__shift__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4349 (all V_xs_2 all V_P_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) & (all B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_y),hAPP(c_List_Oset(T_b),V_xs_2))) & hBOOL(hAPP(V_P_2,B_y)) -> B_x = B_y)) & hBOOL(hAPP(V_P_2,B_x)))) <-> c_List_Olist__ex1(T_b,V_P_2,V_xs_2))) # label(fact_list__ex1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4350 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))) # label(fact_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4351 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OAll(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fimplies),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_c)),c_member(T_b))),V_B_2))))) # label(fact_INTER__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4352 (all V_x_2 all V_f_2 all V_xs_2 all T_b all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_b),c_List_Olinorder__class_Oinsort__insert__key(T_b,T_c,V_f_2,V_x_2,V_xs_2)))))) # label(fact_distinct__insort__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4353 (all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_Divides_OposDivAlg(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l))))) # label(fact_posDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4354 (all V_B_2 all V_A_2 all T_d all V_r_2 all T_c all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Relation_OImage(T_c,T_b,V_r_2)),V_B_2))))) # label(fact_Image__INT__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4355 (all V_g_2 all V_f_2 all V_B_2 all V_A_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) & class_Groups_Ocomm__monoid__mult(T_b) -> (V_B_2 = V_A_2 -> (c_Nat__Transfer_Onat__set(V_B_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_b),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_b),V_g_2),V_B_2)))))) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4356 (all V_ys_2 all V_xs_2 all V_x_2 all T_b ((hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> hAPP(hAPP(c_List_Oappend(T_b),c_List_Oremove1(T_b,V_x_2,V_xs_2)),V_ys_2) = c_List_Oremove1(T_b,V_x_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) & (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> c_List_Oremove1(T_b,V_x_2,hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),c_List_Oremove1(T_b,V_x_2,V_ys_2))))) # label(fact_remove1__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4357 (all V_f_2 all T_c all T_d all T_b c_Product__Type_Oapfst(T_b,T_d,T_c,V_f_2) = c_Product__Type_Omap__pair(T_b,T_d,T_c,T_c,V_f_2,c_Fun_Oid(T_c))) # label(fact_apfst__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4358 (all V_j all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)) = hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_j),c_List_Olist_ONil(tc_Nat_Onat)))))) # label(fact_upt__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4359 (all V_B_2 all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_B_2),c_Complete__Lattice_OInf__class_OInf(T_b,V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_B_2),V_A_2)))) # label(fact_inf__Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4360 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2))))) # label(fact_neg__le__0__iff__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 4361 (all V_f_2 all T_c all T_b (c_Fun_Obij__betw(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_bij__is__inj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4362 (all V_xs_2 all V_g_2 all V_f_2 all T_c all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBC(T_c,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,tc_HOL_Obool),T_c),c_fequal),V_f_2)),hAPP(V_g_2,V_xs_2))),V_xs_2))))) # label(fact_sorted__map__same) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4363 (all V_a_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> V_a_2 = c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_Inf__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4364 (all V_B_2 all V_a_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2))))))) # label(fact_card__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4365 (all V_a_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (c_Groups_Ozero__class_Ozero(T_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_a_2),V_a_2) <-> c_Groups_Ozero__class_Ozero(T_b) = V_a_2))) # label(fact_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4366 (all T_1 all T_2 (class_Enum_Oenum(T_2) & class_Enum_Oenum(T_1) -> class_Enum_Oenum(tc_fun(T_2,T_1)))) # label(arity_fun__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4367 (all V_a all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield(T_a) -> V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_divide__numeral__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4368 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(c_Relation_ODomain(T_b,T_c),V_r_2)) = hAPP(c_Relation_ODomain(T_b,T_c),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4369 (all V_A_2 all V_f_2 all T_c all V_B_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))) <-> (exists B_AA (V_B_2 = hAPP(c_Set_Oimage(T_c,T_b,V_f_2),B_AA) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),B_AA),V_A_2)))))) # label(fact_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4370 (all V_b_2 all V_a_2 all V_fb_2 all V_fa_2 all T_c all T_b c_Nitpick_Opair__box_Opair__box__size(T_b,T_c,V_fa_2,V_fb_2,c_Nitpick_Opair__box_OPairBox(T_b,T_c,V_a_2,V_b_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_fa_2,V_a_2)),hAPP(V_fb_2,V_b_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_pair__box_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4371 (all V_w_2 all T_b (class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_w_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Nat_Osemiring__1__class_ONats(T_b)))))) # label(fact_Nats__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4372 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Groups_Oordered__ab__semigroup__add__imp__le(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_c_2),V_b_2)))))) # label(fact_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4373 (all V_xs all V_x all T_a all V_v ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = V_x) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_nth__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4374 (all V_x_2 all V_B_2 all V_A_2 all T_c all T_b (class_Groups_Ominus(T_b) -> hAPP(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,T_b)),V_A_2),V_B_2),V_x_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(V_A_2,V_x_2)),hAPP(V_B_2,V_x_2)))) # label(fact_fun__diff__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4375 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_c)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_c))))) # label(fact_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4376 (all V_n all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Otake(T_a),V_n),c_List_Olist_ONil(T_a))) # label(fact_take__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4377 (all T_1 all T_2 (class_Enum_Oenum(T_1) & class_Enum_Oenum(T_2) -> class_Enum_Oenum(tc_prod(T_2,T_1)))) # label(arity_prod__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4378 (all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),c_List_Olistsp(T_b,V_A_2)),c_List_Olistsp(T_b,V_B_2))))) # label(fact_listsp__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4379 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4380 (all V_B_2 all V_A_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,B_x))))))) # label(fact_UN__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4381 (all V_z_2 all V_P_2 all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),V_P_2),V_z_2) = hAPP(hAPP(V_P_2,hAPP(c_Product__Type_Ofst(T_c,T_d),V_z_2)),hAPP(c_Product__Type_Osnd(T_c,T_d),V_z_2))) # label(fact_split__beta) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4382 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2)),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),c_Set_Oimage(T_c,T_b,V_f_2)),hAPP(c_Set_OPow(T_c),V_A_2))),hAPP(c_Set_OPow(T_b),V_B_2))))) # label(fact_image__Pow__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4383 (all V_B_2 all V_A_2 all T_b (class_Lattices_Odistrib__lattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_b))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),tc_fun(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBS(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_b,tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_b),hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),tc_fun(tc_fun(T_b,tc_fun(T_b,T_b)),tc_fun(T_b,tc_fun(T_b,tc_HOL_Obool))),T_b),c_COMBB(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool),T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal))),c_Lattices_Osemilattice__inf__class_Oinf(T_b)))))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_B_2))))))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_B_2)))))))) # label(fact_inf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4384 (all V_d all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4385 (all V_B_2 all V_A_2 all T_d all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Set_Ovimage(T_b,T_c,V_f_2)),V_B_2))) # label(fact_vimage__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4386 (all V_x_2 all V_f_2 all T_d all T_c all T_b hAPP(c_Product__Type_Ofst(T_b,T_c),hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),V_x_2)) = hAPP(c_Product__Type_Ofst(T_b,T_d),V_x_2)) # label(fact_fst__apsnd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4387 (all V_j all V_i hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),c_List_Oupt(V_i,V_j)))) # label(fact_distinct__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4388 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_b),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_b),V_u_2))))) # label(fact_ivl__disj__un_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4389 (all V_xs_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) & c_Fun_Oinj__on(T_c,T_b,V_f_2,hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_distinct__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4390 (all V_k ((V_k != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) -> hAPP(c_Code__Numeral_Oint__of,V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Code__Numeral_Oint__of,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_k),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))),c_Groups_Oone__class_Oone(tc_Int_Oint))) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Code__Numeral_Oint__of,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_k),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) = hAPP(c_Code__Numeral_Oint__of,V_k))) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Code__Numeral_Oint__of,V_k)))) # label(fact_int__of__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4391 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add__abs(T_b) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_b) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Groups_Oabs__class_Oabs(T_b),V_a_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4392 (all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)) # label(fact_atLeast__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4393 (all V_x_2 all V_w_2 all T_b (class_Int_Onumber(T_b) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2) = V_x_2 <-> V_x_2 = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) # label(fact_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4394 (all V_xs_2 all T_b V_xs_2 = hAPP(hAPP(c_List_Omap(T_b,T_b),c_COMBI(T_b)),V_xs_2)) # label(fact_map__ident) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4395 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4396 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_b,V_r_2)),c_Transitive__Closure_Ortrancl(T_b,V_s_2))))) # label(fact_rtrancl__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4397 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))) # label(fact_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4398 (all V_v all V_w ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4399 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4400 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x))) # label(fact_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4401 (all V_fa_2 all T_b c_List_Olist_Olist__size(T_b,V_fa_2,c_List_Olist_ONil(T_b)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_list_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4402 (all V_x_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = hAPP(c_List_Oset(T_b),c_List_Oremove1(T_b,V_x_2,V_xs_2)))) # label(fact_set__remove1__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4403 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_u_2)) # label(fact_atLeastPlusOneAtMost__greaterThanAtMost__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4404 (all V_a_2 all V_r_2 all T_b (hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),V_a_2)) <-> (all B_x (hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_b,V_r_2),B_x)))))) # label(fact_accp_Oequations) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4405 (all V_m all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_n)))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)))) # label(fact_negative__zless) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4406 (all V_P_2 all V_B_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(V_P_2,B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)) -> hBOOL(hAPP(V_P_2,B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_ball__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4407 (all V_a_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_a_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)) = V_a_2)))) # label(fact_sup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4408 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_a))) # label(fact_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4409 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4410 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_x = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4411 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4412 (all V_k hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_k) = hAPP(c_Code__Numeral_Onat__of,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),V_k))) # label(fact_nat__of__number) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4413 (all V_f_2 all V_A_2 all T_b all T_c (class_Rings_Olinordered__semidom(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),V_f_2),V_A_2)))))) # label(fact_setprod__pos) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4414 (all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_b,T_b),c_Transitive__Closure_Otrancl(T_b,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))))) # label(fact_trancl__converseI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4415 (all V_f_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> -(all B_k hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(V_f_2,hAPP(c_Nat_OSuc,B_k))),hAPP(V_f_2,B_k))),V_r_2))))) # label(fact_wf__no__infinite__down__chainE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4416 (all V_R_2 all T_c all T_d all T_b c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_R_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)))) # label(fact_rel__comp__empty2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4417 (all V_z all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_z),V_x)))))) # label(fact_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4418 (all V_x_2 all T_b V_x_2 = c_Hilbert__Choice_OEps(T_b,hAPP(c_fequal,V_x_2))) # label(fact_some__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4419 (all T_d all T_c all V_c_2 all V_b_2 all V_B_2 all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(hAPP(V_B_2,V_a_2),V_b_2),V_c_2)) -> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2))))) # label(fact_SUP2__I) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4420 (all V_A_2 all T_b (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_b),V_A_2))) # label(fact_card__infinite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4421 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)),V_a) = V_a)) # label(fact_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4422 (all T_b hAPP(c_Relation_OField(T_b),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Field__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4423 (all V_b all V_a all T_a (class_Rings_Oordered__ring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))))) # label(fact_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4424 (all V_c all V_b all V_a all T_a (class_Rings_Oordered__comm__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))))) # label(fact_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4425 (all V_x_2 all V_n_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),c_List_Oreplicate(T_c,V_n_2,V_x_2)) = c_List_Oreplicate(T_b,V_n_2,hAPP(V_f_2,V_x_2))) # label(fact_map__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4426 (all V_a_2 all V_b_2 all V_A_2 all T_b (class_Lattices_Osemilattice__inf(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Lattices_Osemilattice__inf__class_Oinf(T_b)),V_b_2),V_A_2))))) # label(fact_fold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4427 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),V_i)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_k)),V_j)))))) # label(fact_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4428 (all V_ys_2 all V_xs_2 all T_b (V_xs_2 = c_List_Olist_ONil(T_b) & c_List_Olist_ONil(T_b) = V_ys_2 <-> c_List_Olist_ONil(T_b) = hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_ys_2))) # label(fact_append__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4429 (all V_P_2 all T_b (class_Enum_Oenum(T_b) -> ((all B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_b),V_P_2))))) # label(fact_all__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4430 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_B_2))) # label(fact_vimage__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4431 (all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = V_i) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_i) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i)))) # label(fact_zabs__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4432 (all V_A_2 all T_b c_Fun_Obij__betw(T_b,T_b,c_Fun_Oid(T_b),V_A_2,V_A_2)) # label(fact_bij__betw__id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4433 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2)) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4434 (all V_B_2 all V_a_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4435 (all V_v_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2)),c_Int_Onat))))))) # label(fact_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4436 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4437 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4438 (all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_member(T_b),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)),V_A_2)))))) # label(fact_Max__in) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4439 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> (V_n != V_m -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4440 (all V_a all T_a (class_Groups_Oabs__if(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a))) # label(fact_abs__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4441 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(T_a))))) # label(fact_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4442 (all V_i all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_i))),V_n)))) # label(fact_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4443 (all V_list_2 all V_g_2 all T_d all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_d,T_b),hAPP(c_Fun_Ocomp(T_c,T_b,T_d,V_f_2),V_g_2)),V_list_2) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_list_2))) # label(fact_List_Omap_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4444 (all V_B_2 all V_k_2 all V_A_2 all T_b ((all B_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_n),V_k_2))),V_B_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),V_B_2))))) # label(fact_UN__finite2__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4445 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_b_2),V_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_b,V_r_2)))))) # label(fact_rtrancl__into__trancl1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4446 (all V_a1_2 all V_a2_2 all T_b (class_Orderings_Olinorder(T_b) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_a1_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a2_2),B_x)))) & c_List_Olinorder__class_Osorted(T_b,V_a1_2) <-> c_List_Olinorder__class_Osorted(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_a2_2),V_a1_2))))) # label(fact_sorted_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4447 (all V_x_2 all V_n_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(c_List_Oset(T_b),c_List_Oreplicate(T_b,hAPP(c_Nat_OSuc,V_n_2),V_x_2))) # label(fact_set__replicate__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4448 (all V_B_2 all V_A_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2))) # label(fact_Int__lower1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4449 (all V_m hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,V_m)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m))) # label(fact_int__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4450 (all V_f_2 all V_r_2 all T_b all T_c hAPP(hAPP(c_Relation_Oinv__image(T_c,T_b),V_r_2),V_f_2) = hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_c,T_c)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_c,T_c),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_c,T_c)))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_c),tc_fun(T_b,tc_prod(T_c,T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(tc_fun(T_b,T_c),tc_fun(T_b,tc_prod(T_c,T_c))),T_b),c_COMBB(T_c,tc_prod(T_c,T_c),T_b)),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_prod(T_c,T_c)),T_b),c_Product__Type_OPair(T_c,T_c)),V_f_2))),V_f_2)))),V_r_2)))) # label(fact_inv__image__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4451 (all V_u_2 all V_l_2 all V_less__eq_2 all V_i_2 all T_b (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),c_SetInterval_Oord_OatLeastAtMost(T_b,V_less__eq_2,V_l_2,V_u_2))))) # label(fact_ord_OatLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4452 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)))) # label(fact_inf__sup__aci_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4453 (all V_S_2 all T_b (class_Groups_Olinordered__ab__group__add(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),V_S_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_S_2)))))) # label(fact_minus__Min__eq__Max) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4454 (all V_x_2 all V_A_2 all V_y_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_b),V_y_2),V_A_2),V_x_2)) <-> V_x_2 = V_y_2 | hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4455 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),V_a)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4456 (all V_B_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hAPP(V_g_2,B_x) = V_z_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2))))))) # label(fact_folding__image__simple_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4457 (all V_P_2 all T_b c_List_Olist__all(T_b,V_P_2,c_List_Olist_ONil(T_b))) # label(fact_list__all__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4458 (all V_m_2 all V_q_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> (V_q_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_q_2)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_q_2)),V_m_2))))) # label(fact_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4459 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))))) # label(fact_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4460 (all V_b_2 all V_f_2 all T_c all T_b all V_a_2 all V_x_2 ((V_x_2 != V_a_2 -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_a_2,V_b_2),V_x_2)) & (V_a_2 = V_x_2 -> V_b_2 = hAPP(c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_a_2,V_b_2),V_x_2)))) # label(fact_fun__upd__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4461 (all V_s_2 all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> (c_Wellfounded_Owf(T_b,V_s_2) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_b,T_b),V_r_2)),hAPP(c_Relation_ORange(T_b,T_b),V_s_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Wellfounded_Owf(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)))))) # label(fact_wf__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4462 (all V_Q_2 all V_P_2 (hBOOL(V_Q_2) | -hBOOL(hAPP(hAPP(c_fconj,V_P_2),V_Q_2)))) # label(help_c__fconj__3) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4463 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),hAPP(c_List_Oset(tc_prod(T_b,T_c)),hAPP(hAPP(c_List_Ozip(T_b,T_c),V_xs_2),V_ys_2)))) -> -(hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),V_ys_2)))))) # label(fact_in__set__zipE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4464 (all V_z all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))))) # label(fact_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4465 (all V_m V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4466 (all V_xs all T_a all V_y all V_x ((V_y != V_x -> c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Oremove1(T_a,V_x,V_xs))) & (V_x = V_y -> V_xs = c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs))))) # label(fact_remove1_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4467 (all V_y all V_x all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4468 (all V_k_2 all V_i_2 (hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),V_k_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_i_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4469 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a)))) # label(fact_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4470 (all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_b,T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_b,T_c),tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_finite__Plus__UNIV__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4471 (all V_s_2 all V_r_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_b,T_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),V_s_2)),V_r_2)),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_b,V_r_2)),V_s_2))))) # label(fact_trancl__Int__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4472 (all T_b hAPP(c_Relation_ORange(T_b,T_b),c_Relation_OId(T_b)) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Range__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4473 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),c_Wellfounded_Oless__than)))) # label(fact_less__than__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4474 (all V_B_2 all V_C_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)))) # label(fact_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4475 (all V_B_2 all V_C_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)),V_B_2)) # label(fact_Diff__Int2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4476 (all T_2 all T_1 (class_Lattices_Olattice(T_1) -> class_Lattices_Osemilattice__sup(tc_fun(T_2,T_1)))) # label(arity_fun__Lattices_Osemilattice__sup) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4477 (all V_s2_2 all V_s1_2 all T_c all T_b c_Lazy__Sequence_Oproduct(T_b,T_c,V_s1_2,V_s2_2) = hAPP(c_Lazy__Sequence_Obind(T_b,tc_prod(T_b,T_c),V_s1_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_b,T_c))),tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_b,T_c)),T_b),c_Lazy__Sequence_Obind(T_c,tc_prod(T_b,T_c),V_s2_2)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_b,T_c))),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_b,T_c)),T_c),c_Lazy__Sequence_Osingle(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c))))) # label(fact_) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4478 (all V_A_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),V_A_2))) -> (V_b_2 != V_a_2 -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4479 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 4480 (all V_xs_2 all V_P_2 all T_b (c_List_Olist__ex(T_b,V_P_2,hAPP(c_List_Orev(T_b),V_xs_2)) <-> c_List_Olist__ex(T_b,V_P_2,V_xs_2))) # label(fact_list__ex__rev) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4481 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))))) # label(fact_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4482 (all V_z_2 all V_y_2 all V_x_2 all V_R_2 all T_b (c_Equiv__Relations_Opart__equivp(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(V_R_2,V_y_2),V_z_2)) -> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_z_2)))))) # label(fact_part__equivp__transp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4483 (all V_n_2 all V_z_2 all V_R_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_R_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),V_n_2),V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2)))))) # label(fact_rel__pow__Suc__I2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4484 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) & hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_i) = V_x_2)))) # label(fact_in__set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4485 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Diff__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4486 (all V_x_2 all V_g_2 all V_f_2 all T_e all T_b all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_b),hAPP(c_Fun_Ocomp(T_e,tc_fun(T_d,T_b),T_c,V_f_2),V_g_2)),V_x_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_c,T_d),V_x_2))),hAPP(c_Product__Type_Osnd(T_c,T_d),V_x_2))) # label(fact_split__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4487 (all V_b all V_a all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)))))) # label(fact_less__supI1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4488 (all V_x_2 all V_f_2 all V_A_2 all T_d all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,T_b)),V_A_2),V_f_2),V_x_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_b),V_A_2),hAPP(hAPP(c_COMBC(T_c,T_d,T_b),V_f_2),V_x_2)))) # label(fact_INFI__apply) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4489 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))))) # label(fact_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4490 (all V_m all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m))))) # label(fact_not__int__zless__negative) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4491 (all V_f_2 all V_B_2 all V_A_2 all T_b all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_A_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,T_c),V_B_2),V_f_2)))))) # label(fact_SUP__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4492 (all V_b_2 all V_a_2 all V_P_2 (hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2))) <-> -(-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2)) | (exists B_d (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d) = V_a_2 & -hBOOL(hAPP(V_P_2,B_d))))))) # label(fact_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4493 (all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_i)) -> c_List_Oupt(V_i,V_j) = c_List_Olist_ONil(tc_Nat_Onat))) # label(fact_upt__conv__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4494 (all V_u_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4495 (all V_s_2 all V_r_2 all T_b all T_c hAPP(c_Relation_Oconverse(T_c,T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)),hAPP(c_Relation_Oconverse(T_c,T_b),V_s_2))) # label(fact_converse__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4496 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) | hAPP(c_Nat_OSuc,V_n_2) = V_m_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4497 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_w),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)))))) # label(fact_add__frac__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4498 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4499 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)))) # label(fact_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4500 (all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))))) # label(fact_inj__image__Compl__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4501 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4502 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_b (class_Orderings_Olinorder(T_b) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_c_2),V_d_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),V_d_2)) -> V_c_2 = V_a_2))))) # label(fact_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4503 (all V_r_2 all T_b hAPP(c_Relation_Oconverse(T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_r_2)) = c_Transitive__Closure_Ortrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2))) # label(fact_rtrancl__converse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4504 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_a),V_b)),V_c)) # label(fact_div__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4505 (all V_y all V_x all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)))))) # label(fact_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4506 (all V_R_2 all V_f_2 all T_b hAPP(hAPP(c_Relation_Oinv__image(tc_prod(tc_Nat_Onat,T_b),T_b),c_Wellfounded_Olex__prod(tc_Nat_Onat,T_b,c_Wellfounded_Oless__than,V_R_2)),hAPP(hAPP(c_COMBS(T_b,T_b,tc_prod(tc_Nat_Onat,T_b)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_b,tc_prod(tc_Nat_Onat,T_b)),T_b),c_Product__Type_OPair(tc_Nat_Onat,T_b)),V_f_2)),c_COMBI(T_b))) = c_Wellfounded_Omlex__prod(T_b,V_f_2,V_R_2)) # label(fact_mlex__prod__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4507 (all V_n_2 all V_v_2 (hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),c_Int_Onat)),V_n_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4508 (all V_g_2 all V_h_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image(T_b,T_c,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Fun_Oinj__on(T_c,T_c,V_h_2,V_A_2) -> hAPP(hAPP(V_F_2,hAPP(c_Fun_Ocomp(T_c,T_b,T_c,V_g_2),V_h_2)),V_A_2) = hAPP(hAPP(V_F_2,V_g_2),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_A_2)))))) # label(fact_folding__image_Oreindex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4509 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m_2),V_n_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))))) # label(fact_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4510 (all V_xs_2 all T_b all V_x_2 all V_P_2 (-hBOOL(hAPP(V_P_2,V_x_2)) -> hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2) = hAPP(c_List_OremoveAll(T_b,V_x_2),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2)))) # label(fact_removeAll__filter__not) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4511 (all V_x_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2)) # label(fact_folding__one__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4512 (all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olist_ONil(T_a) != V_xs -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> c_List_Olinorder__class_Osorted(T_a,c_List_Obutlast(T_a,V_xs)))))) # label(fact_sorted__butlast) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4513 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))))) # label(fact_abs__le__D2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4514 (all V_y_2 all V_x_2 all V_P_2 (hBOOL(hAPP(V_P_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2)))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),V_x_2)) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4515 (all V_x_2 V_x_2 = hAPP(c_Int_OAbs__Integ,hAPP(c_Int_ORep__Integ,V_x_2))) # label(fact_Rep__Integ__inverse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4516 (all V_r_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),hAPP(c_Relation_OField(T_b),V_r_2))))) # label(fact_finite__Field) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4517 (all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool))))))) # label(fact_finite__Prod__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4518 (all T_c all T_b all V_b_2 all V_a_2 all V_r_2 (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> hBOOL(hAPP(c_Predicate_ODomainP(T_b,T_c,V_r_2),V_a_2)))) # label(fact_DomainP_Ointros) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4519 (all V_x_2 all T_b c_HOL_OThe(T_b,hAPP(c_fequal,V_x_2)) = V_x_2) # label(fact_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4520 (all V_xs all V_n all T_a hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_length__drop) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4521 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))) # label(fact_diff__minus__eq__add) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4522 (all V_c_2 all T_c all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,hAPP(c_COMBK(T_c,T_b),V_c_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_image__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4523 (all V_x_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> V_x_2 = hAPP(hAPP(V_f_2,V_x_2),V_x_2))) # label(fact_folding__image__simple__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4524 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4525 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2))) # label(fact_Int__Union) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4526 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = V_B_2) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))))) # label(fact_UN__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4527 (all V_f_2 all T_b all T_c c_Lazy__Sequence_Olazy__sequence_OEmpty(T_b) = c_Lazy__Sequence_Omap(T_c,T_b,V_f_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c))) # label(fact__07) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4528 (all V_p_2 all V_g_2 all T_e all V_f_2 all T_b all T_c all T_d hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_b,V_f_2),hAPP(c_Product__Type_Oapfst(T_e,T_b,T_d,V_g_2),V_p_2)) = hAPP(c_Product__Type_Oapfst(T_e,T_b,T_c,V_g_2),hAPP(c_Product__Type_Oapsnd(T_d,T_c,T_e,V_f_2),V_p_2))) # label(fact_apsnd__apfst__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4529 (all V_k all V_j all V_u all V_i hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_u)),V_k)) # label(fact_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4530 (all V_f_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_fun(T_c,tc_HOL_Obool),T_b,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_Set_Oinsert(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_A_2))) # label(fact_image__eq__fold__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4531 (all V_xs_2 all V_x_2 all T_b (class_Orderings_Olinorder(T_b) -> hAPP(c_List_Ofoldl(T_b,T_b,c_Orderings_Oord__class_Omax(T_b),V_x_2),V_xs_2) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))))) # label(fact_min__max_OSup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4532 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)),V_z3) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z2),V_z3))) # label(fact_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4533 (all V_B_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__image__simple__idem(T_b,T_c,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4534 (all V_z_2 all V_g_2 all V_h_2 all T_c all V_A_2 all T_b all T_d (class_Groups_Oab__semigroup__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Fun_Oinj__on(T_b,T_c,V_h_2,V_A_2) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_b,c_Groups_Otimes__class_Otimes(T_d)),hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_g_2),V_h_2)),V_z_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_g_2),V_z_2),hAPP(c_Set_Oimage(T_b,T_c,V_h_2),V_A_2)))))) # label(fact_fold__image__reindex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4535 (all V_n all V_k all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),V_k)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_n)),V_k)))) # label(fact_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4536 (all V_A_2 all T_b V_A_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Set_Oinsert(T_b)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))) # label(fact_UN__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4537 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)))))))) # label(fact_min__max_OSup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4538 (all V_n all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_m))) # label(fact_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4539 (all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y))))))) # label(fact_divide__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4540 (all V_A_2 all T_b (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))))) # label(fact_subset__Compl__self__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4541 (all V_R_2 all V_Q_2 all V_P_2 all T_b all T_d all T_c hAPP(hAPP(hAPP(c_COMBC(T_c,T_d,T_b),V_P_2),V_Q_2),V_R_2) = hAPP(hAPP(V_P_2,V_R_2),V_Q_2)) # label(help_c__COMBC__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4542 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))))) # label(fact_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4543 (all V_c all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b)),V_c)) # label(fact_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4544 (all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_r = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4545 (all V_x all V_z all V_y all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_x)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_z),V_x)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)),V_x))) # label(fact_min__max_Osup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4546 (all V_y_2 all V_x_2 all V_S_2 all V_R_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_R_2),V_S_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_b),c_COMBC(T_c,tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_b,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_c),tc_fun(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_b,T_c)))),c_Product__Type_OPair(T_b,T_c)))),V_S_2)),V_x_2),V_y_2)))) # label(fact_inf__Int__eq2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4547 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4548 (all T_c all T_b c_FunDef_Orp__inv__image(T_b,T_c) = hAPP(c_Product__Type_Oprod_Oprod__case(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),hAPP(hAPP(c_COMBC(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))))),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,T_b),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),c_COMBS(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_b),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,T_b),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),tc_fun(T_c,T_b)),c_Product__Type_OPair(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))),c_Relation_Oinv__image(T_b,T_c))))),c_Relation_Oinv__image(T_b,T_c)))) # label(fact_rp__inv__image__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4549 (all V_B_2 all V_A_2 all T_c all V_C_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_b),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_B_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_C_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Times__subset__cancel2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4550 (all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),V_x_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)))) # label(fact_fold__graph__imp__finite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4551 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) -> (hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2) <-> V_xs_2 = V_ys_2))) # label(fact_inj__map__eq__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4552 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4553 (all V_k_2 all V_P_2 all T_b (class_Orderings_Owellorder(T_b) -> (hBOOL(hAPP(V_P_2,V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Orderings_Oord__class_OLeast(T_b,V_P_2)),V_k_2))))) # label(fact_Least__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4554 (all V_A_2 all T_c all T_b all V_B_2 ((V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(tc_prod(T_c,T_b),T_c,c_Product__Type_Ofst(T_c,T_b)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_B_2))) = V_A_2) & (V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_prod(T_c,T_b),T_c,c_Product__Type_Ofst(T_c,T_b)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_c),V_B_2)))))) # label(fact_fst__image__times) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4555 (all V_c_2 all T_b all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2))) -> -(-hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_B_2)))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4556 (all V_f_2 all V_A_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> c_Nat__Transfer_Ois__nat(hAPP(V_f_2,B_x)))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,T_b),c_Int_Onat),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,tc_Int_Oint),V_f_2),V_A_2))) # label(fact_transfer__int__nat__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4557 (all T_a (class_Orderings_Olinorder(T_a) -> c_List_Olinorder__class_Osorted(T_a,c_List_Olist_ONil(T_a)))) # label(fact_sorted_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4558 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))))) # label(fact_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4559 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4560 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))))) # label(fact_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4561 (all V_f_2 all V_T_2 all V_S_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_S_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_T_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_T_2))))))) # label(fact_setsum__Un__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4562 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4563 (all V_n_2 all V_m_2 c_Nat_Onat_Onat__case(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_n_2),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),V_n_2)),V_m_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4564 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)))) # label(fact_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4565 (all V_l all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l))) # label(fact_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4566 (all V_xs_2 all V_P_2 all T_b hAPP(hAPP(c_List_Otake(T_b),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_OtakeWhile(T_b,V_P_2,V_xs_2))),V_xs_2) = c_List_OtakeWhile(T_b,V_P_2,V_xs_2)) # label(fact_takeWhile__eq__take) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4567 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n_2)),hAPP(c_Code__Numeral_Onat__of,V_m_2))))) # label(fact_less__eq__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4568 (all V_R_2 all T_b c_Relation_OId(T_b) = hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_b,T_b),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2)) # label(fact_relpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4569 (all V_ys all T_a all V_xs ((V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_xs) = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) & (c_List_Olist_ONil(T_a) = V_xs -> hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(c_List_Ohd(T_a),V_ys)))) # label(fact_hd__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4570 (all V_w hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4571 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4572 (all V_a all V_b all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b -> V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))))) # label(fact_divide__eq__imp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4573 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4574 (all V_k_2 all V_i_2 all T_b (class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_i_2),hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_i_2),V_k_2))))) # label(fact_atMost__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4575 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4576 (all T_b hAPP(c_Set_OCollect(tc_prod(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_COMBS(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_fconj)),c_Orderings_Oord__class_Oless(tc_fun(T_b,tc_HOL_Obool))))),c_Finite__Set_Ofinite(T_b)))) = c_Wellfounded_Ofinite__psubset(T_b)) # label(fact_finite__psubset__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4577 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)))))) # label(fact_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4578 (all V_f_2 all V_m_2 all V_n_2 all T_b hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2)),V_f_2) = hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_n_2),hAPP(hAPP(c_Nat_Ocompow(T_b,T_b),V_m_2),V_f_2))) # label(fact_funpow__mult) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4579 (all V_x_2 all V_A_2 all V_f_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_Set_Ovimage(T_b,T_c,V_f_2),V_A_2),V_x_2)) <-> hBOOL(hAPP(V_A_2,hAPP(V_f_2,V_x_2))))) # label(fact_vimage__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4580 (all V_a2_2 all V_r_2 all T_c all T_b (hBOOL(hAPP(c_Predicate_ORangeP(T_b,T_c,V_r_2),V_a2_2)) <-> (exists B_a hBOOL(hAPP(hAPP(V_r_2,B_a),V_a2_2))))) # label(fact_RangeP_Oequations) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4581 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_c,V_f_2),V_z_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))))) # label(fact_fun__left__comm_Ofold__insert__remove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4582 (all T_a (class_Int_Onumber__ring(T_a) -> -c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_nonzero__number__of__Min) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4583 (all V_a all V_y all V_x all T_b all T_a (hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y)) = V_a -> V_a = V_x)) # label(fact_fst__eqD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4584 (all V_A_2 all V_B_2 all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool))),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_B_2)),V_A_2)) # label(fact_Int__Union2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4585 (all V_m_2 all T_c all V_p_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_p_2)) -> hAPP(c_Quotient_OBabs(T_b,T_c,V_p_2,V_m_2),V_x_2) = hAPP(V_m_2,V_x_2))) # label(fact_Babs__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4586 (all V_xs_2 all T_b hAPP(c_List_Oset(T_b),hAPP(c_List_Orotate1(T_b),V_xs_2)) = hAPP(c_List_Oset(T_b),V_xs_2)) # label(fact_set__rotate1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4587 (all V_x_2 all T_b all V_A_2 ((hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)) -> c_List_Osublist(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)),V_A_2) = hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b))) & (-hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)) -> c_List_Olist_ONil(T_b) = c_List_Osublist(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)),V_A_2)))) # label(fact_sublist__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4588 (all V_b_2 all V_a_2 all T_b (class_Groups_Oab__group__add(T_b) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_b) <-> V_a_2 = V_b_2))) # label(fact_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4589 (all V_A_2 all T_b hAPP(c_Relation_Oconverse(T_b,T_b),c_Relation_OId__on(T_b,V_A_2)) = c_Relation_OId__on(T_b,V_A_2)) # label(fact_converse__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4590 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,c_Transitive__Closure_Ortrancl(T_b,V_r_2)))))) # label(fact_listrel__reflcl__if__listrel1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 4591 (all V_a all V_b all T_a (class_Fields_Olinordered__field(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))))) # label(fact_nonzero__abs__divide) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4592 (all V_w_2 all V_c_2 all V_b_2 all T_b (class_Fields_Ofield__inverse__zero(T_b) & class_Int_Onumber(T_b) -> ((V_c_2 = c_Groups_Ozero__class_Ozero(T_b) -> c_Groups_Ozero__class_Ozero(T_b) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)) & (V_c_2 != c_Groups_Ozero__class_Ozero(T_b) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)) <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)))) # label(fact_divide__eq__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4593 (all V_ys all V_xs all V_n all T_b all T_a hAPP(hAPP(c_List_Otake(tc_prod(T_a,T_b)),V_n),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Otake(T_b),V_n),V_ys))) # label(fact_take__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4594 (all V_r_H all V_q_H all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_q = V_q_H)))) # label(fact_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4595 (all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> (exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)) & hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) = V_k)))) # label(fact_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4596 (all T_b all T_c all V_x_2 all V_y_2 all V_P_2 (hBOOL(hAPP(hAPP(V_P_2,V_y_2),V_x_2)) -> hBOOL(hAPP(hAPP(V_P_2,hAPP(c_Product__Type_Osnd(T_c,T_b),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2),V_y_2))),hAPP(c_Product__Type_Ofst(T_c,T_b),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_x_2),V_y_2)))))) # label(fact_exI__realizer) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4597 (all V_Y_2 all V_X_2 all V_f_2 all T_c all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_b,tc_fun(T_c,tc_HOL_Obool),V_r_2,V_f_2) -> (hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_f_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_Y_2),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_b),B_y),V_A_2)) -> (hAPP(V_f_2,B_y) = hAPP(V_f_2,B_x) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_r_2))))))) -> V_Y_2 = V_X_2))))))) # label(fact_UN__equiv__class__inject) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4598 (all V_a all V_x all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_x))))) # label(fact_less__infI2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4599 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__of__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4600 (all V_X_2 (V_X_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_FunDef_Omin__strict)))) # label(fact_smin__emptyI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4601 (all V_a_2 all V_b_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))))) # label(fact_min__max_Ofold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4602 (all V_f_2 all T_c all T_b all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_b) -> hAPP(c_List_Ohd(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)) = hAPP(V_f_2,hAPP(c_List_Ohd(T_b),V_xs_2)))) # label(fact_hd__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4603 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OposDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))))) # label(fact_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4604 (all V_c all V_a all V_b all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_a)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_c),V_b)))))))) # label(fact_divide__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4605 (all V_N_2 all V_M_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_M_2),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_M_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_N_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_M_2)))))))) # label(fact_Min__antimono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4606 (all V_B_2 all T_b c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))),V_B_2)) # label(fact_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4607 (all V_X_2 all V_a_2 all V_f_2 all T_b all V_A_2 (c_Inductive_Ocomplete__lattice__class_Ogfp(tc_fun(T_b,tc_HOL_Obool),V_f_2) = V_A_2 -> (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(V_f_2,c_Inductive_Ocomplete__lattice__class_Olfp(tc_fun(T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),tc_fun(T_b,tc_HOL_Obool)),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool))),V_f_2)),V_X_2))),V_A_2))))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2))))))) # label(fact_def__coinduct3) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4608 (all V_xs_2 all V_x_2 all T_b hAPP(hAPP(c_HOL_OLet(tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),c_Enum_Osublists(T_b,V_xs_2)),hAPP(hAPP(c_COMBS(tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_List_Olist(tc_List_Olist(T_b)),tc_fun(tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),tc_List_Olist(tc_List_Olist(T_b))),c_List_Oappend(tc_List_Olist(T_b))),hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(c_List_Olist_OCons(T_b),V_x_2)))),c_COMBI(tc_List_Olist(tc_List_Olist(T_b))))) = c_Enum_Osublists(T_b,hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) # label(fact_sublists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4609 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_b),V_b_2),V_a_2)),hAPP(c_Relation_Oconverse(T_b,T_c),V_r_2))))) # label(fact_converseI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4610 (all T_b (class_Orderings_Owellorder(T_b) -> c_Wellfounded_Owf(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),c_Orderings_Oord__class_Oless(T_b)))))) # label(fact_wf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4611 (all V_A_2 all V_f_2 all T_c all V_z_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & V_z_2 = hAPP(V_f_2,B_x))) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_z_2),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),V_A_2))))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4612 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n))) # label(fact_div__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4613 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_R_2 c_New__DSequence_Opos__not__seq(hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2)) = c_New__Random__Sequence_Opos__not__random__dseq(V_R_2,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__not__random__dseq__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4614 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2))) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2)))) -> c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(c_Divides_Oadjust(V_b_2),c_Divides_OnegDivAlg(V_a_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b_2)))))) # label(fact_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4615 (all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OInf__class_OInf(T_b,c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(T_b))) # label(fact_Inf__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4616 (all V_xs_2 all T_b (class_Orderings_Olinorder(T_b) -> c_List_Olinorder__class_Osorted(T_b,c_List_Olinorder__class_Osort__key(T_b,T_b,c_COMBI(T_b),V_xs_2)))) # label(fact_sorted__sort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4617 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_b all T_c hAPP(V_f_2,hAPP(V_g_2,V_x_2)) = hAPP(hAPP(c_Fun_Ocomp(T_c,T_b,T_d,V_f_2),V_g_2),V_x_2)) # label(fact_o__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4618 (all V_code__numeral_2 all V_f2_2 all V_f1_2 all T_b c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_b,V_f1_2,V_f2_2,c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2)) = hAPP(hAPP(V_f2_2,V_code__numeral_2),c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_b,V_f1_2,V_f2_2,V_code__numeral_2))) # label(fact_code__numeral_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4619 (all T_b all T_c c_Product__Type_Oapsnd(T_c,T_c,T_b,c_Fun_Oid(T_c)) = c_Fun_Oid(tc_prod(T_b,T_c))) # label(fact_apsnd__id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4620 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Oorder(T_b) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_m_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4621 (all V_z (hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,V_z))) # label(fact_neg__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4622 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_i_2,V_j_2) = hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_j_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_greaterThanLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4623 (all V_u_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Int_Onat,V_u_2))) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))) # label(fact_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4624 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Finite__Set_Ofold1(T_b,c_Orderings_Oord__class_Omin(T_b)),V_A_2)),V_x_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),V_x_2))))))))) # label(fact_fold1__below__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4625 (all V_c_2 all V_b_2 all V_w_2 all T_b (class_Fields_Olinordered__field__inverse__zero(T_b) & class_Int_Onumber(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w_2)),c_Groups_Ozero__class_Ozero(T_b)))))))) # label(fact_le__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4626 (all V_y_2 all T_b (class_Rings_Olinordered__idom(T_b) & class_Int_Onumber__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_y_2))))) # label(fact_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4627 (all V_n all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),V_xs)),hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))))) # label(fact_take__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4628 (all V_n hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_n)))) # label(fact_lessI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4629 (all V_r_2 all V_p_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),V_p_2),c_Transitive__Closure_Otrancl(T_b,V_r_2))))) # label(fact_r__into__trancl_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4630 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4631 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2))))))) # label(fact_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4632 (all V_k all V_v_H all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H))),V_k))))) # label(fact_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4633 (all V_i all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs))))) # label(fact_distinct__drop) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4634 (all V_S_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_S_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_b,c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_S_2)) = V_S_2))) # label(fact_inv__into__image__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4635 (all V_r_2 all T_b (c_Relation_Otrans(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),V_r_2))) <-> c_Predicate_Otransp(T_b,V_r_2))) # label(fact_transp__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4636 (all V_S_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_b) & class_Orderings_Oorder(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> ((exists B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(T_b),B_xa),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),B_x),B_xa)))) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_S_2)))) -> c_Orderings_Oord__class_OLeast(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_S_2))) = hAPP(V_f_2,c_Orderings_Oord__class_OLeast(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2))))))) # label(fact_Least__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4637 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4638 (all V_b_2 all V_a_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2))))))) # label(fact_neg__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4639 (all V_n_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2))),hAPP(V_f_2,hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_setsum__atMost__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4640 (all V_t_2 all V_s_2 all V_b_2 all V_a_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a_2),V_s_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_b_2),V_t_2))),c_FunDef_Opair__leq)))) # label(fact_pair__leqI1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4641 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a)) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4642 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),V_z) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4643 (all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_divmod__int__rel__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4644 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_t_2)))))))) # label(fact_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4645 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)))) # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4646 (all V_zs_2 all T_c all T_b hAPP(hAPP(c_List_Ozip(T_b,T_c),hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_b),c_Product__Type_Ofst(T_b,T_c)),V_zs_2)),hAPP(hAPP(c_List_Omap(tc_prod(T_b,T_c),T_c),c_Product__Type_Osnd(T_b,T_c)),V_zs_2)) = V_zs_2) # label(fact_zip__map__fst__snd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4647 (all V_b_2 all V_B_2 all V_A_2 all T_c all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)))) <-> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_INF1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4648 (all V_m V_m = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4649 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))))) # label(fact_abs__ge__minus__self) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4650 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2)))))))) # label(fact_Min__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4651 (all V_f_2 all T_c all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) -> c_Relation_Otrans(T_c,hAPP(hAPP(c_Relation_Oinv__image(T_b,T_c),V_r_2),V_f_2)))) # label(fact_trans__inv__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4652 (all V_A_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OInf__class_OInf(T_b,hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OAll(T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fimplies),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_A_2)))),hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_b)))))) = c_Complete__Lattice_OSup__class_OSup(T_b,V_A_2))) # label(fact_Sup__Inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4653 (all V_P_2 all V_a_2 all T_b (hBOOL(hAPP(V_P_2,V_a_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_OCollect(T_b),V_P_2))))) # label(fact_mem__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4654 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4655 (all V_j all V_i all V_m hAPP(hAPP(c_List_Odrop(tc_Nat_Onat),V_m),c_List_Oupt(V_i,V_j)) = c_List_Oupt(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m),V_j)) # label(fact_drop__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4656 (all V_P_2 all V_B_2 all V_A_2 all T_b ((exists B_x (hBOOL(hAPP(V_P_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) & hBOOL(hAPP(V_P_2,B_x)))) | (exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)) & hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4657 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_diff__self) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4658 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_b),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2))))))) # label(fact_Sup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4659 (all V_B_2 all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)))) # label(fact_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4660 (all V_xs_2 all V_P_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),c_List_OtakeWhile(T_b,V_P_2,V_xs_2)))) -> hBOOL(hAPP(V_P_2,V_x_2)) & hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_set__takeWhileD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4661 (all V_g_2 all V_f_2 all V_B_2 all V_A_2 all T_b (class_Groups_Ocomm__monoid__mult(T_b) & class_Groups_Ocomm__monoid__add(T_b) -> (V_A_2 = V_B_2 -> (c_Nat__Transfer_Onat__set(V_B_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),V_g_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_b),V_f_2),V_A_2)))))) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4662 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)))))) # label(fact_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4663 (all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Nat_OSuc,V_m)))) # label(fact_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4664 (all T_c all V_f_2 all V_b_2 all V_A_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(V_f_2,V_x_2) = V_b_2 -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2)))))) # label(fact_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4665 (all V_xs_2 all V_n_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Odrop(T_b),V_n_2),V_xs_2))),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_set__drop__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4666 (all V_xs_2 all V_n_2 all T_b hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(c_List_Oset(T_b),hAPP(c_List_Orotate(T_b,V_n_2),V_xs_2))) # label(fact_set__rotate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4667 (all V_n all V_m all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m)),V_n)) -> c_List_Oupt(V_i,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m)) = hAPP(hAPP(c_List_Otake(tc_Nat_Onat),V_m),c_List_Oupt(V_i,V_n)))) # label(fact_take__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4668 (all V_i_2 all V_f_2 all V_x_2 all T_b all T_c hAPP(c_Lazy__Sequence_Obind(T_c,T_b,hAPP(V_x_2,V_i_2)),hAPP(hAPP(c_COMBC(T_c,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_b)),V_f_2),V_i_2)) = hAPP(c_New__DSequence_Opos__bind(T_c,T_b,V_x_2,V_f_2),V_i_2)) # label(fact_New__DSequence_Opos__bind__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4669 (all V_m all V_n all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_n)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)),V_k))))) # label(fact_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4670 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hBOOL(hAPP(c_List_Odistinct(T_a),hAPP(c_List_OremoveAll(T_a,V_x),V_xs))))) # label(fact_distinct__removeAll) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4671 (all V_r_2 all T_b (c_Relation_Otrans(T_b,V_r_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_r_2)) -> (all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_z)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_z)),V_r_2)))))))) # label(fact_trans__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4672 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n))) # label(fact_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4673 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_Osublists(T_a,V_xs)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) # label(fact_length__sublists) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4674 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4675 (all V_x_2 all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))))) # label(fact_card__Diff1__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4676 (all V_xy all V_i all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),hAPP(c_Product__Type_Ofst(T_a,T_b),V_xy))),hAPP(hAPP(c_List_Olist__update(T_b,V_ys),V_i),hAPP(c_Product__Type_Osnd(T_a,T_b),V_xy))) = hAPP(hAPP(c_List_Olist__update(tc_prod(T_a,T_b),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i),V_xy)) # label(fact_update__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4677 (all V_xss_2 all T_b hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Otl(T_b)),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_b)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_b)))),V_xss_2)) = c_List_Oconcat(tc_List_Olist(T_b),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_b)),T_b,c_List_Olist_ONil(tc_List_Olist(T_b)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b))),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_List_Olist(tc_List_Olist(T_b)),tc_List_Olist(tc_List_Olist(T_b))),c_List_Olist_OCons(tc_List_Olist(T_b))),c_List_Olist_ONil(tc_List_Olist(T_b)))))),V_xss_2))) # label(fact_transpose__aux__filter__tail) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4678 (all T_b all V_g_2 all V_f_2 all V_P_2 ((all B_x (hBOOL(hAPP(V_P_2,B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(V_f_2,hAPP(V_g_2,B_x))),hAPP(V_f_2,B_x))))) -> c_Wellfounded_Owf(T_b,hAPP(c_Set_OCollect(tc_prod(T_b,T_b)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_b),hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),V_P_2))),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_fun(T_b,T_b),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBB(T_b,tc_HOL_Obool,T_b)),c_fequal)),V_g_2))))))) # label(fact_wf__if__measure) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4679 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4680 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))))) = c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)))))) # label(fact_Max_Oremove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4681 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2))) # label(fact_listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4682 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2))) <-> -hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) & hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)))) # label(fact_distinct_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4683 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4684 (all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4685 (all V_P all T_a V_P = hAPP(c_COMBI(T_a),V_P)) # label(help_c__COMBI__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4686 (all V_b all V_c all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)))))) # label(fact_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4687 (all V_f_2 all T_d all V_A_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(hAPP(V_f_2,V_a_2),V_b_2)),hAPP(c_Set_Oimage(tc_prod(T_b,T_c),T_d,hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,T_d),V_f_2)),V_A_2))))) # label(fact_pair__imageI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4688 (all V_z_2 -(all B_x all B_y V_z_2 != hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B_x),B_y)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))))))) # label(fact_eq__Abs__Integ) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4689 (all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_abs__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4690 (all V_k all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k))))) # label(fact_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4691 (all V_f_2 all T_c all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_b),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_b))) # label(fact_setsum__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4692 (all V_P_2 all T_c all T_b hAPP(c_Relation_ODomain(T_b,T_c),hAPP(c_Set_OCollect(tc_prod(T_b,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2))) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_b),c_HOL_OEx(T_c)),V_P_2))) # label(fact_Domain__Collect__split) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4693 (all V_xs_2 all T_b all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i_2),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_b)),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),hAPP(c_Random_Opick(T_b,V_xs_2),V_i_2)),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_b),T_b),c_Product__Type_Osnd(tc_Code__Numeral_Ocode__numeral,T_b)),V_xs_2)))))) # label(fact_pick__member) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4694 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w)))) # label(fact_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4695 (all V_n_2 all V_xs_2 all V_x_2 all T_b hAPP(hAPP(c_List_Onth(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),V_n_2) = c_Nat_Onat_Onat__case(T_b,V_x_2,hAPP(c_List_Onth(T_b),V_xs_2),V_n_2)) # label(fact_nth_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4696 (all V_b all V_a all T_a (class_Groups_Oordered__comm__monoid__add(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b))))))) # label(fact_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4697 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_c all T_b all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (V_B_2 = hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hAPP(hAPP(c_Fun_Ocomp(T_c,T_d,T_b,V_h_2),V_f_2),B_x) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_d),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_h_2),V_B_2)))))) # label(fact_strong__setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4698 (all V_a all V_N all V_n all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_N)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),c_Groups_Oone__class_Oone(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_N)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))))) # label(fact_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4699 (all V_n all V_m hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),hAPP(c_Nat_OSuc,V_m)))) # label(fact_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4700 (all V_u_2 all V_m_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_l_2,V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I19_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 4701 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z) = V_z) # label(fact_zmult__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4702 (all V_z V_z = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4703 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4704 (all V_r_2 all T_b (c_Wellfounded_Owf(T_b,V_r_2) -> c_Wellfounded_Owf(tc_fun(T_b,tc_HOL_Obool),c_Wellfounded_Omax__ext(T_b,V_r_2)))) # label(fact_max__ext__wf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4705 (all V_x_2 all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_R_2)),hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),V_S_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4706 (all V_xs all T_a all V_ys (V_ys != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = c_List_Olast(T_a,V_ys))) # label(fact_last__appendR) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4707 (all V_n_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_n_2)) -> (-hBOOL(hAPP(V_P_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> hAPP(c_Nat_OSuc,c_Orderings_Oord__class_OLeast(tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat),V_P_2),c_Nat_OSuc))) = c_Orderings_Oord__class_OLeast(tc_Nat_Onat,V_P_2)))) # label(fact_Least__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4708 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)))) # label(fact_min__max_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4709 (all V_x_2 all V_A_2 all T_b (class_Lattices_Olattice(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_b),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_b,V_A_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Inf__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4710 (all V_ys_2 all V_g_2 all T_e all V_xs_2 all V_f_2 all T_b all T_d all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_b),V_f_2),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_g_2),V_ys_2))) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_e),T_b),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_e,T_b),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_c,T_d)),tc_fun(T_e,T_b),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),T_b,T_e),V_f_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_e,T_d),tc_fun(T_e,tc_prod(T_c,T_d))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(tc_fun(T_e,T_d),tc_fun(T_e,tc_prod(T_c,T_d))),T_c),c_COMBB(T_d,tc_prod(T_c,T_d),T_e)),c_Product__Type_OPair(T_c,T_d))),V_g_2)))),hAPP(hAPP(c_List_Ozip(T_c,T_e),V_xs_2),V_ys_2))) # label(fact_map__zip__map2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4711 (all V_size_2 all V_nrandom_2 all T_b c_Random__Sequence_Oempty(T_b,V_nrandom_2,V_size_2) = hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_b)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_b))) # label(fact_Random__Sequence_Oempty__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4712 (all V_nat_H_2 all V_nat_2 (hAPP(c_Nat_OSuc,V_nat_H_2) = hAPP(c_Nat_OSuc,V_nat_2) <-> V_nat_2 = V_nat_H_2)) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4713 (all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4714 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Orderings_Oorder__class_Omono(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),V_f_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,V_A_2)),hAPP(V_f_2,V_B_2)))))) # label(fact_mono__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4715 (all V_fs_2 all T_b all V_y_2 all V_x_2 all V_f_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,V_fs_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_List_Omeasures(T_b,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_b,tc_Nat_Onat)),V_f_2),V_fs_2))))))) # label(fact_measures__lesseq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4716 (all V_b all V_a all V_c all T_a (class_Rings_Olinordered__semiring(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_c)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4717 (all V_B_2 all V_A_2 all T_b hAPP(c_Relation_OImage(T_b,T_b,c_Relation_OId__on(T_b,V_A_2)),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Image__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4718 (all V_A_2 all V_c_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2))) -> -hBOOL(hAPP(hAPP(c_member(T_b),V_c_2),V_A_2)))) # label(fact_ComplE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4719 (all V_x_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm__idem(T_b,T_c,V_f_2) -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_x_2)))) # label(fact_fun__left__comm__idem_Ofun__comp__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4720 (all V_xs_2 all V_P_2 all T_b ((all B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> hBOOL(hAPP(V_P_2,hAPP(hAPP(c_List_Onth(T_b),V_xs_2),B_n))))) <-> c_List_Olist__all(T_b,V_P_2,V_xs_2))) # label(fact_list__all__length) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4721 (all V_xs_2 all T_b hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),c_List_Oremdups(T_b,V_xs_2)) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_List_Oset(T_b),V_xs_2))) # label(fact_length__remdups__card__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4722 (all V_s_2 all V_r_2 all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_b_2),V_b_H_2)),V_s_2)) & V_a_H_2 = V_a_2 | hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_a_H_2)),V_r_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_a_H_2),V_b_H_2))),c_Wellfounded_Olex__prod(T_b,T_c,V_r_2,V_s_2))))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4723 (all V_S_2 all V_g_2 all V_f_2 all V_P_2 all T_b all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),hAPP(c_Set_OCollect(T_c),V_P_2)))),hAPP(c_Set_Oimage(T_c,T_b,V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_P_2))))) = hAPP(c_Set_Oimage(T_c,T_b,hAPP(hAPP(c_COMBS(T_c,T_b,T_b),hAPP(hAPP(c_COMBS(T_c,T_b,tc_fun(T_b,T_b)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_b,tc_fun(T_b,T_b)),T_c),c_If(T_b)),V_P_2)),V_f_2)),V_g_2)),V_S_2)) # label(fact_if__image__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4724 (all T_c all V_D_2 all V_C_2 all T_b all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_B_2)) -> hAPP(V_C_2,B_x) = hAPP(V_D_2,B_x))) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_D_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)))) # label(fact_INT__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4725 (all V_x_2 -(all B_y (V_x_2 = hAPP(c_Code__Numeral_Oof__nat,B_y) -> -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_y),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))))) # label(fact_of__nat__cases) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4726 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_x = V_y -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y))))) # label(fact_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4727 (all V_f_2 all V_B_2 all T_c all V_A_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2))))) # label(fact_finite__surj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4728 (all V_B_2 all V_X_2 all V_f_2 all T_c all V_r_2 all V_A_2 all T_b (c_Equiv__Relations_Oequiv(T_b,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_b,tc_fun(T_c,tc_HOL_Obool),V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_b,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_b),V_A_2),V_r_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_f_2,B_x)),V_B_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_f_2)),V_B_2))))))) # label(fact_UN__equiv__class__type) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4729 (all V_P_2 all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> ((all B_x hBOOL(hAPP(V_P_2,hAPP(c_Code__Numeral_Onat__of,B_x)))) -> hBOOL(hAPP(V_P_2,V_y_2))))) # label(fact_nat__of__induct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4730 (all V_a all V_m all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4731 (all T_1 all T_2 (class_Enum_Oenum(T_2) & class_Enum_Oenum(T_1) -> class_Enum_Oenum(tc_sum(T_2,T_1)))) # label(arity_sum__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4732 (all V_A_2 all T_b c_Relation_Orefl__on(T_b,V_A_2,c_Relation_OId__on(T_b,V_A_2))) # label(fact_refl__on__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4733 (all V_v_2 all V_A_2 all V_x_2 all V_z_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)),V_v_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> -(all B_y (hAPP(hAPP(V_f_2,V_x_2),B_y) = V_v_2 -> -hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_b,T_c,V_f_2,V_z_2,V_A_2),B_y)))))))) # label(fact_fun__left__comm_Ofold__graph__insertE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4734 (all V_f_2 all T_c all V_B_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_b,V_f_2),V_B_2))))) # label(fact_vimage__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4735 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),V_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4736 (all V_r_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_b,T_b)),V_r_2)) -> (c_Wellfounded_Owf(T_b,V_r_2) <-> c_Wellfounded_Oacyclic(T_b,V_r_2)))) # label(fact_wf__iff__acyclic__if__finite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4737 (all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Finite__Set_Ocard(T_b),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_distinct__card) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4738 (all V_w all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) -> c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) -> c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))))) # label(fact_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4739 (all V_r_2 all V_ys_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,V_r_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),B_x),hAPP(c_List_Oset(tc_prod(T_b,T_b)),hAPP(hAPP(c_List_Ozip(T_b,T_b),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_b,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool)),T_b),c_COMBC(T_b,tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_b,tc_prod(T_b,T_b)),tc_fun(T_b,tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool)),T_b),hAPP(c_COMBB(tc_prod(T_b,T_b),tc_fun(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool),tc_HOL_Obool),T_b),c_member(tc_prod(T_b,T_b)))),c_Product__Type_OPair(T_b,T_b)))),V_r_2)),B_x)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys_2))) # label(fact_listrel__iff__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4740 (all V_n -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4741 (all V_s_2 all V_r_2 all T_b (c_Relation_Osym(T_b,V_r_2) -> (c_Relation_Osym(T_b,V_s_2) -> c_Relation_Osym(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_sym__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4742 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olinorder__class_OMin(T_b,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_b),c_Big__Operators_Olinorder__class_OMin(T_b,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_b,V_B_2))))))) # label(fact_Min_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4743 (all V_g_2 all V_f_2 all V_xs_2 all T_b all T_c (class_Groups_Oordered__ab__semigroup__add(T_c) & class_Groups_Omonoid__add(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_b,T_c),V_g_2),V_xs_2))))))) # label(fact_listsum__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4744 (all V_K_2 all V_f_2 all V_A_2 all T_b all T_c (class_Groups_Oordered__ab__semigroup__add(T_c) & class_Rings_Osemiring__1(T_c) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_b),B_i),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_i)),V_K_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(c_Finite__Set_Ocard(T_b),V_A_2))),V_K_2)))))) # label(fact_setsum__bounded) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4745 (all V_g_2 all V_f_2 all T_c all T_b all T_e all T_d hAPP(c_Fun_Ocomp(tc_prod(T_d,T_e),T_d,tc_prod(T_b,T_c),c_Product__Type_Ofst(T_d,T_e)),c_Product__Type_Omap__pair(T_b,T_d,T_c,T_e,V_f_2,V_g_2)) = hAPP(c_Fun_Ocomp(T_b,T_d,tc_prod(T_b,T_c),V_f_2),c_Product__Type_Ofst(T_b,T_c))) # label(fact_fst__comp__map__pair) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4746 (all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_neg__0__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4747 (all T_a c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a) = c_Lazy__Sequence_Oflat(T_a,c_Lazy__Sequence_Olazy__sequence_OEmpty(tc_Lazy__Sequence_Olazy__sequence(T_a)))) # label(fact__04) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4748 (all V_xs all V_n all T_a hAPP(c_List_Orotate1(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orotate1(T_a),V_xs))) # label(fact_rotate1__rotate__swap) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4749 (all V_k c_Quickcheck_Obeyond(V_k,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) # label(fact_beyond__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4750 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_b (class_Rings_Oordered__ring(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),V_e_2)),V_c_2)),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_b_2),V_e_2)),V_d_2)))))) # label(fact_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4751 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4752 (all V_xs_2 all V_P_2 all T_b (hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2) = V_xs_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),hAPP(c_List_Oset(T_b),V_xs_2))) -> hBOOL(hAPP(V_P_2,B_x)))))) # label(fact_filter__id__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4753 (all V_y_2 all V_x_2 all V_f_2 all T_b all T_c (class_Orderings_Oorder(T_b) & class_Orderings_Oorder(T_c) -> (c_Orderings_Oorder__class_Omono(T_b,T_c,V_f_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))))))) # label(fact_monoD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4754 (all V_n all V_m all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))))) # label(fact_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4755 (all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n))))))) # label(fact_power__gt1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4756 (all V_y_2 all V_x_2 all V_f_2 all V_A_2 all T_c all T_b (hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),V_y_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_b,T_c,V_A_2,V_f_2),V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> V_y_2 = V_x_2)))) # label(fact_inv__into__injective) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4757 (all V_f_2 all T_b (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) -> (hAPP(c_Set_Oimage(T_b,T_b,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) -> c_Fun_Oinj__on(T_b,T_b,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))))) # label(fact_finite__UNIV__surj__inj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4758 (all V_a all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a)) # label(fact_add__0__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4759 (all V_xs_2 all T_b hAPP(c_List_Oset(T_b),V_xs_2) = hAPP(c_List_Oset(T_b),c_List_Oremdups(T_b,V_xs_2))) # label(fact_set__remdups) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4760 (all V_B_2 all T_c all V_A_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(V_B_2,V_a_2))))) # label(fact_INT__lower) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4761 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4762 (all V_B_2 all V_X_2 all T_c all T_b hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_b,tc_HOL_Obool),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_COMBC(tc_fun(T_b,tc_HOL_Obool),tc_fun(T_b,tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),c_Product__Type_OSigma(T_b,T_c)),V_B_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),V_X_2)),V_B_2)) # label(fact_Sigma__Union) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4763 (all V_xs_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),hAPP(c_List_Oset(tc_prod(T_b,T_b)),hAPP(hAPP(c_List_Ozip(T_b,T_b),V_xs_2),V_xs_2)))) <-> V_b_2 = V_a_2 & hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_zip__same) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4764 (all T_b all V_a_2 all V_P_2 (hBOOL(hAPP(V_P_2,V_a_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),hAPP(c_Set_OCollect(T_b),V_P_2))))) # label(fact_CollectI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4765 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_greaterThanAtMost__upto) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4766 (all V_ys all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Otl(T_a),V_xs)),V_ys))) # label(fact_tl__append2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4767 (all V_P_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_c all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2))) -> (hBOOL(hAPP(hAPP(V_P_2,V_bx_2),V_by_2)) -> ((all B_a all B_b all B_aa all B_ba (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_aa),B_ba))),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_b,T_c),tc_prod(T_b,T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_b,T_c),tc_prod(T_b,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_aa),B_ba)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_b,T_c),V_r_2))) -> (hBOOL(hAPP(hAPP(V_P_2,B_aa),B_ba)) -> hBOOL(hAPP(hAPP(V_P_2,B_a),B_b)))))) -> hBOOL(hAPP(hAPP(V_P_2,V_ax_2),V_ay_2)))))) # label(fact_converse__rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4768 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)),V_n) = V_m)) # label(fact_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4769 (all V_k all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_k)) # label(fact_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4770 (all V_B_2 all V_a_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_B_2)) # label(fact_insert__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4771 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),V_z))) # label(fact_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4772 (all V_f_2 all T_b c_FunDef_Ois__measure(T_b,V_f_2)) # label(fact_is__measure_Ointros) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4773 (all T_c all T_b all V_P_2 ((all B_x1 hBOOL(hAPP(V_P_2,B_x1))) <-> (all B_a all B_b hBOOL(hAPP(V_P_2,hAPP(hAPP(c_Product__Type_OPair(T_b,T_c),B_a),B_b)))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4774 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)))) # label(fact_inf__sup__aci_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4775 (all V_y_2 all V_x_2 all V_f_2 all V_r_2 all T_c all T_b (c_Predicate_Oinv__imagep(T_b,T_c,V_r_2,V_f_2,V_x_2,V_y_2) <-> hBOOL(hAPP(hAPP(V_r_2,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))))) # label(fact_inv__imagep__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4776 (all V_a_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_b,T_b,c_Orderings_Oord__class_Omax(T_b)),V_a_2),V_A_2) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),V_A_2))))) # label(fact_Max_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4777 (all V_i_2 all V_xs_2 all T_b (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))),V_xs_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2))) -> hAPP(hAPP(c_List_Onth(tc_List_Olist(T_b)),V_xs_2),V_i_2) = hAPP(hAPP(c_List_Omap(tc_List_Olist(T_b),T_b),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_Nat_Onat,T_b),c_List_Onth(T_b)),V_i_2)),hAPP(c_List_Ofilter(tc_List_Olist(T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),c_List_Otranspose(T_b,V_xs_2)))))) # label(fact_transpose__column) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4778 (all V_y_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_y_2),hAPP(c_Finite__Set_Ocard(T_b),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_b,T_c),hAPP(c_COMBK(T_c,T_b),V_y_2)),V_A_2)))) # label(fact_setprod__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4779 (all V_B_2 all V_a_2 all T_c all V_A_2 all V_u_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_u_2),V_A_2)) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_b,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Set_Oinsert(T_c),V_a_2)),V_B_2)))) # label(fact_UN__insert__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4780 (all T_b (class_Rings_Oring__1(T_b) -> hBOOL(hAPP(hAPP(c_member(T_b),c_Groups_Oone__class_Oone(T_b)),c_Int_Oring__1__class_OInts(T_b))))) # label(fact_Ints__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4781 (all V_A_2 all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),V_xs_2),c_List_Olists(T_b,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),c_List_Olists(T_b,V_A_2))))) # label(fact_Cons__in__lists__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4782 (all V_B_2 all V_A_2 all T_b all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_b,tc_HOL_Obool),hAPP(c_Set_Oimage(T_c,tc_fun(T_b,tc_HOL_Obool),V_B_2),V_A_2))) # label(fact_UNION__eq__Union__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4783 (all V_P_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)))) & hBOOL(hAPP(V_P_2,B_x)))) <-> (exists B_x1 hBOOL(hAPP(V_P_2,B_x1))))) # label(fact_bex__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4784 (all V_k c_Int_OBit0(V_k) != c_Int_OMin) # label(fact_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4785 (all V_xs_2 all V_P_2 all T_b hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_List_Oset(T_b),hAPP(c_List_Ofilter(T_b,V_P_2),V_xs_2))),hAPP(c_List_Oset(T_b),V_xs_2)))) # label(fact_filter__is__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4786 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H))))))))) # label(fact_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4787 (all V_q all V_b all V_r all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_c)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_r),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_q,V_c))),V_r)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),V_c)))))) # label(fact_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4788 (all V_S_2 all V_R_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_R_2),V_S_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_S_2),c_Transitive__Closure_Ortrancl(T_b,V_R_2))) -> c_Transitive__Closure_Ortrancl(T_b,V_R_2) = c_Transitive__Closure_Ortrancl(T_b,V_S_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4789 (all V_x_2 all V_f_2 all T_d all T_c all T_b hAPP(c_Product__Type_Ofst(T_b,T_c),hAPP(c_Product__Type_Oapfst(T_d,T_b,T_c,V_f_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_d,T_c),V_x_2))) # label(fact_fst__apfst) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4790 (all V_x_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_b,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_b,hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)))))) # label(fact_Max__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4791 (all V_r_2 all V_q_2 all V_b_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_r_2),V_b_2)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_If(tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_q_2)),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_q_2)),V_r_2))) = hAPP(c_Divides_Oadjust(V_b_2),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_2),V_r_2))) # label(fact_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4792 (all V_tsa_2 all V_t_2 all V_G_2 all T_b (c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),V_t_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b),tc_HOL_Obool)))) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,V_tsa_2) -> c_Hoare__Mirabelle_Ohoare__derivs(T_b,V_G_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_b)),V_t_2),V_tsa_2))))) # label(fact_hoare__derivs_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4793 (all V_xs_2 all V_f_2 all T_d all V_a_2 all V_g_2 all T_c all T_b hAPP(c_List_Ofoldl(T_b,T_c,V_g_2,V_a_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_b,T_d,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_d,T_c),tc_fun(T_d,T_b)),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_b),tc_fun(tc_fun(T_d,T_c),tc_fun(T_d,T_b)),T_b),c_COMBB(T_c,T_b,T_d)),V_g_2)),V_f_2),V_a_2),V_xs_2)) # label(fact_foldl__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4794 (all V_x_2 all V_F_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofolding__idem(T_b,T_c,V_f_2,V_F_2) -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_x_2)))) # label(fact_folding__idem_Oidem__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4795 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (V_f_2 = c_Fun_Ofun__upd(T_b,T_c,V_f_2,V_x_2,V_y_2) <-> V_y_2 = hAPP(V_f_2,V_x_2))) # label(fact_fun__upd__idem__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4796 (all V_n all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4797 (all V_c all V_b all V_v all T_b (class_Int_Onumber(T_b) & class_Rings_Osemiring(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_c)))) # label(fact_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4798 (all V_vs_2 all V_us_2 all V_r_2 all V_ys_2 all V_xs_2 all T_b (V_ys_2 = V_xs_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_us_2),V_vs_2)),c_List_Olistrel1(T_b,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))) & V_vs_2 = V_us_2 -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),V_us_2)),hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),V_vs_2))),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_append__listrel1I) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4799 (all V_r_2 all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Oappend(T_b),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),c_List_Olist_ONil(T_b)))),hAPP(hAPP(c_List_Oappend(T_b),V_ys_2),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),c_List_Olist_ONil(T_b))))),c_List_Olistrel1(T_b,V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) & V_xs_2 = V_ys_2 | V_y_2 = V_x_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_b,V_r_2))))) # label(fact_snoc__listrel1__snoc__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4800 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b_H),V_c))))) # label(fact_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4801 (all T_b all V_a_2 all V_P_2 ((hBOOL(hAPP(V_P_2,V_a_2)) -> hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_P_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)))) & (-hBOOL(hAPP(V_P_2,V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_b),hAPP(hAPP(c_COMBS(T_b,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_b),c_fconj),hAPP(hAPP(c_COMBC(T_b,T_b,tc_HOL_Obool),c_fequal),V_a_2))),V_P_2))))) # label(fact_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4802 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,V_l_2),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_b),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))))))) # label(fact_ivl__disj__un_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4803 (all V_A_2 all T_c all T_b hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_A_2)))) # label(fact_Compl__Times__UNIV1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4804 (all V_y_2 all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)) = hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_b,V_x_2,V_y_2)))) # label(fact_image__uminus__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4805 (all V_r_2 all V_y_2 all T_b ((exists B_a exists B_x hAPP(hAPP(c_List_Olist_OCons(T_b),B_a),B_x) = V_y_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),c_List_Olist_ONil(T_b)),V_y_2)),c_List_Olexord(T_b,V_r_2))))) # label(fact_lexord__Nil__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4806 (all V_s_2 all V_xs_2 all T_b (hBOOL(hAPP(c_List_Odistinct(T_b),V_xs_2)) -> c_List_Osublist(T_b,V_xs_2,V_s_2) = hAPP(c_List_Ofilter(T_b,hAPP(hAPP(c_COMBC(T_b,tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool),c_member(T_b)),hAPP(c_List_Oset(T_b),c_List_Osublist(T_b,V_xs_2,V_s_2)))),V_xs_2))) # label(fact_filter__in__sublist) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4807 (all V_xs_2 all V_f_2 all T_c all T_b c_List_Oconcat(T_b,hAPP(hAPP(c_List_Omap(T_c,tc_List_Olist(T_b)),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_COMBB(T_b,tc_fun(tc_List_Olist(T_b),tc_List_Olist(T_b)),T_c),c_List_Olist_OCons(T_b)),V_f_2)),c_List_Olist_ONil(T_b))),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)) # label(fact_concat__map__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4808 (all V_C_2 all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4809 (all V_B_2 all V_A_2 all T_c all T_b all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_b),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_INT__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4810 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = V_x)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4811 (all V_a_2 all T_b (class_Rings_Olinordered__idom(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),c_Groups_Ozero__class_Ozero(T_b)),V_a_2)) <-> c_Groups_Oone__class_Oone(T_b) = c_Groups_Osgn__class_Osgn(T_b,V_a_2)))) # label(fact_sgn__1__pos) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4812 (all V_B_2 all V_A_2 all T_b hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),V_B_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4813 (all V_xss_2 all T_b (c_List_Oconcat(T_b,V_xss_2) = c_List_Olist_ONil(T_b) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_b)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_b)),V_xss_2))) -> B_x = c_List_Olist_ONil(T_b))))) # label(fact_concat__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4814 (all V_a_2 all V_b_2 all V_c_2 all T_b (class_Fields_Olinordered__field(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_c_2),c_Groups_Ozero__class_Ozero(T_b))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),V_b_2),V_c_2)),V_a_2)))))) # label(fact_neg__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4815 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_length__tl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4816 (all V_x_2 all V_P_2 ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B_y),c_Int_OInteg)) -> hBOOL(hAPP(V_P_2,hAPP(c_Int_OAbs__Integ,B_y))))) -> hBOOL(hAPP(V_P_2,V_x_2)))) # label(fact_Abs__Integ__induct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4817 (all T_b c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool)) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_b,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_b,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Inter__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4818 (all V_v_H_2 all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2) <-> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2))) -> (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2))) -> V_v_H_2 = V_v_2) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4819 (all V_x_2 all V_n_2 all T_b (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_List_Oreplicate(T_b,V_n_2,V_x_2) = c_List_Olist_ONil(T_b))) # label(fact_empty__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4820 (all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OSup__class_OSup(T_b,c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(T_b))) # label(fact_Sup__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4821 (all V_YS_2 all V_y_2 all V_XS_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_XS_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_x_2),V_y_2)),c_FunDef_Opair__leq)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),V_YS_2)),c_FunDef_Omin__weak)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_XS_2),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_y_2),V_YS_2))),c_FunDef_Omin__weak)))))) # label(fact_wmin__insertI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4822 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4823 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)))) # label(fact_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4824 (all V_k_2 all V_P_2 all T_b (class_Orderings_Owellorder(T_b) -> (hBOOL(hAPP(V_P_2,V_k_2)) -> hBOOL(hAPP(V_P_2,c_Orderings_Oord__class_OLeast(T_b,V_P_2)))))) # label(fact_LeastI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4825 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_inf__sup__aci_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 4826 (all V_P_2 all T_c all T_b c_HOL_OThe(tc_prod(T_b,T_c),hAPP(hAPP(c_COMBS(tc_prod(T_b,T_c),T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_c,tc_HOL_Obool),tc_prod(T_b,T_c)),V_P_2),c_Product__Type_Ofst(T_b,T_c))),c_Product__Type_Osnd(T_b,T_c))) = c_HOL_OThe(tc_prod(T_b,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_b,T_c,tc_HOL_Obool),V_P_2))) # label(fact_The__split) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4827 (all V_P_2 all T_b -c_List_Olist__ex1(T_b,V_P_2,c_List_Olist_ONil(T_b))) # label(fact_list__ex1__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4828 (all V_yq all T_a c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a),V_yq) = V_yq) # label(fact__02) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4829 (all V_b_2 hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_fun(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),hAPP(c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_If(tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),V_b_2))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),V_b_2))))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) = c_Divides_Oadjust(V_b_2)) # label(fact_adjust__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4830 (all V_xs_2 all V_n_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2)),V_n_2)) <-> hAPP(hAPP(c_List_Odrop(T_b),V_n_2),V_xs_2) = c_List_Olist_ONil(T_b))) # label(fact_drop__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4831 (all V_x_2 all V_f_2 all V_c_2 all T_d all T_b all T_c V_c_2 = hAPP(hAPP(c_Fun_Ocomp(T_c,T_b,T_d,hAPP(c_COMBK(T_b,T_c),V_c_2)),V_f_2),V_x_2)) # label(fact_K__record__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4832 (all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> V_n = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4833 (all V_xs_2 all T_c all V_x_2 all V_f_2 all T_b (class_Orderings_Olinorder(T_b) -> (-hBOOL(hAPP(hAPP(c_member(T_b),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_c,T_b,V_f_2),hAPP(c_List_Oset(T_c),V_xs_2)))) -> c_List_Olinorder__class_Oinsort__insert__key(T_c,T_b,V_f_2,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_b,V_f_2),V_x_2),V_xs_2)))) # label(fact_insort__insert__insort__key) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4834 (all V_ys all T_b all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys))) -> hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),hAPP(hAPP(c_List_Onth(T_b),V_ys),V_i)) = hAPP(hAPP(c_List_Onth(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i)))) # label(fact_nth__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4835 (all V_d all V_c all V_b all V_a all V_r all T_a (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_a) -> (V_r != c_Groups_Ozero__class_Ozero(T_a) -> (V_c != V_d & V_b = V_a -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_d)))))) # label(fact_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4836 (all T_b hAPP(c_Relation_Oinv__image(tc_Nat_Onat,T_b),c_Wellfounded_Oless__than) = c_Wellfounded_Omeasure(T_b)) # label(fact_measure__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4837 (all V_r_2 all T_b (-(exists B_f all B_i hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),hAPP(B_f,hAPP(c_Nat_OSuc,B_i))),hAPP(B_f,B_i))),V_r_2))) <-> c_Wellfounded_Owf(T_b,V_r_2))) # label(fact_wf__iff__no__infinite__down__chain) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4838 (all V_y_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),V_A_2))) -> hAPP(V_f_2,hAPP(c_Fun_Othe__inv__into(T_b,T_c,V_A_2,V_f_2),V_y_2)) = V_y_2))) # label(fact_f__the__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4839 (all V_B_2 all V_A_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_b),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_B_2)),c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_b,c_Orderings_Oord__class_Omax(T_b),V_A_2)))))) # label(fact_min__max_OSup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4840 (all V_xs_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_List_Omember(T_b),V_xs_2),V_x_2)))) # label(fact_in__set__member) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4841 (all V_n hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n) # label(fact_max__0L) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4842 (all V_f_2 all V_xs_2 all T_c all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> hAPP(c_List_Ofoldl(T_b,T_c,hAPP(c_COMBC(T_c,T_b,T_b),hAPP(hAPP(c_COMBB(T_b,tc_fun(T_b,T_b),T_c),c_Lattices_Osemilattice__sup__class_Osup(T_b)),V_f_2)),c_Orderings_Obot__class_Obot(T_b)),V_xs_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_b),hAPP(c_List_Oset(T_c),V_xs_2)),V_f_2))) # label(fact_SUPR__set__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4843 (all V_R_2 all V_T_2 all V_S_2 all T_c all T_d all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_S_2),V_R_2)),hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),V_T_2),V_R_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_b,T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_d),tc_HOL_Obool)),V_S_2),V_T_2)),V_R_2)) # label(fact_rel__comp__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4844 (all V_r_2 all V_A_2 all T_b (c_Relation_Ototal__on(T_b,V_A_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_b))) <-> c_Relation_Ototal__on(T_b,V_A_2,V_r_2))) # label(fact_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4845 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) # label(fact_of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4846 (all V_b_2 all V_a_2 hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fequal,V_a_2))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_2)))))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fimplies,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Int_Oint)),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fimplies,hAPP(c_fNot,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) = c_Divides_Odivmod__int__rel(V_a_2,V_b_2)) # label(fact_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4847 (all V_z all V_w all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_w),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))))))))) # label(fact_frac__less2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4848 (all V_N_2 all V_h_2 all V_F_2 all V_f_2 all T_b (c_Finite__Set_Ofolding__one__idem(T_b,V_f_2,V_F_2) -> ((all B_x all B_y hAPP(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool)) != V_N_2 -> hAPP(V_F_2,hAPP(c_Set_Oimage(T_b,T_b,V_h_2),V_N_2)) = hAPP(V_h_2,hAPP(V_F_2,V_N_2))))))) # label(fact_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4849 (all V_b_2 all V_a_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_b)))))) # label(fact_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4850 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_mult__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4851 (all V_ys_2 all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)),c_List_Olistrel(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_b),tc_List_Olist(T_b)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y_2),V_ys_2))),c_List_Olistrel(T_b,V_r_2)))))) # label(fact_listrel_OCons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4852 (all V_B_2 all V_D_2 all V_C_2 all V_A_2 all T_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),V_D_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Diff__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4853 (all V_y_2 all V_x_2 all V_R_2 all T_b (c_Equiv__Relations_Opart__equivp(T_b,V_R_2) -> (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_R_2,V_y_2),V_x_2))))) # label(fact_part__equivp__symp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4854 (all V_R_2 all V_y_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,V_R_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_b,V_R_2))) & V_y_2 != V_x_2 | V_y_2 = V_x_2)) # label(fact_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4855 (all V_u_2 all V_l_2 all T_b (class_Orderings_Olinorder(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatMost(T_b),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_b),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_b,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4856 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_b (c_Finite__Set_Ofun__left__comm(T_b,T_c,V_f_2) -> hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(V_f_2,V_y_2)),hAPP(V_f_2,V_x_2)))) # label(fact_fun__left__comm_Ofun__comp__comm) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4857 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_x != V_y))) # label(fact_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4858 (all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4859 (all V_r_2 all V_x_2 all V_y_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_b,hAPP(c_Relation_Oconverse(T_b,T_b),V_r_2)))))) # label(fact_rtrancl__converseI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4860 (all V_C_2 all V_B_2 all V_A_2 all T_c all T_b hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_b,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2))) = hAPP(hAPP(c_Product__Type_OSigma(T_b,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_b),V_C_2))) # label(fact_Times__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4861 (all V_A_2 all V_x_2 all T_b hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2)) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4862 (all V_xs_2 all V_n_2 all V_x_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),hAPP(hAPP(c_List_Otake(T_b),V_n_2),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),hAPP(c_List_Oset(T_b),V_xs_2))))) # label(fact_in__set__takeD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4863 (all V_a_2 all V_B_2 all V_A_2 all T_c all V_b_2 all T_b (hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_b),V_b_2),hAPP(V_B_2,V_a_2)))))) # label(fact_INT__D) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4864 (all V_c all V_b all V_e all V_a all T_a (class_Rings_Osemiring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_e)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_e)),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_e)),V_c))) # label(fact_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4865 (all V_n all V_m ((V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_n) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n))))) # label(fact_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4866 (all V_y_2 all V_x_2 all T_b (class_Groups_Oordered__ab__group__add(T_b) -> hAPP(c_Set_Oimage(T_b,T_b,c_Groups_Ouminus__class_Ouminus(T_b)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_b,V_x_2,V_y_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_b,hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_y_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_b),V_x_2)))) # label(fact_image__uminus__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4867 (all V_n all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),V_n))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4868 (all V_r_2 all T_b (c_Relation_Oantisym(T_b,V_r_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_x),B_y)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),B_y),B_x)),V_r_2)) -> B_y = B_x))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4869 (all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c c_List_Omaps(T_c,T_b,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Oappend(T_b),hAPP(V_f_2,V_x_2)),c_List_Omaps(T_c,T_b,V_f_2,V_xs_2))) # label(fact_maps__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4870 (all V_y_2 all V_x_2 all V_ys_2 all V_P_2 all T_c all V_xs_2 all T_b all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_xs_2))) -> (c_List_Olist__all2(T_b,T_c,V_P_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(V_P_2,V_x_2),V_y_2)) -> c_List_Olist__all2(T_b,T_c,V_P_2,hAPP(hAPP(c_List_Olist__update(T_b,V_xs_2),V_i_2),V_x_2),hAPP(hAPP(c_List_Olist__update(T_c,V_ys_2),V_i_2),V_y_2)))))) # label(fact_list__all2__update__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4871 (all V_z2 all V_z1 all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2))) # label(fact_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4872 (all V_x all V_z all V_y all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_z),V_x)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)),V_x))) # label(fact_sup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4873 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = V_x)) # label(fact_inf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4874 (all V_j all V_i all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (c_List_Olinorder__class_Osorted(T_a,V_xs) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j)))))))) # label(fact_sorted__nth__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4875 (all V_a_2 all V_xs_2 all V_x_2 all V_f_2 all T_b all T_c c_List_Ofoldr(T_c,T_b,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2),V_a_2) = hAPP(hAPP(V_f_2,V_x_2),c_List_Ofoldr(T_c,T_b,V_f_2,V_xs_2,V_a_2))) # label(fact_foldr_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4876 (all V_R_2 all T_b hAPP(c_Relation_ORange(T_b,T_b),c_Transitive__Closure_Ortrancl(T_b,V_R_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_b,tc_HOL_Obool))) # label(fact_Range__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4877 (all V_z all V_w all V_y all V_x all T_a (class_Fields_Olinordered__field(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_w),V_z)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_w))))))))) # label(fact_frac__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4878 (all V_a (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_a) = c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4879 (all V_B_2 all V_A_2 all V_x_2 all T_b (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_b),V_x_2),V_B_2)) -> (hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_b),V_x_2),V_B_2) <-> V_B_2 = V_A_2)))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4880 (all V_f_2 all T_c all T_b all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_b) -> hAPP(V_f_2,c_List_Olast(T_b,V_xs_2)) = c_List_Olast(T_c,hAPP(hAPP(c_List_Omap(T_b,T_c),V_f_2),V_xs_2)))) # label(fact_last__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4881 (all V_A_2 all T_c all V_I_2 all T_b ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_b),B_x),V_I_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(V_A_2,B_x))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_b,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_finite__INT) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4882 (all V_y_2 all V_k_2 all V_xs_2 all V_f_2 all T_b all T_c hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_k_2),V_y_2)) = hAPP(hAPP(c_List_Olist__update(T_b,hAPP(hAPP(c_List_Omap(T_c,T_b),V_f_2),V_xs_2)),V_k_2),hAPP(V_f_2,V_y_2))) # label(fact_map__update) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4883 (all V_B_2 all T_d all V_A_2 all V_r_2 all T_c all T_b (c_Relation_Osingle__valued(T_b,T_c,hAPP(c_Relation_Oconverse(T_c,T_b),V_r_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_b,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_b,tc_HOL_Obool),T_d),c_Relation_OImage(T_c,T_b,V_r_2)),V_B_2)) = hAPP(c_Relation_OImage(T_c,T_b,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Image__INT__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4884 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4885 (all V_A_2 all V_r_2 all V_b_2 all V_a_2 all T_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_b,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_b,T_b),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_b,V_r_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_b,T_b),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_b,T_b),V_A_2),hAPP(c_COMBK(tc_fun(T_b,tc_HOL_Obool),T_b),V_A_2)))) -> hBOOL(hAPP(hAPP(c_member(T_b),V_a_2),V_A_2)) | V_b_2 = V_a_2))) # label(fact_trancl__subset__Sigma__aux) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4886 (all T_c all T_b c_Fun_Oid(tc_prod(T_b,T_c)) = c_Product__Type_Omap__pair(T_b,T_b,T_c,T_c,c_Fun_Oid(T_b),c_Fun_Oid(T_c))) # label(fact_map__pair_Oid) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4887 (all V_n all V_m V_m = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_n)) # label(fact_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4888 (all V_b_2 all V_a_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),V_a_2))))) # label(fact_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4889 (all V_b all V_a all T_a (class_Rings_Olinordered__semiring__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),c_Groups_Ozero__class_Ozero(T_a))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),c_Groups_Ozero__class_Ozero(T_a))))))) # label(fact_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4890 (all V_ys_2 all V_xs_2 all T_b (c_List_Oconcat(T_b,V_ys_2) = c_List_Oconcat(T_b,V_xs_2) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),B_x),hAPP(c_List_Oset(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_List_Ozip(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),B_x)))) -> V_xs_2 = V_ys_2)))) # label(fact_concat__injective) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4891 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2)) -> (V_n_2 = V_m_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2))))) # label(fact_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4892 (all V_a_2 all V_f_2 all T_c all T_b V_a_2 = hAPP(c_List_Ofoldl(T_b,T_c,V_f_2,V_a_2),c_List_Olist_ONil(T_c))) # label(fact_foldl__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4893 (all V_b_2 all V_a_2 all T_b (hAPP(hAPP(c_Set_Oinsert(T_b),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(T_b),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_b,tc_HOL_Obool))) -> V_a_2 = V_b_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4894 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_b (c_Fun_Oinj__on(T_b,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2)) <-> c_Fun_Oinj__on(T_b,T_c,V_f_2,V_B_2) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Set_Oimage(T_b,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_b,tc_HOL_Obool)),V_B_2),V_A_2))) & c_Fun_Oinj__on(T_b,T_c,V_f_2,V_A_2))) # label(fact_inj__on__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4895 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_b,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4896 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_b,c_Groups_Oplus__class_Oplus(T_c)),V_g_2),c_Groups_Ozero__class_Ozero(T_c)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2)))) # label(fact_setsum_OF__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4897 (all V_ys_2 all V_xs_2 all T_b ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),B_x),hAPP(c_List_Oset(tc_prod(tc_List_Olist(T_b),tc_List_Olist(T_b))),hAPP(hAPP(c_List_Ozip(tc_List_Olist(T_b),tc_List_Olist(T_b)),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_b),tc_List_Olist(T_b),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_b),tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_b),tc_Nat_Onat),tc_fun(tc_List_Olist(T_b),tc_HOL_Obool)),tc_List_Olist(T_b)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_b))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_b)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_b))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_b)))),B_x)))) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_b))),V_ys_2) -> (V_xs_2 = V_ys_2 <-> c_List_Oconcat(T_b,V_ys_2) = c_List_Oconcat(T_b,V_xs_2))))) # label(fact_concat__eq__concat__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4898 (all V_x all V_b all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b)),V_x)) -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_x)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_x)))))) # label(fact_le__supE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4899 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4900 (all T_b all V_g_2 all V_f_2 all T_c (class_Orderings_Oord(T_c) -> ((all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b,T_c)),V_f_2),V_g_2))))) # label(fact_le__funI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4901 (all V_l hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),V_l))) # label(fact_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4902 (all V_g_2 all V_A_2 all T_b all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_b),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_b,T_c),V_g_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4903 (all V_n_2 all V_m_2 all V_f_2 all T_b (class_Groups_Ocomm__monoid__add(T_b) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_b),V_f_2),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_m_2,V_n_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_b),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_b),V_f_2),c_List_Oupt(V_m_2,V_n_2))))) # label(fact_setsum__set__upt__conv__listsum__nat) # label(axiom) # label(non_clause). [assumption]. 12.16/12.17 4904 (all V_n all V_m hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n))) # label(fact_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 12.16/12.17 4905 (all V_f2_2 all V_f1_2 all T_b c_Nat_Onat_Onat__case(T_b,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_f1_2) # label(fact_nat__case__0) # label(axiom) # label(non_clause). [assumption]. 12.16/12.17 4906 (all V_xs_2 all T_b (class_Complete__Lattice_Ocomplete__lattice(T_b) -> c_Complete__Lattice_OSup__class_OSup(T_b,hAPP(c_List_Oset(T_b),V_xs_2)) = hAPP(c_List_Ofoldl(T_b,T_b,c_Lattices_Osemilattice__sup__class_Osup(T_b),c_Orderings_Obot__class_Obot(T_b)),V_xs_2))) # label(fact_Sup__set__fold) # label(axiom) # label(non_clause). [assumption]. 12.16/12.17 12.16/12.17 ============================== end of process non-clausal formulas === 12.16/12.17 12.16/12.17 ============================== PROCESS INITIAL CLAUSES =============== 12.16/12.17 12.16/12.17 ============================== PREDICATE ELIMINATION ================= 12.16/12.17 4907 class_Rings_Osemiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__1) # label(axiom). [assumption]. 12.16/12.17 4908 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_iszero__def) # label(axiom). [clausify(5)]. 12.16/12.17 4909 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_iszero__def) # label(axiom). [clausify(5)]. 12.16/12.17 4910 -class_Rings_Osemiring__1(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_COMBK(A,B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(B),D))),C) # label(fact_setsum__constant) # label(axiom). [clausify(395)]. 12.16/12.17 4911 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__0) # label(axiom). [clausify(470)]. 12.16/12.17 Derived: -c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(4907,a,4908,a)]. 12.16/12.17 Derived: c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(4907,a,4909,a)]. 12.16/12.17 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(4907,a,4910,a)]. 12.16/12.17 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(4907,a,4911,a)]. 12.16/12.17 4912 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__nat__add) # label(axiom). [clausify(798)]. 12.16/12.17 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B)). [resolve(4912,a,4907,a)]. 12.16/12.17 4913 -class_Rings_Osemiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) # label(fact_of__nat__1) # label(axiom). [clausify(954)]. 12.16/12.17 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(4913,a,4907,a)]. 12.16/12.17 4914 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Nat_Osemiring__1__class_ONats(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Nat_Osemiring__1__class_ONats(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__add) # label(axiom). [clausify(1187)]. 12.46/12.45 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(4914,a,4907,a)]. 12.46/12.45 4915 -class_Rings_Osemiring__1(A) | -class_Groups_Ocomm__monoid__mult(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,B),c_Nat_Osemiring__1__class_Oof__nat(A)),C)),D) # label(fact_of__nat__setprod) # label(axiom). [clausify(1354)]. 12.46/12.45 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat)),B)),C). [resolve(4915,a,4907,a)]. 12.46/12.45 4916 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__nat__mult) # label(axiom). [clausify(1787)]. 12.46/12.45 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B)). [resolve(4916,a,4907,a)]. 12.46/12.45 4917 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,c_Groups_Ozero__class_Ozero(A)) # label(fact_iszero__0) # label(axiom). [clausify(1994)]. 12.46/12.45 Derived: c_Int_Oiszero(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(4917,a,4907,a)]. 12.46/12.45 4918 class_Rings_Osemiring__1(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring__1) # label(axiom). [assumption]. 12.46/12.45 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(4918,a,4908,a)]. 12.46/12.45 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A. [resolve(4918,a,4909,a)]. 12.46/12.45 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(4918,a,4910,a)]. 12.46/12.45 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(4918,a,4911,a)]. 12.46/12.45 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B)). [resolve(4918,a,4912,a)]. 12.46/12.45 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(4918,a,4913,a)]. 12.46/12.45 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(4918,a,4914,a)]. 12.46/12.49 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral)),B)),C). [resolve(4918,a,4915,a)]. 12.46/12.49 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B)). [resolve(4918,a,4916,a)]. 12.46/12.49 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(4918,a,4917,a)]. 12.46/12.49 4919 -class_Rings_Osemiring__1(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(c_COMBK(A,B),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),D))),C) # label(fact_listsum__triv) # label(axiom). [clausify(2064)]. 12.46/12.49 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(4919,a,4907,a)]. 12.46/12.49 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(4919,a,4918,a)]. 12.46/12.49 4920 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B))) # label(fact_of__nat__double) # label(axiom). [clausify(2084)]. 12.46/12.49 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,A))). [resolve(4920,a,4907,a)]. 12.46/12.49 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,A))). [resolve(4920,a,4918,a)]. 13.20/13.25 4921 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),C) # label(fact_of__nat__power) # label(axiom). [clausify(2139)]. 13.20/13.25 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),B). [resolve(4921,a,4907,a)]. 13.20/13.25 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),B). [resolve(4921,a,4918,a)]. 13.20/13.25 4922 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,hAPP(hAPP(c_COMBC(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_Groups_Oone__class_Oone(A)),B,c_Groups_Ozero__class_Ozero(A)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) # label(fact_semiring__1__class_Oof__nat__code) # label(axiom). [clausify(2284)]. 13.20/13.25 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A). [resolve(4922,a,4907,a)]. 13.20/13.25 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A). [resolve(4922,a,4918,a)]. 13.20/13.25 4923 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,c_Groups_Oone__class_Oone(A)) # label(fact_not__iszero__1) # label(axiom). [clausify(2718)]. 13.20/13.25 Derived: -c_Int_Oiszero(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(4923,a,4907,a)]. 13.20/13.25 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)). [resolve(4923,a,4918,a)]. 13.20/13.25 4924 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_of__nat__in__Nats) # label(axiom). [clausify(3229)]. 13.20/13.25 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(4924,a,4907,a)]. 13.20/13.25 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(4924,a,4918,a)]. 13.20/13.25 4925 -class_Rings_Osemiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_one__power2) # label(axiom). [clausify(3246)]. 13.20/13.25 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4925,a,4907,a)]. 13.20/13.25 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4925,a,4918,a)]. 13.20/13.25 4926 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,B),c_Nat_Osemiring__1__class_Oof__nat(A)),C)),D) # label(fact_of__nat__setsum) # label(axiom). [clausify(3367)]. 13.52/13.53 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat)),B)),C). [resolve(4926,a,4907,a)]. 13.52/13.53 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral)),B)),C). [resolve(4926,a,4918,a)]. 13.52/13.53 4927 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Oone__class_Oone(A)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__1) # label(axiom). [clausify(3390)]. 13.52/13.53 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(4927,a,4907,a)]. 13.52/13.53 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(4927,a,4918,a)]. 13.52/13.53 4928 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Nat_Osemiring__1__class_ONats(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Nat_Osemiring__1__class_ONats(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__mult) # label(axiom). [clausify(3395)]. 13.52/13.53 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(4928,a,4907,a)]. 13.52/13.53 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(4928,a,4918,a)]. 13.52/13.53 4929 class_Rings_Osemiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__1) # label(axiom). [assumption]. 13.52/13.53 Derived: -c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(4929,a,4908,a)]. 13.52/13.53 Derived: c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(4929,a,4909,a)]. 13.52/13.53 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(4929,a,4910,a)]. 13.52/13.53 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(4929,a,4911,a)]. 13.52/13.53 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(4929,a,4912,a)]. 13.52/13.53 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(4929,a,4913,a)]. 13.94/13.94 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(4929,a,4914,a)]. 13.94/13.94 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)),C). [resolve(4929,a,4915,a)]. 13.94/13.94 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(4929,a,4916,a)]. 13.94/13.94 Derived: c_Int_Oiszero(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(4929,a,4917,a)]. 13.94/13.94 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(4929,a,4919,a)]. 13.94/13.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A))). [resolve(4929,a,4920,a)]. 13.94/13.94 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)),A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A). [resolve(4929,a,4922,a)]. 13.94/13.94 Derived: -c_Int_Oiszero(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(4929,a,4923,a)]. 13.94/13.94 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(4929,a,4924,a)]. 13.94/13.94 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4929,a,4925,a)]. 13.94/13.94 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)),C). [resolve(4929,a,4926,a)]. 13.94/13.94 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(4929,a,4927,a)]. 13.94/13.94 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(4929,a,4928,a)]. 13.94/13.94 4930 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,hAPP(c_Nat_OSuc,C),D) = c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,C,hAPP(B,D)) # label(fact_of__nat__aux_Osimps_I2_J) # label(axiom). [clausify(4075)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,B,hAPP(A,C)). [resolve(4930,a,4907,a)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,B,hAPP(A,C)). [resolve(4930,a,4918,a)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,B,hAPP(A,C)). [resolve(4930,a,4929,a)]. 14.34/14.33 4931 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) = C # label(fact_of__nat__aux_Osimps_I1_J) # label(axiom). [clausify(4222)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(4931,a,4907,a)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(4931,a,4918,a)]. 14.34/14.33 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(4931,a,4929,a)]. 14.34/14.33 4932 -class_Rings_Osemiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_zero__power2) # label(axiom). [clausify(4290)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4932,a,4907,a)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4932,a,4918,a)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4932,a,4929,a)]. 14.34/14.33 4933 -class_Rings_Osemiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_of__nat__0) # label(axiom). [clausify(4547)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(4933,a,4907,a)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(4933,a,4918,a)]. 14.34/14.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(4933,a,4929,a)]. 14.34/14.33 4934 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) # label(fact_of__nat__Suc) # label(axiom). [clausify(4663)]. 14.34/14.33 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)). [resolve(4934,a,4907,a)]. 14.34/14.33 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)). [resolve(4934,a,4918,a)]. 14.34/14.38 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)). [resolve(4934,a,4929,a)]. 14.34/14.38 4935 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(B),f576(C,D,E,B,A)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(B),E))),C))) # label(fact_setsum__bounded) # label(axiom). [clausify(4744)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f576(B,C,D,A,tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(4935,b,4907,a)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f576(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(4935,b,4918,a)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f576(B,C,D,A,tc_Int_Oint)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(4935,b,4929,a)]. 14.34/14.38 4936 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f576(C,B,D,E,A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(E),D))),C))) # label(fact_setsum__bounded) # label(axiom). [clausify(4744)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f576(B,A,C,D,tc_Nat_Onat))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(4936,b,4907,a)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,f576(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(4936,b,4918,a)]. 14.34/14.38 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f576(B,A,C,D,tc_Int_Oint))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(4936,b,4929,a)]. 21.41/21.46 4937 -class_Nat_Osize(A) | c_FunDef_Ois__measure(A,c_Nat_Osize__class_Osize(A)) # label(fact_measure__size) # label(axiom). [clausify(2220)]. 21.41/21.46 4938 class_Nat_Osize(tc_Lazy__Sequence_Olazy__sequence(A)) # label(arity_Lazy____Sequence__Olazy____sequence__Nat_Osize) # label(axiom). [clausify(7)]. 21.41/21.46 4939 class_Nat_Osize(tc_Option_Ooption(A)) # label(arity_Option__Ooption__Nat_Osize) # label(axiom). [clausify(679)]. 21.41/21.46 4940 class_Nat_Osize(tc_prod(A,B)) # label(arity_prod__Nat_Osize) # label(axiom). [clausify(1814)]. 21.41/21.46 4941 class_Nat_Osize(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osize) # label(axiom). [assumption]. 21.41/21.46 4942 class_Nat_Osize(tc_Nitpick_Opair__box(A,B)) # label(arity_Nitpick__Opair____box__Nat_Osize) # label(axiom). [clausify(2499)]. 21.41/21.46 4943 class_Nat_Osize(tc_List_Olist(A)) # label(arity_List__Olist__Nat_Osize) # label(axiom). [clausify(3040)]. 21.41/21.46 4944 class_Nat_Osize(tc_Hoare__Mirabelle_Otriple(A)) # label(arity_Hoare____Mirabelle__Otriple__Nat_Osize) # label(axiom). [clausify(3331)]. 21.41/21.46 4945 class_Nat_Osize(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osize) # label(axiom). [assumption]. 21.41/21.46 4946 class_Nat_Osize(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Nat_Osize) # label(axiom). [assumption]. 21.41/21.46 4947 class_Nat_Osize(tc_sum(A,B)) # label(arity_sum__Nat_Osize) # label(axiom). [clausify(3556)]. 21.41/21.46 4948 class_Nat_Osize(tc_Code__Evaluation_Oterm) # label(arity_Code____Evaluation__Oterm__Nat_Osize) # label(axiom). [assumption]. 21.41/21.46 4949 class_Nat_Osize(tc_HOL_Obool) # label(arity_HOL__Obool__Nat_Osize) # label(axiom). [assumption]. 21.41/21.46 Derived: c_FunDef_Ois__measure(tc_HOL_Obool,c_Nat_Osize__class_Osize(tc_HOL_Obool)). [resolve(4949,a,4937,a)]. 21.41/21.46 4950 class_Rings_Oring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1) # label(axiom). [assumption]. 21.41/21.46 4951 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),C) # label(fact_of__int__power) # label(axiom). [clausify(16)]. 21.41/21.46 4952 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_of__int__of__nat) # label(axiom). [clausify(474)]. 21.41/21.46 4953 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B)) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) # label(fact_of__int__of__nat) # label(axiom). [clausify(474)]. 21.41/21.46 4954 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) # label(fact_of__int__num) # label(axiom). [clausify(956)]. 21.41/21.46 4955 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) # label(fact_of__int__num) # label(axiom). [clausify(956)]. 21.41/21.46 4956 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,A),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(A,A)),tc_fun(tc_Int_Oint,A)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,A),A),tc_fun(tc_fun(tc_Int_Oint,tc_fun(A,A)),tc_fun(tc_Int_Oint,A)),tc_Int_Oint),c_COMBB(tc_fun(A,A),A,tc_Int_Oint)),hAPP(hAPP(c_COMBB(A,tc_fun(tc_fun(A,A),A),tc_Int_Oint),c_HOL_OLet(A,A)),c_Int_Oring__1__class_Oof__int(A)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,A),tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,A)),tc_fun(tc_fun(A,A),tc_fun(A,A)),tc_Int_Oint),c_COMBS(A,A,A)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,A),tc_fun(A,tc_fun(A,A))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,A)),tc_fun(tc_fun(A,A),tc_fun(A,tc_fun(A,A))),tc_Int_Oint),c_COMBB(A,tc_fun(A,A),A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),tc_Int_Oint),c_If(A)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_COMBI(A))))),hAPP(hAPP(c_COMBC(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Oplus__class_Oplus(A)),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_COMBI(A)))),c_Groups_Oone__class_Oone(A))))),c_Divides_Odivmod__int(B,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_of__int__num) # label(axiom). [clausify(956)]. 21.41/21.46 4957 -class_Rings_Oring__1(A) | -class_Rings_Oinverse(A) | c_Nitpick_Oof__frac(B,A,C) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Onum(B),C))),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Odenom(B),C))) # label(fact_Nitpick_Oof__frac__def) # label(axiom). [clausify(1001)]. 21.41/21.46 4958 -class_Rings_Oring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) # label(fact_power__minus1__even) # label(axiom). [clausify(1210)]. 21.41/21.46 4959 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B)) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) # label(fact_of__nat__nat) # label(axiom). [clausify(1546)]. 21.41/21.46 4960 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__add) # label(axiom). [clausify(1749)]. 21.41/21.46 4961 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__mult) # label(axiom). [clausify(1887)]. 21.41/21.46 4962 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__of__int) # label(axiom). [clausify(1892)]. 21.41/21.46 4963 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__minus) # label(axiom). [clausify(2131)]. 21.41/21.46 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),B). [resolve(4950,a,4951,a)]. 21.41/21.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(4950,a,4952,a)]. 21.41/21.46 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A)) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A). [resolve(4950,a,4953,a)]. 21.41/21.47 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A). [resolve(4950,a,4954,a)]. 21.41/21.47 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(4950,a,4955,a)]. 21.41/21.47 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),tc_Int_Oint),c_HOL_OLet(tc_Int_Oint,tc_Int_Oint)),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_COMBI(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_COMBI(tc_Int_Oint)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),c_Divides_Odivmod__int(A,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(4950,a,4956,a)]. 21.41/21.47 Derived: -class_Rings_Oinverse(tc_Int_Oint) | c_Nitpick_Oof__frac(A,tc_Int_Oint,B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),B))),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nitpick_Odenom(A),B))). [resolve(4950,a,4957,a)]. 21.41/21.47 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A)). [resolve(4950,a,4958,a)]. 21.41/21.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A)) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A). [resolve(4950,a,4959,a)]. 21.41/21.47 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(4950,a,4960,a)]. 21.68/21.68 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4950,a,4961,a)]. 21.68/21.68 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4950,a,4962,a)]. 21.68/21.68 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4950,a,4963,a)]. 21.68/21.68 4964 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__mult) # label(axiom). [clausify(2222)]. 21.68/21.68 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(4964,a,4950,a)]. 21.68/21.68 4965 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_of__int__0) # label(axiom). [clausify(2323)]. 21.68/21.68 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(4965,a,4950,a)]. 21.68/21.68 4966 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__add) # label(axiom). [clausify(2387)]. 21.68/21.68 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4966,a,4950,a)]. 21.68/21.68 4967 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)) # label(fact_of__int__minus) # label(axiom). [clausify(2494)]. 21.68/21.68 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)). [resolve(4967,a,4950,a)]. 21.68/21.68 4968 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,B),c_Int_Oring__1__class_Oof__int(A)),C)),D) # label(fact_of__int__setsum) # label(axiom). [clausify(2533)]. 21.68/21.68 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)),B)),C). [resolve(4968,a,4950,a)]. 21.68/21.68 4969 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__power) # label(axiom). [clausify(2627)]. 22.19/22.22 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4969,a,4950,a)]. 22.19/22.22 4970 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Ozero__class_Ozero(A)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__0) # label(axiom). [clausify(2721)]. 22.19/22.22 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4970,a,4950,a)]. 22.19/22.22 4971 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__of__nat) # label(axiom). [clausify(2733)]. 22.19/22.22 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4971,a,4950,a)]. 22.19/22.22 4972 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__diff) # label(axiom). [clausify(2913)]. 22.19/22.22 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(4972,a,4950,a)]. 22.19/22.22 4973 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) # label(fact_power__minus__even) # label(axiom). [clausify(3190)]. 22.19/22.22 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)). [resolve(4973,a,4950,a)]. 22.19/22.22 4974 -class_Rings_Oring__1(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) # label(fact_power__minus1__odd) # label(axiom). [clausify(3290)]. 22.19/22.22 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))). [resolve(4974,a,4950,a)]. 22.19/22.22 4975 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__minus) # label(axiom). [clausify(3620)]. 22.19/22.22 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(4975,a,4950,a)]. 22.71/22.76 4976 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__diff) # label(axiom). [clausify(3680)]. 22.71/22.76 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4976,a,4950,a)]. 22.71/22.76 4977 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B),C)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__int) # label(axiom). [clausify(4152)]. 22.71/22.76 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),A),B)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(4977,a,4950,a)]. 22.71/22.76 4978 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) # label(fact_of__nat__diff) # label(axiom). [clausify(4197)]. 22.71/22.76 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)). [resolve(4978,a,4950,a)]. 22.71/22.76 4979 -class_Groups_Ocomm__monoid__mult(A) | -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,B),c_Int_Oring__1__class_Oof__int(A)),C)),D) # label(fact_of__int__setprod) # label(axiom). [clausify(4268)]. 22.71/22.76 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)),B)),C). [resolve(4979,b,4950,a)]. 22.71/22.76 4980 -class_Rings_Oring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),c_Groups_Oone__class_Oone(tc_Int_Oint)) # label(fact_of__int__1) # label(axiom). [clausify(4283)]. 22.71/22.76 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(4980,a,4950,a)]. 23.79/23.87 4981 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__minus) # label(axiom). [clausify(4332)]. 23.79/23.87 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4981,a,4950,a)]. 23.79/23.87 4982 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Oone__class_Oone(A)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__1) # label(axiom). [clausify(4780)]. 23.79/23.87 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(4982,a,4950,a)]. 23.79/23.87 4983 -class_Rings_Oring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) = hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)) # label(fact_of__int__of__nat__eq) # label(axiom). [clausify(4845)]. 23.79/23.87 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)). [resolve(4983,a,4950,a)]. 23.79/23.87 4984 class_Rings_Olinordered__idom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__idom) # label(axiom). [assumption]. 23.79/23.87 4985 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_of__int__le__iff) # label(axiom). [clausify(17)]. 23.79/23.87 4986 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_of__int__le__iff) # label(axiom). [clausify(17)]. 23.79/23.87 4987 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),B)) # label(fact_mult__right__le__one__le) # label(axiom). [clausify(170)]. 23.79/23.87 4988 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I2_J) # label(axiom). [clausify(234)]. 23.79/23.87 4989 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I2_J) # label(axiom). [clausify(234)]. 23.79/23.87 4990 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_le__number__of) # label(axiom). [clausify(295)]. 23.79/23.87 4991 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_le__number__of) # label(axiom). [clausify(295)]. 23.89/23.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(4984,a,4985,a)]. 23.89/23.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(4984,a,4986,a)]. 23.89/23.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),A)). [resolve(4984,a,4987,a)]. 23.89/23.93 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(4984,a,4988,a)]. 23.89/23.93 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(4984,a,4989,a)]. 23.89/23.93 4992 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Osgn__class_Osgn(A,B)) # label(fact_abs__sgn) # label(axiom). [clausify(605)]. 23.89/23.93 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)). [resolve(4992,a,4984,a)]. 23.89/23.93 4993 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sgn__0__0) # label(axiom). [clausify(635)]. 23.89/23.93 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(4993,a,4984,a)]. 23.89/23.93 4994 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_sgn__0__0) # label(axiom). [clausify(635)]. 23.89/23.93 4995 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__less__iff) # label(axiom). [clausify(640)]. 23.89/23.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(4995,a,4984,a)]. 23.89/23.93 4996 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__less__iff) # label(axiom). [clausify(640)]. 23.89/23.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(4996,a,4984,a)]. 23.89/23.97 4997 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OPls)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__special_I3_J) # label(axiom). [clausify(648)]. 23.89/23.97 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(4997,b,4984,a)]. 23.89/23.97 4998 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OPls)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__special_I3_J) # label(axiom). [clausify(648)]. 23.89/23.97 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(4998,b,4984,a)]. 23.89/23.97 4999 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_abs__power2) # label(axiom). [clausify(735)]. 23.89/23.97 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(4999,a,4984,a)]. 23.89/23.97 5000 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)))),c_Groups_Ozero__class_Ozero(A))) # label(fact_odd__power__less__zero) # label(axiom). [clausify(812)]. 23.89/23.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5000,a,4984,a)]. 23.89/23.97 5001 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(816)]. 23.89/23.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5001,a,4984,a)]. 23.89/23.97 5002 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(816)]. 23.98/24.00 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5002,a,4984,a)]. 23.98/24.00 5003 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(816)]. 23.98/24.00 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5003,a,4984,a)]. 23.98/24.00 5004 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Osgn__class_Osgn(A,B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__less) # label(axiom). [clausify(826)]. 23.98/24.00 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5004,a,4984,a)]. 23.98/24.00 5005 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Osgn__class_Osgn(A,B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__less) # label(axiom). [clausify(826)]. 23.98/24.00 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5005,a,4984,a)]. 23.98/24.00 5006 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),B)),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_Ints__odd__less__0) # label(axiom). [clausify(858)]. 23.98/24.00 5007 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),B)),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_Ints__odd__less__0) # label(axiom). [clausify(858)]. 23.98/24.00 5008 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) # label(fact_less__special_I4_J) # label(axiom). [clausify(861)]. 23.98/24.00 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5008,b,4984,a)]. 23.98/24.00 5009 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) # label(fact_less__special_I4_J) # label(axiom). [clausify(861)]. 23.98/24.00 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5009,b,4984,a)]. 23.98/24.08 5010 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__special_I2_J) # label(axiom). [clausify(888)]. 23.98/24.08 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5010,b,4984,a)]. 23.98/24.08 5011 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__special_I2_J) # label(axiom). [clausify(888)]. 23.98/24.08 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5011,b,4984,a)]. 23.98/24.08 5012 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_abs__number__of) # label(axiom). [clausify(1058)]. 23.98/24.08 5013 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_abs__number__of) # label(axiom). [clausify(1058)]. 23.98/24.08 5014 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),B)) # label(fact_less__special_I1_J) # label(axiom). [clausify(1152)]. 23.98/24.08 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)). [resolve(5014,b,4984,a)]. 23.98/24.08 5015 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),B)) # label(fact_less__special_I1_J) # label(axiom). [clausify(1152)]. 23.98/24.08 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)). [resolve(5015,b,4984,a)]. 23.98/24.08 5016 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_zero__le__power2) # label(axiom). [clausify(1163)]. 23.98/24.08 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))). [resolve(5016,a,4984,a)]. 24.09/24.20 5017 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__le__0__iff) # label(axiom). [clausify(1467)]. 24.09/24.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5017,a,4984,a)]. 24.09/24.20 5018 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__le__0__iff) # label(axiom). [clausify(1467)]. 24.09/24.20 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5018,a,4984,a)]. 24.09/24.20 5019 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_abs__power__minus) # label(axiom). [clausify(1519)]. 24.09/24.20 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(5019,a,4984,a)]. 24.09/24.20 5020 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__self__iff) # label(axiom). [clausify(1552)]. 24.09/24.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5020,a,4984,a)]. 24.09/24.20 5021 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__self__iff) # label(axiom). [clausify(1552)]. 24.09/24.20 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5021,a,4984,a)]. 24.09/24.20 5022 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) != B # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1560)]. 24.09/24.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5022,a,4984,a)]. 24.19/24.25 5023 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1560)]. 24.19/24.25 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5023,a,4984,a)]. 24.19/24.25 5024 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1560)]. 24.19/24.25 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5024,a,4984,a)]. 24.19/24.25 5025 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Osgn__class_Osgn(A,B))) # label(fact_sgn__greater) # label(axiom). [clausify(1629)]. 24.19/24.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A))). [resolve(5025,a,4984,a)]. 24.19/24.25 5026 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Osgn__class_Osgn(A,B))) # label(fact_sgn__greater) # label(axiom). [clausify(1629)]. 24.19/24.25 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A))). [resolve(5026,a,4984,a)]. 24.19/24.25 5027 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1708)]. 24.28/24.36 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5027,a,4984,a)]. 24.28/24.36 5028 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1708)]. 24.28/24.36 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5028,a,4984,a)]. 24.28/24.36 5029 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1708)]. 24.28/24.36 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5029,a,4984,a)]. 24.28/24.36 5030 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_abs__minus__one) # label(axiom). [clausify(1805)]. 24.28/24.36 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5030,b,4984,a)]. 24.28/24.36 5031 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_power2__less__0) # label(axiom). [clausify(1907)]. 24.28/24.36 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5031,a,4984,a)]. 24.28/24.36 5032 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) # label(fact_of__int__less__iff) # label(axiom). [clausify(1967)]. 24.53/24.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5032,a,4984,a)]. 24.53/24.59 5033 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) # label(fact_of__int__less__iff) # label(axiom). [clausify(1967)]. 24.53/24.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5033,a,4984,a)]. 24.53/24.59 5034 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) != hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__1__neg) # label(axiom). [clausify(1995)]. 24.53/24.59 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5034,a,4984,a)]. 24.53/24.59 5035 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__1__neg) # label(axiom). [clausify(1995)]. 24.53/24.59 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5035,a,4984,a)]. 24.53/24.59 5036 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Osgn__class_Osgn(A,B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) # label(fact_sgn__neg) # label(axiom). [clausify(2000)]. 24.53/24.59 5037 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)))) # label(fact_zero__le__even__power_H) # label(axiom). [clausify(2057)]. 24.53/24.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)))). [resolve(5037,a,4984,a)]. 24.53/24.59 5038 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Oone__class_Oone(A) # label(fact_sgn__pos) # label(axiom). [clausify(2101)]. 24.53/24.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = c_Groups_Oone__class_Oone(tc_Int_Oint). [resolve(5038,a,4984,a)]. 24.53/24.59 5039 -class_Rings_Olinordered__idom(A) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__neqE__linordered__idom) # label(axiom). [clausify(2112)]. 24.53/24.59 5040 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_abs__mult__self) # label(axiom). [clausify(2376)]. 24.76/24.86 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A). [resolve(5040,a,4984,a)]. 24.76/24.86 5041 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_odd__0__le__power__imp__0__le) # label(axiom). [clausify(2379)]. 24.76/24.86 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5041,a,4984,a)]. 24.76/24.86 5042 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)) # label(fact_abs__mult) # label(axiom). [clausify(2536)]. 24.76/24.86 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)). [resolve(5042,a,4984,a)]. 24.76/24.86 5043 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Osgn__class_Osgn(A,B)),c_Groups_Osgn__class_Osgn(A,C)) # label(fact_sgn__times) # label(axiom). [clausify(2724)]. 24.76/24.86 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,B)). [resolve(5043,a,4984,a)]. 24.76/24.86 5044 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C) # label(fact_power__abs) # label(axiom). [clausify(2756)]. 24.76/24.86 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B). [resolve(5044,a,4984,a)]. 24.76/24.86 5045 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__abs) # label(axiom). [clausify(2765)]. 24.76/24.86 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5045,a,4984,a)]. 24.76/24.86 5046 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OPls)) # label(fact_le__special_I3_J) # label(axiom). [clausify(2892)]. 25.08/25.15 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OPls)). [resolve(5046,a,4984,a)]. 25.08/25.15 5047 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OPls)) # label(fact_le__special_I3_J) # label(axiom). [clausify(2892)]. 25.08/25.15 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OPls)). [resolve(5047,a,4984,a)]. 25.08/25.15 5048 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),B)) # label(fact_abs__power__minus__one) # label(axiom). [clausify(3015)]. 25.08/25.15 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A)). [resolve(5048,a,4984,a)]. 25.08/25.15 5049 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) # label(fact_abs__of__nat) # label(axiom). [clausify(3046)]. 25.08/25.15 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A). [resolve(5049,a,4984,a)]. 25.08/25.15 5050 -class_Rings_Olinordered__idom(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Groups_Oabs__class_Oabs(A),c_Groups_Oone__class_Oone(A)) # label(fact_abs__one) # label(axiom). [clausify(3088)]. 25.08/25.15 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5050,a,4984,a)]. 25.08/25.15 5051 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,c_Groups_Osgn__class_Osgn(A,B)) = c_Groups_Osgn__class_Osgn(A,B) # label(fact_sgn__sgn) # label(axiom). [clausify(3137)]. 25.08/25.15 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)) = c_Groups_Osgn__class_Osgn(tc_Int_Oint,A). [resolve(5051,a,4984,a)]. 25.08/25.15 5052 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_abs__mult__less) # label(axiom). [clausify(3157)]. 25.08/25.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5052,a,4984,a)]. 25.08/25.15 5053 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),C)),B) # label(fact_abs__mult__pos) # label(axiom). [clausify(3350)]. 25.27/25.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)),A). [resolve(5053,a,4984,a)]. 25.27/25.35 5054 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),B)) # label(fact_mult__left__le__one__le) # label(axiom). [clausify(3411)]. 25.27/25.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),A)). [resolve(5054,a,4984,a)]. 25.27/25.35 5055 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_less__number__of) # label(axiom). [clausify(3488)]. 25.27/25.35 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5055,a,4984,a)]. 25.27/25.35 5056 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_less__number__of) # label(axiom). [clausify(3488)]. 25.27/25.35 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5056,a,4984,a)]. 25.27/25.35 5057 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_zero__less__power2) # label(axiom). [clausify(3706)]. 25.27/25.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5057,a,4984,a)]. 25.27/25.35 5058 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_zero__less__power2) # label(axiom). [clausify(3706)]. 25.27/25.35 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5058,a,4984,a)]. 25.51/25.61 5059 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Osgn__class_Osgn(A,B)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) = B # label(fact_mult__sgn__abs) # label(axiom). [clausify(3999)]. 25.51/25.61 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)) = A. [resolve(5059,a,4984,a)]. 25.51/25.61 5060 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) # label(fact_of__int__0__le__iff) # label(axiom). [clausify(4011)]. 25.51/25.61 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5060,a,4984,a)]. 25.51/25.61 5061 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) # label(fact_of__int__0__le__iff) # label(axiom). [clausify(4011)]. 25.51/25.61 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5061,a,4984,a)]. 25.51/25.61 5062 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(4104)]. 25.51/25.61 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5062,a,4984,a)]. 25.51/25.61 5063 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(4104)]. 25.51/25.61 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5063,a,4984,a)]. 25.51/25.61 5064 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(4104)]. 25.59/25.67 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5064,a,4984,a)]. 25.59/25.67 5065 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__less__0__iff) # label(axiom). [clausify(4129)]. 25.59/25.67 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5065,a,4984,a)]. 25.59/25.67 5066 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__less__0__iff) # label(axiom). [clausify(4129)]. 25.59/25.67 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5066,a,4984,a)]. 25.59/25.67 5067 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) # label(fact_le__special_I4_J) # label(axiom). [clausify(4148)]. 25.59/25.67 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))). [resolve(5067,a,4984,a)]. 25.59/25.67 5068 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) # label(fact_le__special_I4_J) # label(axiom). [clausify(4148)]. 25.59/25.67 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))). [resolve(5068,a,4984,a)]. 25.59/25.67 5069 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C))) # label(fact_zero__le__power__abs) # label(axiom). [clausify(4175)]. 25.59/25.67 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B))). [resolve(5069,a,4984,a)]. 25.59/25.67 5070 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) # label(fact_sum__power2__ge__zero) # label(axiom). [clausify(4195)]. 25.88/25.97 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))). [resolve(5070,a,4984,a)]. 25.88/25.97 5071 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_even__less__0__iff) # label(axiom). [clausify(4288)]. 25.88/25.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5071,a,4984,a)]. 25.88/25.97 5072 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_even__less__0__iff) # label(axiom). [clausify(4288)]. 25.88/25.97 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5072,a,4984,a)]. 25.88/25.97 5073 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__sum__power2__lt__zero) # label(axiom). [clausify(4441)]. 25.88/25.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5073,a,4984,a)]. 25.88/25.97 5074 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) # label(fact_le__special_I1_J) # label(axiom). [clausify(4626)]. 25.88/25.97 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5074,a,4984,a)]. 25.88/25.97 5075 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) # label(fact_le__special_I1_J) # label(axiom). [clausify(4626)]. 29.88/30.06 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5075,a,4984,a)]. 29.88/30.06 5076 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Oone__class_Oone(A) # label(fact_sgn__1__pos) # label(axiom). [clausify(4811)]. 29.88/30.06 5077 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Osgn__class_Osgn(A,B) != c_Groups_Oone__class_Oone(A) # label(fact_sgn__1__pos) # label(axiom). [clausify(4811)]. 29.88/30.06 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != c_Groups_Oone__class_Oone(tc_Int_Oint). [resolve(5077,a,4984,a)]. 29.88/30.06 5078 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 29.88/30.06 5079 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__less__imp__less__left) # label(axiom). [clausify(22)]. 29.88/30.06 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(5078,a,5079,a)]. 29.88/30.06 5080 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 29.88/30.06 5081 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__strict__mono_H) # label(axiom). [clausify(845)]. 29.88/30.06 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5081,a,5078,a)]. 29.88/30.06 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(5081,a,5080,a)]. 29.88/30.06 5082 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__less__mult__pos2) # label(axiom). [clausify(930)]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5082,a,5078,a)]. 30.18/30.31 5083 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_mult__strict__right__mono) # label(axiom). [clausify(1144)]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))). [resolve(5083,a,5078,a)]. 30.18/30.31 5084 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__imp__less__right) # label(axiom). [clausify(1762)]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5084,a,5078,a)]. 30.18/30.31 5085 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__pos__pos) # label(axiom). [clausify(1905)]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5085,a,5078,a)]. 30.18/30.31 5086 class_Rings_Olinordered__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5086,a,5079,a)]. 30.18/30.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5086,a,5081,a)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5086,a,5082,a)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5086,a,5083,a)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5086,a,5084,a)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5086,a,5085,a)]. 30.28/30.37 5087 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__neg__pos) # label(axiom). [clausify(2152)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5087,a,5078,a)]. 30.28/30.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5087,a,5086,a)]. 30.61/30.71 5088 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__less__le__imp__less) # label(axiom). [clausify(2450)]. 30.61/30.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5088,a,5078,a)]. 30.61/30.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(5088,a,5080,a)]. 30.61/30.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5088,a,5086,a)]. 30.61/30.71 5089 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_mult__left__le__imp__le) # label(axiom). [clausify(2644)]. 30.61/30.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5089,a,5078,a)]. 30.61/30.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5089,a,5086,a)]. 30.61/30.71 5090 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__le__less__imp__less) # label(axiom). [clausify(2824)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5090,a,5078,a)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(5090,a,5080,a)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5090,a,5086,a)]. 31.17/31.29 5091 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_zero__less__mult__pos) # label(axiom). [clausify(3497)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5091,a,5078,a)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)). [resolve(5091,a,5080,a)]. 31.17/31.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5091,a,5086,a)]. 31.17/31.29 5092 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__strict__mono) # label(axiom). [clausify(3779)]. 31.87/31.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5092,a,5078,a)]. 31.87/31.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(5092,a,5080,a)]. 31.87/31.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5092,a,5086,a)]. 31.87/31.99 5093 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__strict__left__mono) # label(axiom). [clausify(4112)]. 31.87/31.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(5093,a,5078,a)]. 31.87/31.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5093,a,5086,a)]. 31.87/31.99 5094 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_mult__right__le__imp__le) # label(axiom). [clausify(4686)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5094,a,5078,a)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5094,a,5080,a)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5094,a,5086,a)]. 32.06/32.17 5095 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__pos__neg2) # label(axiom). [clausify(4761)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5095,a,5080,a)]. 32.06/32.17 5096 -class_Rings_Olinordered__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__pos__neg) # label(axiom). [clausify(4889)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5096,a,5078,a)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5096,a,5080,a)]. 32.06/32.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5096,a,5086,a)]. 32.96/33.06 5097 class_Groups_Ocomm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 32.96/33.06 5098 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum__diff1_H) # label(axiom). [clausify(42)]. 32.96/33.06 5099 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),F),hAPP(c_Set_Oimage(B,C,D),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(C,A,B,F),D)),E) # label(fact_setsum__reindex) # label(axiom). [clausify(129)]. 32.96/33.06 5100 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),C)),D)),E) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)) # label(fact_setsum__addf) # label(axiom). [clausify(177)]. 32.96/33.06 5101 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Int_Oint,A),B),c_List_Oupto(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),B),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(C,D))) # label(fact_setsum__set__upto__conv__listsum__int) # label(axiom). [clausify(269)]. 32.96/33.06 5102 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),C))) # label(fact_setsum__head__upt__Suc) # label(axiom). [clausify(491)]. 32.96/33.06 5103 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f65(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,F),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,F,A),D)),hAPP(hAPP(c_Product__Type_OSigma(B,F),C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(F,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(F,A),tc_fun(tc_fun(F,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(F,A)),D)),E)),C) # label(fact_setsum__Sigma) # label(axiom). [clausify(537)]. 32.96/33.06 5104 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f65(F,E,D,C,B,A)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,D),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,D,A),F)),hAPP(hAPP(c_Product__Type_OSigma(B,D),C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(tc_fun(D,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A)),F)),E)),C) # label(fact_setsum__Sigma) # label(axiom). [clausify(537)]. 32.96/33.06 5105 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C))) # label(fact_setsum__head__Suc) # label(axiom). [clausify(550)]. 32.96/33.06 5106 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f85(E,D,C,B,A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum_Ounion__inter__neutral) # label(axiom). [clausify(667)]. 32.96/33.06 5107 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f85(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum_Ounion__inter__neutral) # label(axiom). [clausify(667)]. 32.96/33.06 5108 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(c_fequal,D))),E)),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__delta_H) # label(axiom). [clausify(691)]. 32.96/33.06 5109 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(c_fequal,D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(E,D) # label(fact_setsum__delta_H) # label(axiom). [clausify(691)]. 32.96/33.06 5110 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),D)),E)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum_Odistrib) # label(axiom). [clausify(710)]. 32.96/33.06 5111 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(c_Set_OCollect(B),hAPP(hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C))),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D)))) # label(fact_setsum__restrict__set_H) # label(axiom). [clausify(807)]. 32.96/33.06 5112 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hBOOL(hAPP(hAPP(c_member(B),f106(V6,V7,F,E,D,C,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(869)]. 32.96/33.06 5113 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hAPP(V6,hAPP(D,f106(V6,V7,F,E,D,C,B,A))) != hAPP(V7,f106(V6,V7,F,E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(869)]. 32.96/33.06 5114 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum__insert) # label(axiom). [clausify(882)]. 32.96/33.06 5115 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) = C | hBOOL(hAPP(hAPP(c_member(B),f111(D,E,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C))) # label(fact_setsum__strict__mono) # label(axiom). [clausify(896)]. 32.96/33.06 5116 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Oordered__cancel__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(D,f111(E,D,C,B,A))),hAPP(E,f111(E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__strict__mono) # label(axiom). [clausify(896)]. 32.96/33.06 5117 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,B,tc_HOL_Obool),c_fequal),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__delta) # label(axiom). [clausify(940)]. 32.96/33.06 5118 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,B,tc_HOL_Obool),c_fequal),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(E,D) # label(fact_setsum__delta) # label(axiom). [clausify(940)]. 32.96/33.06 5119 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_member(B),C),hAPP(c_List_Oset(B),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),E),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,C)),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),E),c_List_Oremove1(B,C,D)))) # label(fact_listsum__map__remove1) # label(axiom). [clausify(1000)]. 32.96/33.06 5120 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),C)) # label(fact_setsum__shift__lb__Suc0__0__upt) # label(axiom). [clausify(1027)]. 32.96/33.06 5121 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),C)),D)),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))) # label(fact_listsum__addf) # label(axiom). [clausify(1088)]. 32.96/33.06 5122 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E)))) # label(fact_setsum__ub__add__nat) # label(axiom). [clausify(1288)]. 32.96/33.06 5123 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D)) # label(fact_setsum__restrict__set) # label(axiom). [clausify(1457)]. 32.96/33.06 5124 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Nitpick_Osetsum_H(B,A,D,C) = c_Groups_Ozero__class_Ozero(A) # label(fact_Nitpick_Osetsum_H__def) # label(axiom). [clausify(1466)]. 32.96/33.06 5125 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Nitpick_Osetsum_H(B,A,D,C) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),c_Hilbert__Choice_OEps(tc_List_Olist(B),hAPP(hAPP(c_COMBS(tc_List_Olist(B),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(B)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(B),tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(B)),c_fequal),c_List_Oset(B))),C))),c_List_Odistinct(B))))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom). [clausify(1466)]. 32.96/33.06 5126 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(hAPP(c_Set_Oinsert(B),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),E),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum_Oinsert__remove) # label(axiom). [clausify(1470)]. 32.96/33.06 5127 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f183(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__left) # label(axiom). [clausify(1480)]. 32.96/33.06 5128 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f183(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__left) # label(axiom). [clausify(1480)]. 32.96/33.06 5129 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_List_Odistinct(B),C)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(c_List_Oset(B),C)) # label(fact_listsum__distinct__conv__setsum__set) # label(axiom). [clausify(1506)]. 32.96/33.06 5130 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | -c_Nat__Transfer_Onat__set(B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,tc_Nat_Onat),C),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),B)) # label(fact_transfer__int__nat__sum__prod_I1_J) # label(axiom). [clausify(1524)]. 32.96/33.06 5131 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add_Ocomm__neutral) # label(axiom). [clausify(1547)]. 32.96/33.06 5132 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum_Oremove) # label(axiom). [clausify(1562)]. 32.96/33.06 5133 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(B),C),f209(V6,V7,E,F,C,D,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1687)]. 32.96/33.06 5134 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(B),f209(V6,V7,E,F,C,D,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1687)]. 32.96/33.06 5135 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hAPP(V6,f209(V6,V7,E,F,C,D,B,A)) != hAPP(V7,f209(V6,V7,E,F,C,D,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1687)]. 32.96/33.06 5136 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f214(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(1740)]. 32.96/33.06 5137 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f215(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(1740)]. 32.96/33.06 5138 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | f215(D,E,F,C,B,A) != f214(D,E,F,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(1740)]. 32.96/33.07 5139 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f215(E,F,D,C,B,A)) = hAPP(D,f214(E,F,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),F),hAPP(c_Set_Oimage(B,E,D),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(E,A,B,F),D)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(1740)]. 32.96/33.07 5140 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,hAPP(E,f214(F,D,E,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(F,A),D),hAPP(c_Set_Oimage(B,F,E),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(F,A,B,D),E)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(1740)]. 32.96/33.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5097,a,5098,a)]. 32.96/33.07 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,E),C)),D). [resolve(5097,a,5099,a)]. 32.96/33.07 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)). [resolve(5097,a,5100,a)]. 32.96/33.07 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Int_Oint),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(5097,a,5101,a)]. 32.96/33.07 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(5097,a,5102,a)]. 32.96/33.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f65(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Int_Oint),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Int_Oint),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Int_Oint)),C)),D)),B). [resolve(5097,a,5103,a)]. 32.96/33.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f65(E,D,C,B,A,tc_Int_Oint)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Int_Oint),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint)),E)),D)),B). [resolve(5097,a,5104,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5097,a,5105,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f85(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5097,a,5106,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f85(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5097,a,5107,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(5097,a,5108,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(D,C). [resolve(5097,a,5109,a)]. 32.96/33.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5097,a,5111,a)]. 32.96/33.08 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f106(F,V6,E,D,C,B,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D). [resolve(5097,a,5112,a)]. 32.96/33.09 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f106(F,V6,E,D,C,B,A,tc_Int_Oint))) != hAPP(V6,f106(F,V6,E,D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D). [resolve(5097,a,5113,a)]. 32.96/33.09 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)). [resolve(5097,a,5114,a)]. 32.96/33.09 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f111(C,D,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B))). [resolve(5097,a,5115,a)]. 32.96/33.09 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(C,f111(D,C,B,A,tc_Int_Oint))),hAPP(D,f111(D,C,B,A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B))). [resolve(5097,a,5116,a)]. 32.96/33.09 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(5097,a,5117,a)]. 32.96/33.09 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(D,C). [resolve(5097,a,5118,a)]. 32.96/33.09 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),D),c_List_Oremove1(A,B,C)))). [resolve(5097,a,5119,a)]. 32.96/33.09 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(5097,a,5120,a)]. 32.96/33.09 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))). [resolve(5097,a,5121,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(5097,a,5122,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5097,a,5123,a)]. 33.00/33.10 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Int_Oint,C,B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5097,a,5124,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Int_Oint,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(5097,a,5125,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5097,a,5126,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f183(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(5097,a,5127,a)]. 33.00/33.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f183(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(5097,a,5128,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(c_List_Oset(A),B)). [resolve(5097,a,5129,a)]. 33.00/33.11 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5097,a,5130,a)]. 33.00/33.11 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f209(F,V6,D,E,B,C,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5097,a,5133,a)]. 33.00/33.11 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f209(F,V6,D,E,B,C,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5097,a,5134,a)]. 33.00/33.11 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f209(F,V6,D,E,B,C,A,tc_Int_Oint)) != hAPP(V6,f209(F,V6,D,E,B,C,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5097,a,5135,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f214(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5097,a,5136,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f215(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5097,a,5137,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f215(C,D,E,B,A,tc_Int_Oint) != f214(C,D,E,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5097,a,5138,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f215(D,E,C,B,A,tc_Int_Oint)) = hAPP(C,f214(D,E,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(D,tc_Int_Oint,A,E),C)),B). [resolve(5097,a,5139,a)]. 33.00/33.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,hAPP(D,f214(E,C,D,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Int_Oint),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(E,tc_Int_Oint,A,C),D)),B). [resolve(5097,a,5140,a)]. 33.00/33.11 5141 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Int_Oint,A),B),c_List_Oupto(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),B),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(C,D))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom). [clausify(1744)]. 33.05/33.17 5142 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Oplus__class_Oplus(A)),D),c_Groups_Ozero__class_Ozero(A)),C) # label(fact_setsum_Oeq__fold) # label(axiom). [clausify(1748)]. 33.05/33.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint)),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B). [resolve(5142,a,5097,a)]. 33.05/33.17 5143 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),A,D),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),c_Set_OCollect(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C)))),hAPP(c_COMBC(B,D,tc_HOL_Obool),hAPP(hAPP(c_COMBB(D,tc_fun(D,tc_HOL_Obool),B),c_fequal),F)))))),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) # label(fact_setsum__image__gen) # label(axiom). [clausify(1826)]. 33.05/33.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Int_Oint,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B). [resolve(5143,a,5097,a)]. 33.05/33.17 5144 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D))) # label(fact_setsum__add__nat__ivl) # label(axiom). [clausify(1933)]. 33.05/33.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(5144,a,5097,a)]. 33.18/33.32 5145 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,C),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,C,A),D)),hAPP(hAPP(c_Product__Type_OSigma(B,C),E),hAPP(c_COMBK(tc_fun(C,tc_HOL_Obool),B),F))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,tc_fun(C,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(C,A),tc_fun(tc_fun(C,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A)),D)),F)),E) # label(fact_setsum__cartesian__product) # label(axiom). [clausify(2157)]. 33.18/33.32 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Int_Oint),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Int_Oint),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint)),C)),E)),D). [resolve(5145,a,5097,a)]. 33.18/33.32 5146 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f257(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono3) # label(axiom). [clausify(2158)]. 33.18/33.32 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f257(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B))). [resolve(5146,b,5097,a)]. 33.18/33.32 5147 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(E,f257(E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono3) # label(axiom). [clausify(2158)]. 33.18/33.32 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(D,f257(D,C,B,A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B))). [resolve(5147,b,5097,a)]. 33.18/33.32 5148 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,D,E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),E),C))) # label(fact_setsum__shift__bounds__cl__nat__ivl) # label(axiom). [clausify(2275)]. 33.18/33.37 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5148,a,5097,a)]. 33.18/33.37 5149 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(c_List_Onth(A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) # label(fact_listsum__setsum__nth) # label(axiom). [clausify(2276)]. 33.18/33.37 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(c_List_Onth(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))). [resolve(5149,a,5097,a)]. 33.18/33.37 5150 class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 33.18/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5150,a,5098,a)]. 33.18/33.37 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,E),C)),D). [resolve(5150,a,5099,a)]. 33.18/33.37 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)). [resolve(5150,a,5100,a)]. 33.18/33.37 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(5150,a,5101,a)]. 33.18/33.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(5150,a,5102,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f65(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Code__Numeral_Ocode__numeral)),C)),D)),B). [resolve(5150,a,5103,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f65(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Code__Numeral_Ocode__numeral),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral)),E)),D)),B). [resolve(5150,a,5104,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5150,a,5105,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f85(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5150,a,5106,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f85(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5150,a,5107,a)]. 33.28/33.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5150,a,5108,a)]. 33.28/33.38 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(5150,a,5109,a)]. 33.28/33.38 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5150,a,5111,a)]. 33.28/33.38 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f106(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5150,a,5112,a)]. 33.28/33.38 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f106(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(V6,f106(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5150,a,5113,a)]. 33.28/33.38 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)). [resolve(5150,a,5114,a)]. 33.28/33.38 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f111(C,D,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5150,a,5115,a)]. 33.28/33.38 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(C,f111(D,C,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f111(D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5150,a,5116,a)]. 33.28/33.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5150,a,5117,a)]. 33.28/33.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(5150,a,5118,a)]. 33.28/33.39 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),D),c_List_Oremove1(A,B,C)))). [resolve(5150,a,5119,a)]. 33.28/33.39 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(5150,a,5120,a)]. 33.28/33.39 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),D))). [resolve(5150,a,5121,a)]. 33.28/33.39 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(5150,a,5122,a)]. 33.31/33.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5150,a,5123,a)]. 33.31/33.40 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Code__Numeral_Ocode__numeral,C,B) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5150,a,5124,a)]. 33.31/33.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Code__Numeral_Ocode__numeral,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(5150,a,5125,a)]. 33.31/33.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5150,a,5126,a)]. 33.31/33.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f183(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(5150,a,5127,a)]. 33.31/33.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f183(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(5150,a,5128,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_List_Oset(A),B)). [resolve(5150,a,5129,a)]. 33.31/33.41 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5150,a,5130,a)]. 33.31/33.41 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5150,a,5131,a)]. 33.31/33.41 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f209(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5150,a,5133,a)]. 33.31/33.41 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f209(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5150,a,5134,a)]. 33.31/33.41 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f209(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(V6,f209(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5150,a,5135,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f214(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5150,a,5136,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f215(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5150,a,5137,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f215(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) != f214(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5150,a,5138,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f215(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) = hAPP(C,f214(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(D,tc_Code__Numeral_Ocode__numeral,A,E),C)),B). [resolve(5150,a,5139,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,hAPP(D,f214(E,C,D,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(E,tc_Code__Numeral_Ocode__numeral,A,C),D)),B). [resolve(5150,a,5140,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5150,a,5142,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5150,a,5143,a)]. 33.31/33.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(5150,a,5144,a)]. 33.31/33.41 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral)),C)),E)),D). [resolve(5150,a,5145,a)]. 33.31/33.41 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f257(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5150,a,5146,b)]. 33.34/33.43 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(D,f257(D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5150,a,5147,b)]. 33.34/33.43 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5150,a,5148,a)]. 33.34/33.43 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))). [resolve(5150,a,5149,a)]. 33.34/33.43 5151 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_COMBK(A,B),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__0) # label(axiom). [clausify(2394)]. 33.34/33.43 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(5151,a,5097,a)]. 33.34/33.43 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5151,a,5150,a)]. 33.34/33.43 5152 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__Int) # label(axiom). [clausify(2412)]. 33.34/33.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5152,a,5097,a)]. 33.34/33.45 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5152,a,5150,a)]. 33.34/33.45 5153 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,C),hAPP(c_Nat_OSuc,D))) # label(fact_setsum__shift__bounds__cl__Suc__ivl) # label(axiom). [clausify(2418)]. 33.34/33.45 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(5153,a,5097,a)]. 33.34/33.45 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(5153,a,5150,a)]. 33.34/33.45 5154 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Int_Oint),B),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),C) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom). [clausify(2444)]. 33.34/33.45 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),B). [resolve(5154,b,5097,a)]. 33.34/33.45 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,b,5150,a)]. 33.49/33.61 5155 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f315(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__right) # label(axiom). [clausify(2629)]. 33.49/33.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f315(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(5155,a,5097,a)]. 33.49/33.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f315(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(5155,a,5150,a)]. 33.49/33.61 5156 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f315(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__right) # label(axiom). [clausify(2629)]. 33.49/33.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f315(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(5156,a,5097,a)]. 33.49/33.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f315(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(5156,a,5150,a)]. 33.49/33.61 5157 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum_Oempty) # label(axiom). [clausify(2630)]. 33.49/33.61 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5157,a,5097,a)]. 33.49/33.61 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5157,a,5150,a)]. 33.49/33.61 5158 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),C))),hAPP(B,C)) # label(fact_setsum__lessThan__Suc) # label(axiom). [clausify(2827)]. 33.65/33.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(5158,a,5097,a)]. 33.65/33.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(5158,a,5150,a)]. 33.65/33.75 5159 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | -c_Nat__Transfer_Onat__set(B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,tc_Nat_Onat),C),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),B)) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom). [clausify(3037)]. 33.65/33.75 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5159,b,5097,a)]. 33.65/33.75 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5159,b,5150,a)]. 33.65/33.75 5160 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,B)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,B))),hAPP(D,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(3105)]. 33.65/33.75 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5160,a,5097,a)]. 33.65/33.75 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5160,a,5150,a)]. 33.65/33.75 5161 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,B)),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(3105)]. 33.80/33.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5161,a,5097,a)]. 33.80/33.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5161,a,5150,a)]. 33.80/33.91 5162 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,A),B),c_List_Oupt(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(C,D))) # label(fact_interv__listsum__conv__setsum__set__nat) # label(axiom). [clausify(3149)]. 33.80/33.91 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Int_Oint),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(5162,a,5097,a)]. 33.80/33.91 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(5162,a,5150,a)]. 33.80/33.91 5163 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum_Oinsert) # label(axiom). [clausify(3298)]. 33.80/33.91 5164 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum__infinite) # label(axiom). [clausify(3323)]. 33.80/33.91 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B). [resolve(5164,a,5097,a)]. 33.80/33.91 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5164,a,5150,a)]. 33.80/33.91 5165 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_COMBI(A)),hAPP(c_List_Oset(A),B)) # label(fact_distinct__listsum__conv__Setsum) # label(axiom). [clausify(3441)]. 33.80/33.91 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),c_COMBI(tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A)). [resolve(5165,a,5097,a)]. 33.80/33.91 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_COMBI(tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5165,a,5150,a)]. 33.80/33.94 5166 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Int_Oint),B),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),B),C) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom). [clausify(3445)]. 33.80/33.94 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Int_Oint),A),B). [resolve(5166,b,5097,a)]. 33.80/33.94 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5166,b,5150,a)]. 33.80/33.94 5167 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5167,a,5098,a)]. 33.80/33.94 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,E),C)),D). [resolve(5167,a,5099,a)]. 33.80/33.94 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)). [resolve(5167,a,5100,a)]. 33.80/33.94 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(5167,a,5101,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(5167,a,5102,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f65(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Nat_Onat),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Nat_Onat),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Nat_Onat)),C)),D)),B). [resolve(5167,a,5103,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f65(E,D,C,B,A,tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Nat_Onat),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat)),E)),D)),B). [resolve(5167,a,5104,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5167,a,5105,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f85(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5167,a,5106,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f85(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5167,a,5107,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5167,a,5108,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(5167,a,5109,a)]. 33.80/33.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5167,a,5111,a)]. 33.80/33.95 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f106(F,V6,E,D,C,B,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D). [resolve(5167,a,5112,a)]. 33.80/33.95 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f106(F,V6,E,D,C,B,A,tc_Nat_Onat))) != hAPP(V6,f106(F,V6,E,D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D). [resolve(5167,a,5113,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)). [resolve(5167,a,5114,a)]. 33.80/33.95 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f111(C,D,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B))). [resolve(5167,a,5115,a)]. 33.80/33.95 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(C,f111(D,C,B,A,tc_Nat_Onat))),hAPP(D,f111(D,C,B,A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B))). [resolve(5167,a,5116,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5167,a,5117,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(5167,a,5118,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),D),c_List_Oremove1(A,B,C)))). [resolve(5167,a,5119,a)]. 33.80/33.95 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(5167,a,5120,a)]. 33.80/33.95 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))). [resolve(5167,a,5121,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(5167,a,5122,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5167,a,5123,a)]. 33.80/33.95 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Nat_Onat,C,B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5167,a,5124,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Nat_Onat,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(5167,a,5125,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5167,a,5126,a)]. 33.80/33.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f183(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(5167,a,5127,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f183(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(5167,a,5128,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(c_List_Oset(A),B)). [resolve(5167,a,5129,a)]. 33.80/33.96 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5167,a,5130,a)]. 33.80/33.96 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f209(F,V6,D,E,B,C,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5167,a,5133,a)]. 33.80/33.96 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f209(F,V6,D,E,B,C,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5167,a,5134,a)]. 33.80/33.96 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f209(F,V6,D,E,B,C,A,tc_Nat_Onat)) != hAPP(V6,f209(F,V6,D,E,B,C,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(5167,a,5135,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f214(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5167,a,5136,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f215(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5167,a,5137,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f215(C,D,E,B,A,tc_Nat_Onat) != f214(C,D,E,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5167,a,5138,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f215(D,E,C,B,A,tc_Nat_Onat)) = hAPP(C,f214(D,E,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(D,tc_Nat_Onat,A,E),C)),B). [resolve(5167,a,5139,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,hAPP(D,f214(E,C,D,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Nat_Onat),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(E,tc_Nat_Onat,A,C),D)),B). [resolve(5167,a,5140,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B). [resolve(5167,a,5142,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B). [resolve(5167,a,5143,a)]. 33.80/33.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(5167,a,5144,a)]. 33.80/33.96 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Nat_Onat),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Nat_Onat),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat)),C)),E)),D). [resolve(5167,a,5145,a)]. 33.80/33.96 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f257(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B))). [resolve(5167,a,5146,b)]. 33.80/33.96 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(D,f257(D,C,B,A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B))). [resolve(5167,a,5147,b)]. 33.80/33.97 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5167,a,5148,a)]. 33.80/33.97 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_List_Onth(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))). [resolve(5167,a,5149,a)]. 33.80/33.97 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5167,a,5151,a)]. 33.80/33.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5167,a,5152,a)]. 33.80/33.97 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(5167,a,5153,a)]. 33.80/33.97 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),B). [resolve(5167,a,5154,b)]. 33.80/33.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f315(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(5167,a,5155,a)]. 33.80/33.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f315(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(5167,a,5156,a)]. 33.80/33.97 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5167,a,5157,a)]. 33.80/33.97 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(5167,a,5158,a)]. 34.04/34.15 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(5167,a,5159,b)]. 34.04/34.15 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5167,a,5160,a)]. 34.04/34.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5167,a,5161,a)]. 34.04/34.15 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(5167,a,5162,a)]. 34.04/34.15 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B). [resolve(5167,a,5164,a)]. 34.04/34.15 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_COMBI(tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A)). [resolve(5167,a,5165,a)]. 34.04/34.15 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Nat_Onat),A),B). [resolve(5167,a,5166,b)]. 34.04/34.15 5168 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),hAPP(hAPP(c_COMBS(D,tc_fun(B,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(B,A),tc_fun(tc_fun(B,tc_HOL_Obool),A),D),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A)),hAPP(c_COMBC(B,D,A),F))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),c_Set_OCollect(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C)))),hAPP(c_COMBC(B,D,tc_HOL_Obool),V6))))),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(tc_fun(D,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A)),F)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),c_Set_OCollect(D)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),hAPP(c_COMBS(D,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),D),c_fconj),hAPP(hAPP(c_COMBC(D,tc_fun(D,tc_HOL_Obool),tc_HOL_Obool),c_member(D)),E)))),V6)))),C) # label(fact_setsum__setsum__restrict) # label(axiom). [clausify(3820)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Int_Oint),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint)),hAPP(c_COMBC(A,C,tc_Int_Oint),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5168,a,5097,a)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_COMBC(A,C,tc_Code__Numeral_Ocode__numeral),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5168,a,5150,a)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Nat_Onat),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat)),hAPP(c_COMBC(A,C,tc_Nat_Onat),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5168,a,5167,a)]. 34.04/34.16 5169 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f464(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5169,a,5097,a)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5169,a,5150,a)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5169,a,5167,a)]. 34.04/34.16 5170 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f465(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5170,a,5097,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5170,a,5150,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5170,a,5167,a)]. 34.04/34.17 5171 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | f465(D,C,B,A) != f464(D,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | f465(C,B,A,tc_Int_Oint) != f464(C,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5171,a,5097,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | f465(C,B,A,tc_Code__Numeral_Ocode__numeral) != f464(C,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5171,a,5150,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | f465(C,B,A,tc_Nat_Onat) != f464(C,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5171,a,5167,a)]. 34.04/34.17 5172 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f466(D,C,B,A)),f464(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Int_Oint)),f464(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5172,a,5097,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),f464(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5172,a,5150,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Nat_Onat)),f464(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5172,a,5167,a)]. 34.04/34.17 5173 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f466(D,C,B,A)),f465(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Int_Oint)),f465(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5173,a,5097,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),f465(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5173,a,5150,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Nat_Onat)),f465(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5173,a,5167,a)]. 34.04/34.17 5174 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f463(D,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f466(D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f466(C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5174,a,5097,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f466(C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5174,a,5150,a)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f463(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f466(C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5174,a,5167,a)]. 34.04/34.17 5175 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f464(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5175,a,5097,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5175,a,5150,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f464(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5175,a,5167,a)]. 34.04/34.18 5176 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f465(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5176,a,5097,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5176,a,5150,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f465(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5176,a,5167,a)]. 34.04/34.18 5177 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | f465(D,C,B,A) != f464(D,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | f465(C,B,A,tc_Int_Oint) != f464(C,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5177,a,5097,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | f465(C,B,A,tc_Code__Numeral_Ocode__numeral) != f464(C,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5177,a,5150,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | f465(C,B,A,tc_Nat_Onat) != f464(C,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5177,a,5167,a)]. 34.04/34.18 5178 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f466(D,C,B,A)),f464(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Int_Oint)),f464(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5178,a,5097,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),f464(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5178,a,5150,a)]. 34.04/34.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Nat_Onat)),f464(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5178,a,5167,a)]. 34.11/34.19 5179 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f466(D,C,B,A)),f465(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.11/34.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Int_Oint)),f465(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5179,a,5097,a)]. 34.11/34.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),f465(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5179,a,5150,a)]. 34.11/34.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f466(C,B,A,tc_Nat_Onat)),f465(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5179,a,5167,a)]. 34.11/34.19 5180 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f463(D,C,B,A))) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f466(D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3829)]. 34.11/34.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f466(C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5180,a,5097,a)]. 34.11/34.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Code__Numeral_Ocode__numeral))) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f466(C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5180,a,5150,a)]. 34.11/34.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f463(C,B,A,tc_Nat_Onat))) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f466(C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(5180,a,5167,a)]. 34.11/34.21 5181 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,B,C))) # label(fact_setsum__head) # label(axiom). [clausify(3855)]. 34.11/34.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5181,a,5097,a)]. 34.11/34.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5181,a,5150,a)]. 34.11/34.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5181,a,5167,a)]. 34.11/34.21 5182 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,A,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_Fun_Oid(A)),hAPP(c_Set_Oimage(B,A,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),D) # label(fact_setsum__reindex__id) # label(axiom). [clausify(3865)]. 34.11/34.21 Derived: -c_Fun_Oinj__on(A,tc_Int_Oint,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),c_Fun_Oid(tc_Int_Oint)),hAPP(c_Set_Oimage(A,tc_Int_Oint,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C). [resolve(5182,a,5097,a)]. 34.11/34.21 Derived: -c_Fun_Oinj__on(A,tc_Code__Numeral_Ocode__numeral,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_Fun_Oid(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Set_Oimage(A,tc_Code__Numeral_Ocode__numeral,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5182,a,5150,a)]. 34.11/34.21 Derived: -c_Fun_Oinj__on(A,tc_Nat_Onat,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_Fun_Oid(tc_Nat_Onat)),hAPP(c_Set_Oimage(A,tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C). [resolve(5182,a,5167,a)]. 34.28/34.36 5183 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(c_List_Orev(A),B)) = hAPP(c_List_Omonoid__add__class_Olistsum(A),B) # label(fact_listsum__rev) # label(axiom). [clausify(3881)]. 34.28/34.36 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(c_List_Orev(tc_Int_Oint),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A). [resolve(5183,a,5097,a)]. 34.28/34.36 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A). [resolve(5183,a,5150,a)]. 34.28/34.36 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(c_List_Orev(tc_Nat_Onat),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A). [resolve(5183,a,5167,a)]. 34.28/34.36 5184 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__disjoint) # label(axiom). [clausify(3985)]. 34.28/34.36 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5184,a,5097,a)]. 34.28/34.36 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5184,a,5150,a)]. 34.28/34.36 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5184,a,5167,a)]. 34.28/34.36 5185 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -c_Fun_Oinj__on(B,B,D,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(B,A,B,E),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(c_Set_Oimage(B,B,D),C)) # label(fact_setsum_Oreindex) # label(axiom). [clausify(4060)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(A,tc_Int_Oint,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5185,a,5097,a)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(A,tc_Code__Numeral_Ocode__numeral,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5185,a,5150,a)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(A,tc_Nat_Onat,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5185,a,5167,a)]. 34.28/34.39 5186 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),D)),E)),F)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),hAPP(c_Set_OCollect(B),D)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),F),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(B,tc_HOL_Obool)),hAPP(c_Set_OCollect(B),D))))) # label(fact_setsum__cases) # label(axiom). [clausify(4082)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(5186,a,5097,a)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(5186,a,5150,a)]. 34.28/34.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(5186,a,5167,a)]. 34.37/34.47 5187 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,tc_fun(C,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(C,A),tc_fun(tc_fun(C,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A)),hAPP(c_COMBC(C,B,A),D))),E)),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),hAPP(hAPP(c_COMBC(C,tc_fun(B,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(B,A),tc_fun(tc_fun(B,tc_HOL_Obool),A),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A)),D)),F)),E) # label(fact_setsum__commute) # label(axiom). [clausify(4083)]. 34.37/34.47 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Int_Oint),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint)),hAPP(c_COMBC(B,A,tc_Int_Oint),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Int_Oint),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint)),C)),E)),D). [resolve(5187,a,5097,a)]. 34.37/34.47 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_COMBC(B,A,tc_Code__Numeral_Ocode__numeral),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral)),C)),E)),D). [resolve(5187,a,5150,a)]. 34.37/34.47 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Nat_Onat),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat)),hAPP(c_COMBC(B,A,tc_Nat_Onat),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Nat_Onat),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat)),C)),E)),D). [resolve(5187,a,5167,a)]. 34.37/34.47 5188 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),B))),hAPP(D,B)) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(4173)]. 34.37/34.47 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5188,a,5097,a)]. 34.37/34.47 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5188,a,5150,a)]. 34.37/34.47 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5188,a,5167,a)]. 34.37/34.47 5189 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),hAPP(c_Nat_OSuc,B))) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(4173)]. 34.37/34.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5189,a,5097,a)]. 34.37/34.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5189,a,5150,a)]. 34.37/34.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5189,a,5167,a)]. 34.37/34.47 5190 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)),hAPP(c_Nat_OSuc,D))) # label(fact_setsum__shift__bounds__Suc__ivl) # label(axiom). [clausify(4181)]. 34.37/34.47 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(5190,a,5097,a)]. 34.37/34.47 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(5190,a,5150,a)]. 34.53/34.64 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(5190,a,5167,a)]. 34.53/34.64 5191 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | B != C | -c_Nat__Transfer_Onat__set(B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f528(D,E,B,C,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),E),C) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4355)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f528(C,D,A,B,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),D),B). [resolve(5191,a,5097,a)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f528(C,D,A,B,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5191,a,5150,a)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f528(C,D,A,B,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),D),B). [resolve(5191,a,5167,a)]. 34.53/34.64 5192 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | B != C | -c_Nat__Transfer_Onat__set(B) | hAPP(D,f528(D,E,B,C,A)) != hAPP(E,f528(D,E,B,C,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),E),C) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4355)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f528(C,D,A,B,tc_Int_Oint)) != hAPP(D,f528(C,D,A,B,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),D),B). [resolve(5192,a,5097,a)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f528(C,D,A,B,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f528(C,D,A,B,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5192,a,5150,a)]. 34.53/34.64 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f528(C,D,A,B,tc_Nat_Onat)) != hAPP(D,f528(C,D,A,B,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),D),B). [resolve(5192,a,5167,a)]. 34.71/34.84 5193 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f546(E,D,C,B,A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__zero) # label(axiom). [clausify(4561)]. 34.71/34.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f546(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5193,a,5097,a)]. 34.71/34.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f546(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5193,a,5150,a)]. 34.71/34.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f546(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5193,a,5167,a)]. 34.71/34.84 5194 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f546(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__zero) # label(axiom). [clausify(4561)]. 34.71/34.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f546(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5194,a,5097,a)]. 34.85/34.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f546(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5194,a,5150,a)]. 34.85/34.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f546(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5194,a,5167,a)]. 34.85/34.95 5195 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),C))),hAPP(B,hAPP(c_Nat_OSuc,C))) # label(fact_setsum__atMost__Suc) # label(axiom). [clausify(4639)]. 34.85/34.95 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(5195,a,5097,a)]. 34.85/34.95 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(5195,a,5150,a)]. 34.85/34.95 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(5195,a,5167,a)]. 34.85/34.95 5196 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_add__0) # label(axiom). [clausify(4641)]. 34.85/34.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = A. [resolve(5196,a,5097,a)]. 34.85/34.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A) = A. [resolve(5196,a,5150,a)]. 34.85/34.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A) = A. [resolve(5196,a,5167,a)]. 34.85/34.95 5197 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f560(D,E,B,C,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),E),C) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(4661)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f560(C,D,A,B,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),D),B). [resolve(5197,b,5097,a)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f560(C,D,A,B,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5197,b,5150,a)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f560(C,D,A,B,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),D),B). [resolve(5197,b,5167,a)]. 35.08/35.18 5198 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(B) | hAPP(D,f560(D,E,B,C,A)) != hAPP(E,f560(D,E,B,C,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),E),C) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(4661)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f560(C,D,A,B,tc_Int_Oint)) != hAPP(D,f560(C,D,A,B,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),D),B). [resolve(5198,b,5097,a)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f560(C,D,A,B,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f560(C,D,A,B,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5198,b,5150,a)]. 35.08/35.18 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(A) | hAPP(C,f560(C,D,A,B,tc_Nat_Onat)) != hAPP(D,f560(C,D,A,B,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),C),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),D),B). [resolve(5198,b,5167,a)]. 35.08/35.18 5199 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum__empty) # label(axiom). [clausify(4691)]. 35.08/35.18 5200 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,D),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),E),C))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom). [clausify(4895)]. 36.66/36.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5200,a,5097,a)]. 36.66/36.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5200,a,5150,a)]. 36.66/36.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5200,a,5167,a)]. 36.66/36.75 5201 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Oplus__class_Oplus(A)),D),c_Groups_Ozero__class_Ozero(A)),C) # label(fact_setsum_OF__eq) # label(axiom). [clausify(4896)]. 36.66/36.75 5202 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum_OF__eq) # label(axiom). [clausify(4896)]. 36.66/36.75 5203 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum_Oinfinite) # label(axiom). [clausify(4902)]. 36.66/36.75 5204 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,A),B),c_List_Oupt(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(C,D))) # label(fact_setsum__set__upt__conv__listsum__nat) # label(axiom). [clausify(4903)]. 36.66/36.75 5205 -class_Groups_Olinordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMin(A,hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(A)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj)),hAPP(hAPP(c_COMBC(A,tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(A,tc_HOL_Obool,A)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_add__Min__commute) # label(axiom). [clausify(2621)]. 36.66/36.75 5206 class_Groups_Olinordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 36.66/36.75 5207 class_Groups_Olinordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 36.66/36.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5205,a,5206,a)]. 36.66/36.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_OCollect(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral),c_HOL_OEx(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBS(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_fconj)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B))))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5205,a,5207,a)]. 37.26/37.35 5208 class_Groups_Olinordered__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 37.26/37.35 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool,tc_Int_Oint),c_HOL_OEx(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5208,a,5205,a)]. 37.26/37.35 5209 -class_Groups_Olinordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMax(A,hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(A)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj)),hAPP(hAPP(c_COMBC(A,tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(A,tc_HOL_Obool,A)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_add__Max__commute) # label(axiom). [clausify(3962)]. 37.26/37.35 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)). [resolve(5209,a,5206,a)]. 41.87/41.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_OCollect(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral),c_HOL_OEx(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBS(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_fconj)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B))))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5209,a,5207,a)]. 41.87/41.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool,tc_Int_Oint),c_HOL_OEx(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5209,a,5208,a)]. 41.87/41.95 5210 class_Groups_Oab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__group__add) # label(axiom). [assumption]. 41.87/41.95 5211 -class_Groups_Oab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__add__distrib) # label(axiom). [clausify(52)]. 41.90/42.00 5212 -class_Groups_Oab__group__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Ominus__class_Ominus(A)),C)),D)),E)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))) # label(fact_listsum__subtractf) # label(axiom). [clausify(489)]. 41.90/42.00 5213 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_ab__diff__minus) # label(axiom). [clausify(584)]. 41.90/42.00 5214 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | D != E | C = B # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(711)]. 41.90/42.00 5215 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | D = E | C != B # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(711)]. 41.90/42.00 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),B)),C)),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))). [resolve(5210,a,5212,a)]. 41.90/42.00 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | C != D | B = A. [resolve(5210,a,5214,a)]. 41.90/42.00 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | C = D | B != A. [resolve(5210,a,5215,a)]. 41.90/42.00 5216 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Ominus__class_Ominus(A)),C)),D)),E) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)) # label(fact_setsum__subtractf) # label(axiom). [clausify(727)]. 41.90/42.00 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),B)),C)),D) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)). [resolve(5216,a,5210,a)]. 41.90/42.00 5217 -class_Groups_Oab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B) # label(fact_minus__diff__eq) # label(axiom). [clausify(842)]. 41.90/42.00 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A). [resolve(5217,a,5210,a)]. 41.90/42.00 5218 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) # label(fact_setsum__Un) # label(axiom). [clausify(947)]. 42.29/42.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))). [resolve(5218,a,5210,a)]. 42.29/42.43 5219 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__diff) # label(axiom). [clausify(1252)]. 42.29/42.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5219,a,5210,a)]. 42.29/42.43 5220 -class_Groups_Oab__group__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(c_Fun_Ocomp(A,A,B,c_Groups_Ouminus__class_Ouminus(A)),C)),D)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),D))) # label(fact_uminus__listsum__map) # label(axiom). [clausify(1746)]. 42.29/42.43 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(tc_Int_Oint,tc_Int_Oint,A,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),B)),C)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C))). [resolve(5220,a,5210,a)]. 42.29/42.43 5221 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),D))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))) # label(fact_setsum__diff__nat__ivl) # label(axiom). [clausify(2277)]. 42.29/42.43 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))). [resolve(5221,a,5210,a)]. 43.72/43.80 5222 -class_Groups_Oab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B) # label(fact_ab__left__minus) # label(axiom). [clausify(2302)]. 43.72/43.80 5223 -class_Groups_Oab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),c_Groups_Ouminus__class_Ouminus(A)),C)),D) # label(fact_setsum__negf) # label(axiom). [clausify(3456)]. 43.72/43.80 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),B)),C). [resolve(5223,a,5210,a)]. 43.72/43.80 5224 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(E,D)) # label(fact_setsum__diff1) # label(axiom). [clausify(3544)]. 43.72/43.80 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(D,C)). [resolve(5224,a,5210,a)]. 43.72/43.80 5225 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) # label(fact_setsum__diff1) # label(axiom). [clausify(3544)]. 43.72/43.80 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B). [resolve(5225,a,5210,a)]. 43.72/43.80 5226 -class_Groups_Oab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBS(tc_Nat_Onat,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(A)),D)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),D),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(D,B)),hAPP(D,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_setsum__natinterval__difff) # label(axiom). [clausify(4296)]. 43.72/43.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),C)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),C),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(C,A)),hAPP(C,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5226,a,5210,a)]. 45.19/45.29 5227 -class_Groups_Oab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBS(tc_Nat_Onat,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(A)),D)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),D),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) # label(fact_setsum__natinterval__difff) # label(axiom). [clausify(4296)]. 45.19/45.29 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),C)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),C),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)). [resolve(5227,a,5210,a)]. 45.19/45.29 5228 -class_Groups_Oab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B = C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(4588)]. 45.19/45.29 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) | A = B. [resolve(5228,a,5210,a)]. 45.19/45.29 5229 -class_Groups_Oab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B != C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(4588)]. 45.19/45.29 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) | A != B. [resolve(5229,a,5210,a)]. 45.19/45.29 5230 class_Rings_Ocomm__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 45.19/45.29 5231 -class_Rings_Ocomm__semiring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_dvd__triv__right) # label(axiom). [clausify(58)]. 45.19/45.29 5232 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(101)]. 45.19/45.29 5233 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(114)]. 45.19/45.29 5234 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(210)]. 45.19/45.29 5235 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(571)]. 45.19/45.29 5236 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(615)]. 45.19/45.29 5237 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_dvd__mult) # label(axiom). [clausify(685)]. 45.19/45.29 5238 -class_Rings_Ocomm__semiring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_dvd__0__right) # label(axiom). [clausify(726)]. 45.19/45.29 5239 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(744)]. 45.19/45.29 5240 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(1011)]. 45.19/45.29 5241 -class_Rings_Ocomm__semiring__1(A) | -class_Rings_Ono__zero__divisors(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,D) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(1018)]. 45.19/45.29 5242 -class_Rings_Ocomm__semiring__1(A) | -class_Rings_Ono__zero__divisors(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f132(D,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(1018)]. 45.19/45.29 5243 -class_Rings_Ocomm__semiring__1(A) | -class_Rings_Ono__zero__divisors(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(D,f132(D,C,B,A)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(1018)]. 45.19/45.29 5244 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom). [clausify(1134)]. 45.19/45.29 5245 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Ozero__class_Ozero(A)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(1305)]. 45.19/45.29 5246 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),D)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(A,A,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(A),C)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(A)),c_Nat_Osemiring__1__class_Oof__nat(A))),D))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))) # label(fact_arith__series__general) # label(axiom). [clausify(1460)]. 45.19/45.29 5247 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_dvd__mult2) # label(axiom). [clausify(1595)]. 45.19/45.29 5248 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(1822)]. 45.19/45.29 5249 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(1917)]. 45.19/45.29 5250 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(1923)]. 45.19/45.29 5251 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom). [clausify(1949)]. 45.19/45.29 5252 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(2075)]. 45.19/45.29 5253 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(2092)]. 45.19/45.29 5254 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(2128)]. 45.19/45.29 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A))). [resolve(5230,a,5231,a)]. 45.19/45.29 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),B). [resolve(5230,a,5232,a)]. 45.19/45.29 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(5230,a,5234,a)]. 45.19/45.29 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(5230,a,5235,a)]. 45.19/45.30 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(5230,a,5237,a)]. 45.19/45.30 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5230,a,5238,a)]. 45.19/45.30 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A). [resolve(5230,a,5239,a)]. 45.19/45.30 Derived: -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,C) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B). [resolve(5230,a,5241,a)]. 45.19/45.30 Derived: -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f132(C,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5230,a,5242,a)]. 45.19/45.30 Derived: -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(C,f132(C,B,A,tc_Int_Oint)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5230,a,5243,a)]. 45.19/45.30 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),A). [resolve(5230,a,5244,a)]. 45.19/45.30 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5230,a,5245,a)]. 45.19/45.30 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),C)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),C))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A))). [resolve(5230,a,5246,a)]. 45.19/45.30 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))). [resolve(5230,a,5247,a)]. 45.19/45.30 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(5230,a,5248,a)]. 45.19/45.30 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)). [resolve(5230,a,5250,a)]. 45.38/45.47 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5230,a,5251,a)]. 45.38/45.47 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),D)). [resolve(5230,a,5253,a)]. 45.38/45.47 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)). [resolve(5230,a,5254,a)]. 45.38/45.47 5255 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(2337)]. 45.38/45.47 5256 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Groups_Oone__class_Oone(A))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),c_Nat_Osemiring__1__class_Oof__nat(A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),B))) # label(fact_gauss__sum) # label(axiom). [clausify(2340)]. 45.38/45.47 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A))). [resolve(5256,a,5230,a)]. 45.38/45.47 5257 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(2349)]. 45.38/45.47 5258 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(2420)]. 45.38/45.47 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5258,a,5230,a)]. 45.38/45.47 5259 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(2460)]. 45.38/45.47 5260 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),D) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(2571)]. 45.79/45.95 5261 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) # label(fact_dvd__mult__right) # label(axiom). [clausify(2794)]. 45.79/45.95 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)). [resolve(5261,a,5230,a)]. 45.79/45.95 5262 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom). [clausify(3007)]. 45.79/45.95 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(5262,a,5230,a)]. 45.79/45.95 5263 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(3279)]. 45.79/45.95 5264 class_Rings_Ocomm__semiring__1(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 45.79/45.95 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A))). [resolve(5264,a,5231,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5264,a,5232,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5264,a,5233,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5264,a,5234,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),A). [resolve(5264,a,5235,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5264,a,5236,a)]. 45.79/45.95 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5264,a,5237,a)]. 45.79/45.95 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5264,a,5238,a)]. 45.79/45.95 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),A). [resolve(5264,a,5239,a)]. 45.87/45.95 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5264,a,5240,a)]. 45.87/45.95 Derived: -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,C) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5264,a,5241,a)]. 45.87/45.95 Derived: -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f132(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5264,a,5242,a)]. 45.87/45.95 Derived: -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(C,f132(C,B,A,tc_Code__Numeral_Ocode__numeral)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5264,a,5243,a)]. 45.87/45.95 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),A). [resolve(5264,a,5244,a)]. 45.87/45.95 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(5264,a,5245,a)]. 45.87/45.95 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),C)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral))),C))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A))). [resolve(5264,a,5246,a)]. 45.87/45.95 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5264,a,5247,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5264,a,5248,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5264,a,5249,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5264,a,5250,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5264,a,5251,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5264,a,5252,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5264,a,5253,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5264,a,5254,a)]. 45.87/45.96 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A))). [resolve(5264,a,5256,a)]. 46.08/46.23 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A) = A. [resolve(5264,a,5257,a)]. 46.08/46.23 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5264,a,5258,a)]. 46.08/46.23 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5264,a,5260,a)]. 46.08/46.23 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5264,a,5261,a)]. 46.08/46.23 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5264,a,5262,a)]. 46.08/46.23 5265 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom). [clausify(3532)]. 46.08/46.23 5266 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A))),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(3594)]. 46.08/46.23 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))),B). [resolve(5266,a,5230,a)]. 46.08/46.23 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5266,a,5264,a)]. 46.08/46.23 5267 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(3604)]. 46.08/46.23 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),B). [resolve(5267,a,5230,a)]. 46.08/46.23 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5267,a,5264,a)]. 46.08/46.23 5268 -class_Rings_Ocomm__semiring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_dvd__triv__left) # label(axiom). [clausify(3710)]. 46.08/46.23 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5268,a,5230,a)]. 46.08/46.23 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5268,a,5264,a)]. 46.39/46.48 5269 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,E) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero) # label(axiom). [clausify(3757)]. 46.39/46.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5269,a,5230,a)]. 46.39/46.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5269,a,5264,a)]. 46.39/46.48 5270 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(3802)]. 46.39/46.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C)). [resolve(5270,a,5230,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5270,a,5264,a)]. 46.39/46.48 5271 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(3957)]. 46.39/46.48 5272 class_Rings_Ocomm__semiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 46.39/46.48 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A))). [resolve(5272,a,5231,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),B). [resolve(5272,a,5232,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A). [resolve(5272,a,5233,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5272,a,5234,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),A). [resolve(5272,a,5235,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5272,a,5236,a)]. 46.39/46.48 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))). [resolve(5272,a,5237,a)]. 46.39/46.48 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5272,a,5238,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),A). [resolve(5272,a,5239,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),B). [resolve(5272,a,5240,a)]. 46.39/46.48 Derived: -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,C) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B). [resolve(5272,a,5241,a)]. 46.39/46.48 Derived: -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f132(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5272,a,5242,a)]. 46.39/46.48 Derived: -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(C,f132(C,B,A,tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5272,a,5243,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),A). [resolve(5272,a,5244,a)]. 46.39/46.48 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5272,a,5245,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),C)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat))),C))),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A))). [resolve(5272,a,5246,a)]. 46.39/46.48 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))). [resolve(5272,a,5247,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)). [resolve(5272,a,5248,a)]. 46.39/46.48 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5272,a,5249,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)). [resolve(5272,a,5250,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5272,a,5251,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5272,a,5252,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)). [resolve(5272,a,5253,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D)). [resolve(5272,a,5254,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),A))). [resolve(5272,a,5256,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A) = A. [resolve(5272,a,5257,a)]. 46.39/46.49 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5272,a,5258,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5272,a,5260,a)]. 46.39/46.49 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),C)). [resolve(5272,a,5261,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)). [resolve(5272,a,5262,a)]. 46.39/46.49 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat))),B). [resolve(5272,a,5266,a)]. 46.68/46.81 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),B). [resolve(5272,a,5267,a)]. 46.68/46.81 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))). [resolve(5272,a,5268,a)]. 46.68/46.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5272,a,5269,a)]. 46.68/46.81 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C)). [resolve(5272,a,5270,a)]. 46.68/46.81 5273 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(4120)]. 46.68/46.81 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A). [resolve(5273,a,5230,a)]. 46.68/46.81 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5273,a,5264,a)]. 46.68/46.81 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A). [resolve(5273,a,5272,a)]. 46.68/46.81 5274 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),D),E)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__dvd__mono) # label(axiom). [clausify(4228)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),C),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5274,a,5230,a)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),C),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5274,a,5264,a)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D))). [resolve(5274,a,5272,a)]. 46.68/46.81 5275 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__mult__left) # label(axiom). [clausify(4375)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5275,a,5230,a)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5275,a,5264,a)]. 46.68/46.81 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5275,a,5272,a)]. 46.78/46.88 5276 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(4384)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5276,a,5264,a)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)). [resolve(5276,a,5272,a)]. 46.78/46.88 5277 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(4410)]. 46.78/46.88 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5277,a,5230,a)]. 46.78/46.88 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5277,a,5264,a)]. 46.78/46.88 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5277,a,5272,a)]. 46.78/46.88 5278 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),E)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(4411)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),D)). [resolve(5278,a,5230,a)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),D)). [resolve(5278,a,5264,a)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),D)). [resolve(5278,a,5272,a)]. 46.78/46.88 5279 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(4437)]. 46.78/46.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),B). [resolve(5279,a,5230,a)]. 47.00/47.16 5280 -class_Rings_Ocomm__semiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(4571)]. 47.00/47.16 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5280,a,5230,a)]. 47.00/47.16 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5280,a,5264,a)]. 47.00/47.16 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5280,a,5272,a)]. 47.00/47.16 5281 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(4573)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D))). [resolve(5281,a,5230,a)]. 47.00/47.16 5282 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(4575)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)). [resolve(5282,a,5230,a)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5282,a,5264,a)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)). [resolve(5282,a,5272,a)]. 47.00/47.16 5283 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(4730)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(5283,a,5230,a)]. 47.00/47.16 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),A). [resolve(5283,a,5264,a)]. 51.18/51.35 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat))),A). [resolve(5283,a,5272,a)]. 51.18/51.35 5284 class_Divides_Osemiring__div(tc_Nat_Onat) # label(arity_Nat__Onat__Divides_Osemiring__div) # label(axiom). [assumption]. 51.18/51.35 5285 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),B) = C # label(fact_div__mult__self2__is__id) # label(axiom). [clausify(85)]. 51.18/51.35 5286 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),C)),C)),c_Divides_Odiv__class_Omod(A,B,C)) = B # label(fact_mod__div__decomp) # label(axiom). [clausify(119)]. 51.18/51.35 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)),A) = B. [resolve(5284,a,5285,a)]. 51.18/51.35 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)) = A. [resolve(5284,a,5286,a)]. 51.18/51.35 5287 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),B) # label(fact_div__self) # label(axiom). [clausify(610)]. 51.18/51.35 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),A). [resolve(5287,a,5284,a)]. 51.18/51.35 5288 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_mod__add__left__eq) # label(axiom). [clausify(628)]. 51.18/51.35 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C),B). [resolve(5288,a,5284,a)]. 51.18/51.35 5289 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_div__0) # label(axiom). [clausify(718)]. 51.18/51.35 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5289,a,5284,a)]. 51.18/51.35 5290 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),D) = c_Divides_Odiv__class_Omod(A,B,D) # label(fact_mod__mult__self1) # label(axiom). [clausify(754)]. 51.18/51.35 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,C). [resolve(5290,a,5284,a)]. 51.18/51.35 5291 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),c_Groups_Oone__class_Oone(A)) # label(fact_div__add__self1) # label(axiom). [clausify(851)]. 51.18/51.35 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5291,a,5284,a)]. 51.18/51.35 5292 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_zmod__simps_I4_J) # label(axiom). [clausify(981)]. 51.53/51.65 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C),B). [resolve(5292,a,5284,a)]. 51.53/51.65 5293 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),C)),C)),c_Divides_Odiv__class_Omod(A,B,C)) = B # label(fact_mod__div__equality) # label(axiom). [clausify(1189)]. 51.53/51.65 5294 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),D) # label(fact_div__mult__mult1) # label(axiom). [clausify(1318)]. 51.53/51.65 5295 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) # label(fact_div__mult__self1) # label(axiom). [clausify(1366)]. 51.53/51.65 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)). [resolve(5295,a,5284,a)]. 51.53/51.65 5296 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),D) # label(fact_div__mult__mult2) # label(axiom). [clausify(1390)]. 51.53/51.65 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),C). [resolve(5296,a,5284,a)]. 51.53/51.65 5297 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__mult__self1__is__0) # label(axiom). [clausify(1404)]. 51.53/51.65 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5297,a,5284,a)]. 51.53/51.65 5298 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__self) # label(axiom). [clausify(1554)]. 51.53/51.65 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5298,a,5284,a)]. 51.53/51.65 5299 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_zmod__simps_I1_J) # label(axiom). [clausify(1680)]. 51.53/51.65 5300 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),B) = C # label(fact_div__mult__self1__is__id) # label(axiom). [clausify(1959)]. 51.53/51.65 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),A) = B. [resolve(5300,a,5284,a)]. 51.53/51.65 5301 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D),C) # label(fact_mod__add__eq) # label(axiom). [clausify(1996)]. 51.56/51.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C),B). [resolve(5301,a,5284,a)]. 51.56/51.69 5302 class_Divides_Osemiring__div(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Divides_Osemiring__div) # label(axiom). [assumption]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A)),A) = B. [resolve(5302,a,5285,a)]. 51.56/51.69 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)) = A. [resolve(5302,a,5286,a)]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),A). [resolve(5302,a,5287,a)]. 51.56/51.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C),B). [resolve(5302,a,5288,a)]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5302,a,5289,a)]. 51.56/51.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,C). [resolve(5302,a,5290,a)]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)). [resolve(5302,a,5291,a)]. 51.56/51.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C),B). [resolve(5302,a,5292,a)]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5302,a,5294,a)]. 51.56/51.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5302,a,5295,a)]. 51.63/51.79 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5302,a,5296,a)]. 51.63/51.79 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5302,a,5297,a)]. 51.63/51.79 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5302,a,5298,a)]. 51.63/51.79 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),A) = B. [resolve(5302,a,5300,a)]. 51.63/51.79 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C),B). [resolve(5302,a,5301,a)]. 51.63/51.79 5303 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B))),c_Divides_Odiv__class_Omod(A,C,B))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D) # label(fact_div__mod__equality2) # label(axiom). [clausify(2093)]. 51.63/51.79 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C). [resolve(5303,a,5284,a)]. 51.63/51.79 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A))),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5303,a,5302,a)]. 51.63/51.79 5304 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = C # label(fact_dvd__mult__div__cancel) # label(axiom). [clausify(2232)]. 51.63/51.79 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)) = B. [resolve(5304,a,5284,a)]. 51.63/51.79 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)) = B. [resolve(5304,a,5302,a)]. 51.63/51.79 5305 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),F),C) # label(fact_mod__add__cong) # label(axiom). [clausify(2250)]. 51.63/51.79 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E),B). [resolve(5305,a,5284,a)]. 51.76/51.91 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,D,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,E,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),D),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),E),B). [resolve(5305,a,5302,a)]. 51.76/51.91 5306 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B) != D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B) = C # label(fact_dvd__div__eq__mult) # label(axiom). [clausify(2305)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A) = B. [resolve(5306,a,5284,a)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A) = B. [resolve(5306,a,5302,a)]. 51.76/51.91 5307 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B) = D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B) != C # label(fact_dvd__div__eq__mult) # label(axiom). [clausify(2305)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A) != B. [resolve(5307,a,5284,a)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A) != B. [resolve(5307,a,5302,a)]. 51.76/51.91 5308 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),c_Divides_Odiv__class_Omod(A,B,C)),C) # label(fact_mod__div__trivial) # label(axiom). [clausify(2308)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),B). [resolve(5308,a,5284,a)]. 51.76/51.91 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),B). [resolve(5308,a,5302,a)]. 51.76/51.91 5309 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_mod__mult__left__eq) # label(axiom). [clausify(2451)]. 51.76/51.91 5310 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,c_Groups_Ozero__class_Ozero(A)) = B # label(fact_mod__by__0) # label(axiom). [clausify(2479)]. 51.76/51.91 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(5310,a,5284,a)]. 51.76/51.91 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5310,a,5302,a)]. 52.03/52.14 5311 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,c_Groups_Ozero__class_Ozero(A),B) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__0) # label(axiom). [clausify(2789)]. 52.03/52.14 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5311,a,5284,a)]. 52.03/52.14 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral),A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5311,a,5302,a)]. 52.03/52.14 5312 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) # label(fact_div__mult__self2) # label(axiom). [clausify(2898)]. 52.03/52.14 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)). [resolve(5312,a,5284,a)]. 52.03/52.14 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5312,a,5302,a)]. 52.03/52.14 5313 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),B) = C # label(fact_dvd__div__mult__self) # label(axiom). [clausify(2944)]. 52.03/52.14 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),A) = B. [resolve(5313,a,5284,a)]. 52.03/52.14 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),A) = B. [resolve(5313,a,5302,a)]. 52.03/52.14 5314 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(2951)]. 52.03/52.14 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B). [resolve(5314,a,5284,a)]. 52.03/52.14 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5314,a,5302,a)]. 52.11/52.23 5315 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(2951)]. 52.11/52.23 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B). [resolve(5315,a,5284,a)]. 52.11/52.23 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5315,a,5302,a)]. 52.11/52.23 5316 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),c_Groups_Ozero__class_Ozero(A)) # label(fact_div__by__0) # label(axiom). [clausify(2994)]. 52.11/52.23 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5316,a,5284,a)]. 52.11/52.23 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(5316,a,5302,a)]. 52.11/52.23 5317 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,c_Divides_Odiv__class_Omod(A,B,C),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__mod__trivial) # label(axiom). [clausify(3085)]. 52.11/52.23 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(5317,a,5284,a)]. 52.11/52.23 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B). [resolve(5317,a,5302,a)]. 52.11/52.23 5318 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),D),E)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),D)) # label(fact_div__mult__div__if__dvd) # label(axiom). [clausify(3089)]. 52.11/52.23 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),C),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),C)). [resolve(5318,a,5284,a)]. 52.11/52.23 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),C),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5318,a,5302,a)]. 52.15/52.31 5319 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),D) # label(fact_dvd__div__mult) # label(axiom). [clausify(3124)]. 52.15/52.31 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),C). [resolve(5319,a,5284,a)]. 52.15/52.31 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),C). [resolve(5319,a,5302,a)]. 52.15/52.31 5320 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_div__by__1) # label(axiom). [clausify(3125)]. 52.15/52.31 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5320,a,5284,a)]. 52.15/52.31 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5320,a,5302,a)]. 52.15/52.31 5321 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) # label(fact_div__mult__swap) # label(axiom). [clausify(3217)]. 52.15/52.31 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)). [resolve(5321,a,5284,a)]. 52.15/52.31 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5321,a,5302,a)]. 52.15/52.31 5322 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),D) # label(fact_div__mult__mult1__if) # label(axiom). [clausify(3247)]. 52.15/52.31 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),C). [resolve(5322,a,5284,a)]. 52.15/52.31 5323 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) # label(fact_div__mult__mult1__if) # label(axiom). [clausify(3247)]. 52.58/52.69 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)). [resolve(5323,a,5284,a)]. 52.58/52.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5323,a,5302,a)]. 52.58/52.69 5324 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),c_Groups_Oone__class_Oone(A)) # label(fact_div__add__self2) # label(axiom). [clausify(3656)]. 52.58/52.69 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5324,a,5284,a)]. 52.58/52.69 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)). [resolve(5324,a,5302,a)]. 52.58/52.69 5325 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__add__self2) # label(axiom). [clausify(3672)]. 52.58/52.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(5325,a,5284,a)]. 52.58/52.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B). [resolve(5325,a,5302,a)]. 52.58/52.69 5326 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,D)),C) # label(fact_mod__mult__mult2) # label(axiom). [clausify(3824)]. 52.58/52.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,C)),B). [resolve(5326,a,5302,a)]. 52.58/52.69 5327 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,c_Groups_Oone__class_Oone(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__by__1) # label(axiom). [clausify(3869)]. 52.58/52.69 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5327,a,5284,a)]. 52.58/52.69 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5327,a,5302,a)]. 52.58/52.69 5328 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),D) # label(fact_zmod__simps_I2_J) # label(axiom). [clausify(3877)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),C). [resolve(5328,a,5284,a)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B),C). [resolve(5328,a,5302,a)]. 52.84/52.98 5329 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),D) # label(fact_mod__mult__right__eq) # label(axiom). [clausify(3939)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),C). [resolve(5329,a,5284,a)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B),C). [resolve(5329,a,5302,a)]. 52.84/52.98 5330 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),D) # label(fact_mod__add__right__eq) # label(axiom). [clausify(3949)]. 52.84/52.98 5331 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C),B) = c_Divides_Odiv__class_Omod(A,C,B) # label(fact_mod__add__self1) # label(axiom). [clausify(4151)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B),A) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A). [resolve(5331,a,5284,a)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B),A) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A). [resolve(5331,a,5302,a)]. 52.84/52.98 5332 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Divides_Odiv__class_Omod(A,C,D)) # label(fact_mod__mult__mult1) # label(axiom). [clausify(4245)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)). [resolve(5332,a,5284,a)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5332,a,5302,a)]. 52.84/52.98 5333 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),C) = c_Divides_Odiv__class_Omod(A,B,C) # label(fact_mod__mult__self2) # label(axiom). [clausify(4259)]. 52.84/52.98 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B). [resolve(5333,a,5284,a)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B). [resolve(5333,a,5302,a)]. 53.21/53.32 5334 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C),C) = c_Groups_Ozero__class_Ozero(A) # label(fact_mod__mult__self2__is__0) # label(axiom). [clausify(4340)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5334,a,5284,a)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5334,a,5302,a)]. 53.21/53.32 5335 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),C)),C)),c_Divides_Odiv__class_Omod(A,B,C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) # label(fact_div__mod__equality) # label(axiom). [clausify(4379)]. 53.21/53.32 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C). [resolve(5335,a,5284,a)]. 53.21/53.32 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C). [resolve(5335,a,5302,a)]. 53.21/53.32 5336 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D),C) # label(fact_mod__mult__eq) # label(axiom). [clausify(4501)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C),B). [resolve(5336,a,5284,a)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C),B). [resolve(5336,a,5302,a)]. 53.21/53.32 5337 class_Divides_Osemiring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Osemiring__div) # label(axiom). [assumption]. 53.21/53.32 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),A) = B. [resolve(5337,a,5285,a)]. 53.21/53.32 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)) = A. [resolve(5337,a,5286,a)]. 53.21/53.32 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C),B). [resolve(5337,a,5288,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5337,a,5289,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,C). [resolve(5337,a,5290,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5337,a,5291,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),C),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),B). [resolve(5337,a,5292,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),C). [resolve(5337,a,5294,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)). [resolve(5337,a,5295,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),C). [resolve(5337,a,5296,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5337,a,5297,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5337,a,5298,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),A) = B. [resolve(5337,a,5300,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C),B). [resolve(5337,a,5301,a)]. 53.21/53.33 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A))),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C). [resolve(5337,a,5303,a)]. 53.21/53.33 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)) = B. [resolve(5337,a,5304,a)]. 53.21/53.33 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),E),B). [resolve(5337,a,5305,a)]. 53.21/53.33 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A) = B. [resolve(5337,a,5306,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A) != B. [resolve(5337,a,5307,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),B). [resolve(5337,a,5308,a)]. 53.21/53.34 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5337,a,5310,a)]. 53.21/53.34 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5337,a,5311,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)). [resolve(5337,a,5312,a)]. 53.21/53.34 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),A) = B. [resolve(5337,a,5313,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B). [resolve(5337,a,5314,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B). [resolve(5337,a,5315,a)]. 53.21/53.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5337,a,5316,a)]. 53.21/53.34 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(5337,a,5317,a)]. 53.21/53.34 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),C),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),C)). [resolve(5337,a,5318,a)]. 53.21/53.34 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),C). [resolve(5337,a,5319,a)]. 53.21/53.34 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(5337,a,5320,a)]. 53.21/53.34 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)). [resolve(5337,a,5321,a)]. 53.21/53.38 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)). [resolve(5337,a,5323,a)]. 53.21/53.38 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A)),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5337,a,5324,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(5337,a,5325,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,C)),B). [resolve(5337,a,5326,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5337,a,5327,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),C). [resolve(5337,a,5328,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C). [resolve(5337,a,5329,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B),A) = c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A). [resolve(5337,a,5331,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)). [resolve(5337,a,5332,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B). [resolve(5337,a,5333,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5337,a,5334,a)]. 53.21/53.38 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C). [resolve(5337,a,5335,a)]. 53.21/53.38 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C),B). [resolve(5337,a,5336,a)]. 53.21/53.38 5338 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Divides_Odiv__class_Omod(A,B,C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),B),C)),C)) = B # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom). [clausify(4645)]. 53.21/53.38 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),B)),B)) = A. [resolve(5338,a,5284,a)]. 53.21/53.38 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),A),B)),B)) = A. [resolve(5338,a,5302,a)]. 55.16/55.30 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),B)),B)) = A. [resolve(5338,a,5337,a)]. 55.16/55.30 5339 -class_Divides_Osemiring__div(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B))),c_Divides_Odiv__class_Omod(A,C,B)) = C # label(fact_mod__div__equality2) # label(axiom). [clausify(4703)]. 55.16/55.30 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A))),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A)) = B. [resolve(5339,a,5284,a)]. 55.16/55.30 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A))),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A)) = B. [resolve(5339,a,5302,a)]. 55.16/55.30 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A))),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A)) = B. [resolve(5339,a,5337,a)]. 55.16/55.30 5340 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),F),C) # label(fact_mod__mult__cong) # label(axiom). [clausify(4800)]. 55.16/55.30 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),E),B). [resolve(5340,a,5284,a)]. 55.16/55.30 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,D,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,E,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),E),B). [resolve(5340,a,5302,a)]. 55.16/55.30 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),E),B). [resolve(5340,a,5337,a)]. 55.16/55.30 5341 class_Int_Oring__char__0(tc_Int_Oint) # label(arity_Int__Oint__Int_Oring__char__0) # label(axiom). [assumption]. 55.16/55.30 5342 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(B))) # label(fact_iszero__number__of__Bit1) # label(axiom). [clausify(90)]. 55.16/55.30 5343 -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Oring__1__class_Oof__int(A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B # label(fact_of__int__0__eq__iff) # label(axiom). [clausify(203)]. 55.16/55.30 5344 -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B # label(fact_of__int__0__eq__iff) # label(axiom). [clausify(203)]. 55.46/55.58 5345 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | B != C | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) # label(fact_eq__number__of) # label(axiom). [clausify(498)]. 55.46/55.58 5346 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | B = C | hAPP(c_Int_Onumber__class_Onumber__of(A),B) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) # label(fact_eq__number__of) # label(axiom). [clausify(498)]. 55.46/55.58 5347 -class_Int_Oring__char__0(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_Ints__double__eq__0__iff) # label(axiom). [clausify(1430)]. 55.46/55.58 5348 -class_Int_Oring__char__0(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_Ints__double__eq__0__iff) # label(axiom). [clausify(1430)]. 55.46/55.58 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(A))). [resolve(5341,a,5342,b)]. 55.46/55.58 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5341,a,5343,a)]. 55.46/55.58 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | A != B | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B). [resolve(5341,a,5345,a)]. 55.46/55.58 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | A = B | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B). [resolve(5341,a,5346,a)]. 55.46/55.58 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5341,a,5347,a)]. 55.46/55.58 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5341,a,5348,a)]. 55.46/55.58 5349 -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Oring__1__class_Oof__int(A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B # label(fact_of__int__eq__0__iff) # label(axiom). [clausify(2606)]. 55.46/55.58 5350 -class_Int_Oring__char__0(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B # label(fact_of__int__eq__0__iff) # label(axiom). [clausify(2606)]. 55.46/55.58 5351 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),B)) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B))) # label(fact_iszero__number__of__Bit0) # label(axiom). [clausify(2784)]. 55.46/55.58 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) | c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A))). [resolve(5351,a,5341,a)]. 55.46/55.58 5352 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),B)) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B))) # label(fact_iszero__number__of__Bit0) # label(axiom). [clausify(2784)]. 55.46/55.58 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A))). [resolve(5352,a,5341,a)]. 55.46/55.58 5353 -class_Int_Oring__char__0(A) | B != C | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Int_Oring__1__class_Oof__int(A),C) # label(fact_of__int__eq__iff) # label(axiom). [clausify(3168)]. 60.60/60.71 Derived: A != B | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B). [resolve(5353,a,5341,a)]. 60.60/60.71 5354 -class_Int_Oring__char__0(A) | B = C | hAPP(c_Int_Oring__1__class_Oof__int(A),B) != hAPP(c_Int_Oring__1__class_Oof__int(A),C) # label(fact_of__int__eq__iff) # label(axiom). [clausify(3168)]. 60.60/60.71 Derived: A = B | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) != hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B). [resolve(5354,a,5341,a)]. 60.60/60.71 5355 -class_Int_Oring__char__0(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),B)),B) # label(fact_Ints__odd__nonzero) # label(axiom). [clausify(3415)]. 60.60/60.71 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)),A). [resolve(5355,a,5341,a)]. 60.60/60.71 5356 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 60.60/60.71 5357 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__strict__increasing) # label(axiom). [clausify(116)]. 60.60/60.71 5358 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__nonpos__nonpos) # label(axiom). [clausify(284)]. 60.60/60.71 5359 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__strict__increasing2) # label(axiom). [clausify(365)]. 60.60/60.71 5360 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__neg__nonpos) # label(axiom). [clausify(662)]. 60.60/60.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))). [resolve(5356,a,5357,a)]. 60.60/60.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5356,a,5358,a)]. 60.60/60.71 5361 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_add__increasing) # label(axiom). [clausify(1295)]. 60.79/60.91 5362 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__pos__pos) # label(axiom). [clausify(1866)]. 60.79/60.91 5363 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(5363,a,5357,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5363,a,5358,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(5363,a,5359,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5363,a,5360,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(5363,a,5361,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))). [resolve(5363,a,5362,a)]. 60.79/60.91 5364 class_Groups_Oordered__comm__monoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5364,a,5357,a)]. 60.79/60.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5364,a,5358,a)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5364,a,5359,a)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5364,a,5360,a)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5364,a,5361,a)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5364,a,5362,a)]. 61.39/61.53 5365 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__nonneg__pos) # label(axiom). [clausify(2680)]. 61.39/61.53 5366 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__neg__neg) # label(axiom). [clausify(2694)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5366,a,5363,a)]. 61.39/61.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5366,a,5364,a)]. 61.39/61.53 5367 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__pos__nonneg) # label(axiom). [clausify(3307)]. 61.51/61.63 5368 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B))) # label(fact_add__increasing2) # label(axiom). [clausify(3351)]. 61.51/61.63 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A))). [resolve(5368,a,5363,a)]. 61.51/61.63 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(5368,a,5364,a)]. 61.51/61.63 5369 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f410(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono2) # label(axiom). [clausify(3465)]. 61.51/61.63 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f410(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B))). [resolve(5369,a,5356,a)]. 61.51/61.63 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f410(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B))). [resolve(5369,a,5363,a)]. 61.51/61.63 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f410(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5369,a,5364,a)]. 61.51/61.63 5370 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(E,f410(E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono2) # label(axiom). [clausify(3465)]. 61.67/61.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(D,f410(D,C,B,A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B))). [resolve(5370,a,5356,a)]. 61.67/61.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(D,f410(D,C,B,A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B))). [resolve(5370,a,5363,a)]. 61.67/61.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(D,f410(D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5370,a,5364,a)]. 61.67/61.82 5371 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3755)]. 61.67/61.82 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5371,a,5356,a)]. 61.67/61.82 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5371,a,5363,a)]. 61.67/61.82 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5371,a,5364,a)]. 61.67/61.82 5372 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3755)]. 61.67/61.82 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5372,a,5356,a)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5372,a,5363,a)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B. [resolve(5372,a,5364,a)]. 61.85/62.02 5373 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3755)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5373,a,5356,a)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5373,a,5363,a)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != B. [resolve(5373,a,5364,a)]. 61.85/62.02 5374 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_add__nonpos__neg) # label(axiom). [clausify(4044)]. 61.85/62.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5374,a,5363,a)]. 64.09/64.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5374,a,5364,a)]. 64.09/64.20 5375 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))) # label(fact_add__nonneg__nonneg) # label(axiom). [clausify(4696)]. 64.09/64.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B))). [resolve(5375,a,5356,a)]. 64.09/64.20 5376 class_Power_Opower(tc_Int_Oint) # label(arity_Int__Oint__Power_Opower) # label(axiom). [assumption]. 64.09/64.20 5377 -class_Power_Opower(A) | c_Power_Opower_Opower(A,c_Groups_Oone__class_Oone(A),c_Groups_Otimes__class_Otimes(A)) = c_Power_Opower__class_Opower(A) # label(fact_power__power__power) # label(axiom). [clausify(121)]. 64.09/64.20 5378 -class_Power_Opower(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_power__0) # label(axiom). [clausify(251)]. 64.09/64.20 Derived: c_Power_Opower_Opower(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),c_Groups_Otimes__class_Otimes(tc_Int_Oint)) = c_Power_Opower__class_Opower(tc_Int_Oint). [resolve(5376,a,5377,a)]. 64.09/64.20 5379 -class_Power_Opower(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__Suc) # label(axiom). [clausify(369)]. 64.09/64.20 5380 class_Power_Opower(tc_Nat_Onat) # label(arity_Nat__Onat__Power_Opower) # label(axiom). [assumption]. 64.09/64.20 Derived: c_Power_Opower_Opower(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)) = c_Power_Opower__class_Opower(tc_Nat_Onat). [resolve(5380,a,5377,a)]. 64.09/64.20 5381 class_Power_Opower(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Power_Opower) # label(axiom). [assumption]. 64.09/64.20 Derived: c_Power_Opower_Opower(tc_Code__Numeral_Ocode__numeral,c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)) = c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral). [resolve(5381,a,5377,a)]. 64.09/64.20 5382 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_OSuc,B)) # label(fact_power__0__Suc) # label(axiom). [clausify(2547)]. 64.09/64.20 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_OSuc,A)). [resolve(5382,a,5376,a)]. 64.09/64.20 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_OSuc,A)). [resolve(5382,a,5380,a)]. 64.09/64.20 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_OSuc,A)). [resolve(5382,a,5381,a)]. 64.23/64.36 5383 -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Power_Opower(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_power__eq__0__iff) # label(axiom). [clausify(2922)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5383,d,5376,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5383,d,5380,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5383,d,5381,a)]. 64.23/64.36 5384 -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Power_Opower(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C # label(fact_power__eq__0__iff) # label(axiom). [clausify(2922)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5384,d,5376,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5384,d,5380,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5384,d,5381,a)]. 64.23/64.36 5385 -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Power_Opower(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C # label(fact_power__eq__0__iff) # label(axiom). [clausify(2922)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5385,d,5376,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5385,d,5380,a)]. 64.23/64.36 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5385,d,5381,a)]. 67.70/67.82 5386 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_power__0__left) # label(axiom). [clausify(3953)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5386,a,5376,a)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5386,a,5380,a)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5386,a,5381,a)]. 67.70/67.82 5387 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_power__0__left) # label(axiom). [clausify(3953)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5387,a,5376,a)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5387,a,5380,a)]. 67.70/67.82 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5387,a,5381,a)]. 67.70/67.82 5388 class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 67.70/67.82 5389 -class_Groups_Oordered__ab__group__add__abs(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),c_Groups_Oabs__class_Oabs(A)),C)),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),c_Groups_Oabs__class_Oabs(A)),C)),D) # label(fact_abs__setsum__abs) # label(axiom). [clausify(127)]. 67.70/67.82 5390 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C))),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B)))) # label(fact_abs__triangle__ineq2__sym) # label(axiom). [clausify(256)]. 67.70/67.82 5391 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),D))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),c_Groups_Oabs__class_Oabs(A)),C)),D))) # label(fact_setsum__abs) # label(axiom). [clausify(1002)]. 67.70/67.82 5392 -class_Groups_Oordered__ab__group__add__abs(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__idempotent) # label(axiom). [clausify(1051)]. 67.70/67.82 5393 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E)))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D))),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),E))))) # label(fact_abs__diff__triangle__ineq) # label(axiom). [clausify(1165)]. 67.70/67.82 5394 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_abs__of__neg) # label(axiom). [clausify(1207)]. 67.70/67.82 5395 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__leI) # label(axiom). [clausify(1271)]. 67.70/67.82 5396 -class_Groups_Oordered__ab__group__add__abs(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__minus__cancel) # label(axiom). [clausify(1483)]. 67.70/67.82 5397 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B))),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__minus__le__zero) # label(axiom). [clausify(1620)]. 67.70/67.82 5398 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Oabs__class_Oabs(A),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__eq__0) # label(axiom). [clausify(2074)]. 67.70/67.82 5399 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Oabs__class_Oabs(A),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__eq__0) # label(axiom). [clausify(2074)]. 67.70/67.82 5400 -class_Groups_Oordered__ab__group__add__abs(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)) # label(fact_abs__add__abs) # label(axiom). [clausify(2096)]. 67.70/67.82 5401 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_zero__less__abs__iff) # label(axiom). [clausify(2255)]. 67.70/67.82 5402 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_zero__less__abs__iff) # label(axiom). [clausify(2255)]. 67.70/67.82 5403 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),c_Groups_Oabs__class_Oabs(A)),C)),D))) # label(fact_setsum__abs__ge__zero) # label(axiom). [clausify(2390)]. 67.70/67.82 5404 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(2541)]. 67.70/67.82 5405 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(2541)]. 67.71/67.83 5406 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(2541)]. 67.71/67.83 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint)),B)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint)),B)),C). [resolve(5388,a,5389,a)]. 67.71/67.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B))),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A)))). [resolve(5388,a,5390,a)]. 67.71/67.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint)),B)),C))). [resolve(5388,a,5391,a)]. 67.71/67.83 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5388,a,5392,a)]. 67.71/67.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D)))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C))),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),D))))). [resolve(5388,a,5393,a)]. 67.71/67.83 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5388,a,5395,a)]. 67.71/67.83 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5388,a,5396,a)]. 67.71/67.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5388,a,5397,a)]. 67.71/67.83 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5388,a,5398,a)]. 67.71/67.83 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5388,a,5399,a)]. 67.71/67.83 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)). [resolve(5388,a,5400,a)]. 67.71/67.83 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5388,a,5401,a)]. 68.15/68.34 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5388,a,5402,a)]. 68.15/68.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Groups_Oabs__class_Oabs(tc_Int_Oint)),B)),C))). [resolve(5388,a,5403,a)]. 68.15/68.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5388,a,5405,a)]. 68.15/68.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5388,a,5406,a)]. 68.15/68.34 5407 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),B))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(A,A),c_Groups_Oabs__class_Oabs(A)),B)))) # label(fact_listsum__abs) # label(axiom). [clausify(2803)]. 68.15/68.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Int_Oint),c_Groups_Oabs__class_Oabs(tc_Int_Oint)),A)))). [resolve(5407,a,5388,a)]. 68.15/68.34 5408 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)))),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)))) # label(fact_abs__triangle__ineq3) # label(axiom). [clausify(2941)]. 68.15/68.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)))),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)))). [resolve(5408,a,5388,a)]. 68.15/68.34 5409 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = B # label(fact_abs__of__nonneg) # label(axiom). [clausify(3152)]. 68.15/68.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = A. [resolve(5409,a,5388,a)]. 68.15/68.34 5410 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_abs__le__D1) # label(axiom). [clausify(3295)]. 68.15/68.34 5411 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__not__less__zero) # label(axiom). [clausify(3610)]. 68.15/68.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5411,a,5388,a)]. 68.15/68.34 5412 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C))),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)))) # label(fact_abs__triangle__ineq2) # label(axiom). [clausify(3778)]. 68.72/68.88 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B))),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)))). [resolve(5412,a,5388,a)]. 68.72/68.88 5413 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_abs__ge__zero) # label(axiom). [clausify(3834)]. 68.72/68.88 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5413,a,5388,a)]. 68.72/68.88 5414 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = B # label(fact_abs__of__pos) # label(axiom). [clausify(3863)]. 68.72/68.88 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = A. [resolve(5414,a,5388,a)]. 68.72/68.88 5415 -class_Groups_Oordered__ab__group__add__abs(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B)) # label(fact_abs__minus__commute) # label(axiom). [clausify(4260)]. 68.72/68.88 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A)). [resolve(5415,a,5388,a)]. 68.72/68.88 5416 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__le__zero__iff) # label(axiom). [clausify(4391)]. 68.72/68.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5416,a,5388,a)]. 68.72/68.88 5417 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__le__zero__iff) # label(axiom). [clausify(4391)]. 68.72/68.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5417,a,5388,a)]. 68.72/68.88 5418 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)))) # label(fact_abs__triangle__ineq) # label(axiom). [clausify(4481)]. 68.72/68.88 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)))). [resolve(5418,a,5388,a)]. 68.72/68.88 5419 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) # label(fact_abs__le__D2) # label(axiom). [clausify(4513)]. 68.72/68.88 5420 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_abs__ge__self) # label(axiom). [clausify(4542)]. 76.44/76.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5420,a,5388,a)]. 76.44/76.59 5421 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)))) # label(fact_abs__triangle__ineq4) # label(axiom). [clausify(4558)]. 76.44/76.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B))),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)))). [resolve(5421,a,5388,a)]. 76.44/76.59 5422 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_abs__of__nonpos) # label(axiom). [clausify(4599)]. 76.44/76.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5422,a,5388,a)]. 76.44/76.59 5423 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_abs__ge__minus__self) # label(axiom). [clausify(4649)]. 76.44/76.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5423,a,5388,a)]. 76.44/76.59 5424 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Oabs__class_Oabs(A),c_Groups_Ozero__class_Ozero(A)) # label(fact_abs__zero) # label(axiom). [clausify(4689)]. 76.44/76.59 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5424,a,5388,a)]. 76.44/76.59 5425 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 76.44/76.59 5426 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(174)]. 76.44/76.59 5427 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_add__le__cancel__right) # label(axiom). [clausify(174)]. 76.44/76.59 5428 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C) # label(fact_min__add__distrib__left) # label(axiom). [clausify(1501)]. 76.44/76.59 5429 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(2149)]. 76.44/76.59 5430 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(2149)]. 76.65/76.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5425,a,5426,a)]. 76.65/76.81 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5425,a,5427,a)]. 76.65/76.81 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5425,a,5428,a)]. 76.65/76.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5425,a,5429,a)]. 76.65/76.81 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5425,a,5430,a)]. 76.65/76.81 5431 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_add__less__imp__less__left) # label(axiom). [clausify(2292)]. 76.65/76.81 5432 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_add__le__imp__le__right) # label(axiom). [clausify(2504)]. 76.65/76.81 5433 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__less__cancel__right) # label(axiom). [clausify(2531)]. 76.65/76.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5433,a,5425,a)]. 76.65/76.81 5434 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__less__cancel__right) # label(axiom). [clausify(2531)]. 76.65/76.81 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5434,a,5425,a)]. 77.48/77.67 5435 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C) # label(fact_max__add__distrib__left) # label(axiom). [clausify(2711)]. 77.48/77.67 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5435,a,5425,a)]. 77.48/77.67 5436 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_add__less__imp__less__right) # label(axiom). [clausify(2837)]. 77.48/77.68 5437 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_add__le__imp__le__left) # label(axiom). [clausify(3268)]. 77.48/77.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5437,a,5425,a)]. 77.48/77.68 5438 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 77.48/77.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5438,a,5426,a)]. 77.48/77.68 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5438,a,5427,a)]. 77.48/77.68 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B). [resolve(5438,a,5428,a)]. 77.48/77.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(5438,a,5429,a)]. 77.48/77.68 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(5438,a,5430,a)]. 77.48/77.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))). [resolve(5438,a,5433,a)]. 77.48/77.68 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))). [resolve(5438,a,5434,a)]. 77.96/78.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),B). [resolve(5438,a,5435,a)]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5438,a,5437,a)]. 77.96/78.14 5439 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5439,a,5426,a)]. 77.96/78.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5439,a,5427,a)]. 77.96/78.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B). [resolve(5439,a,5428,a)]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))). [resolve(5439,a,5429,a)]. 77.96/78.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))). [resolve(5439,a,5430,a)]. 77.96/78.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C))). [resolve(5439,a,5434,a)]. 77.96/78.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)),B). [resolve(5439,a,5435,a)]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5439,a,5437,a)]. 77.96/78.14 5440 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__le__cancel__left) # label(axiom). [clausify(4372)]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5440,a,5425,a)]. 77.96/78.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(5440,a,5438,a)]. 80.38/80.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))). [resolve(5440,a,5439,a)]. 80.38/80.52 5441 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__le__cancel__left) # label(axiom). [clausify(4372)]. 80.38/80.52 5442 -class_Lattices_Obounded__lattice__top(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),c_Orderings_Otop__class_Otop(A)) = B # label(fact_inf__top__right) # label(axiom). [clausify(961)]. 80.38/80.52 5443 class_Lattices_Obounded__lattice__top(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__top) # label(axiom). [assumption]. 80.38/80.52 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = A. [resolve(5442,a,5443,a)]. 80.38/80.52 5444 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),c_Orderings_Otop__class_Otop(A)),B) # label(fact_sup__top__left) # label(axiom). [clausify(1710)]. 80.38/80.52 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_HOL_Obool)),A). [resolve(5444,a,5443,a)]. 80.38/80.52 5445 -class_Lattices_Obounded__lattice__top(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),c_Orderings_Otop__class_Otop(A)),B) = B # label(fact_inf__top__left) # label(axiom). [clausify(2698)]. 80.38/80.52 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_HOL_Obool)),A) = A. [resolve(5445,a,5443,a)]. 80.38/80.52 5446 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),c_Orderings_Otop__class_Otop(A)) # label(fact_sup__top__right) # label(axiom). [clausify(2914)]. 80.38/80.52 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),c_Orderings_Otop__class_Otop(tc_HOL_Obool)). [resolve(5446,a,5443,a)]. 80.38/80.52 5447 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) | c_Orderings_Otop__class_Otop(A) = C # label(fact_inf__eq__top__iff) # label(axiom). [clausify(3500)]. 80.38/80.52 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B) | c_Orderings_Otop__class_Otop(tc_HOL_Obool) = B. [resolve(5447,a,5443,a)]. 80.38/80.52 5448 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) | c_Orderings_Otop__class_Otop(A) = B # label(fact_inf__eq__top__iff) # label(axiom). [clausify(3500)]. 80.38/80.52 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B) | c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A. [resolve(5448,a,5443,a)]. 80.38/80.52 5449 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) | c_Orderings_Otop__class_Otop(A) != C | c_Orderings_Otop__class_Otop(A) != B # label(fact_inf__eq__top__iff) # label(axiom). [clausify(3500)]. 80.38/80.52 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B) | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != B | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != A. [resolve(5449,a,5443,a)]. 80.38/80.52 5450 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__top(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom). [clausify(3847)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),c_Orderings_Otop__class_Otop(tc_fun(B,A))) = C. [resolve(5450,b,5442,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),c_Orderings_Otop__class_Otop(tc_fun(B,A))),C). [resolve(5450,b,5444,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),c_Orderings_Otop__class_Otop(tc_fun(B,A))),C) = C. [resolve(5450,b,5445,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),C),c_Orderings_Otop__class_Otop(tc_fun(B,A))). [resolve(5450,b,5446,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),D) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = D. [resolve(5450,b,5447,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),D) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = C. [resolve(5450,b,5448,a)]. 82.59/82.79 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),D) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != D | c_Orderings_Otop__class_Otop(tc_fun(B,A)) != C. [resolve(5450,b,5449,a)]. 82.59/82.79 5451 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 82.59/82.79 5452 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) # label(fact_of__nat__0__less__iff) # label(axiom). [clausify(222)]. 82.59/82.79 5453 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) # label(fact_of__nat__0__less__iff) # label(axiom). [clausify(222)]. 82.59/82.79 5454 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)))) # label(fact_power__gt1__lemma) # label(axiom). [clausify(329)]. 82.59/82.79 5455 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_power__less__imp__less__base) # label(axiom). [clausify(892)]. 82.59/82.79 5456 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Oone__class_Oone(A))) # label(fact_zero__le__one) # label(axiom). [clausify(969)]. 82.59/82.79 5457 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) # label(fact_zero__le__imp__of__nat) # label(axiom). [clausify(1072)]. 82.59/82.79 5458 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) # label(fact_power__strict__increasing) # label(axiom). [clausify(1092)]. 82.59/82.79 5459 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f156(C,D,B,A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),D))) # label(fact_setprod__nonneg) # label(axiom). [clausify(1219)]. 82.59/82.79 5460 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f156(B,C,D,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(D,A),B),C))) # label(fact_setprod__nonneg) # label(axiom). [clausify(1219)]. 82.59/82.79 5461 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) # label(fact_of__nat__less__imp__less) # label(axiom). [clausify(1284)]. 82.59/82.79 5462 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_one__le__power) # label(axiom). [clausify(1326)]. 82.59/82.79 5463 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) != hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | B = C # label(fact_power2__eq__imp__eq) # label(axiom). [clausify(1397)]. 82.59/82.79 5464 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) # label(fact_of__nat__le__iff) # label(axiom). [clausify(1406)]. 82.59/82.79 5465 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) # label(fact_of__nat__le__iff) # label(axiom). [clausify(1406)]. 82.59/82.79 5466 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D))) # label(fact_power__strict__mono) # label(axiom). [clausify(1419)]. 82.59/82.79 5467 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),D))) # label(fact_power__mono) # label(axiom). [clausify(1559)]. 82.59/82.79 5468 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__one__less__zero) # label(axiom). [clausify(1574)]. 82.59/82.79 5469 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | B = D # label(fact_power__eq__imp__eq__base) # label(axiom). [clausify(1632)]. 82.59/82.80 5470 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C))) # label(fact_power__increasing) # label(axiom). [clausify(1785)]. 82.59/82.80 5471 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_less__imp__of__nat__less) # label(axiom). [clausify(1879)]. 82.59/82.80 5472 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_power2__less__imp__less) # label(axiom). [clausify(1990)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A))). [resolve(5451,a,5452,a)]. 82.59/82.80 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A))). [resolve(5451,a,5453,a)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)))). [resolve(5451,a,5454,a)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5451,a,5455,a)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B))). [resolve(5451,a,5458,a)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5451,a,5461,a)]. 82.59/82.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5451,a,5462,a)]. 82.59/82.80 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | A = B. [resolve(5451,a,5463,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5451,a,5464,a)]. 82.75/82.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5451,a,5465,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C))). [resolve(5451,a,5466,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C))). [resolve(5451,a,5467,a)]. 82.75/82.91 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5451,a,5469,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B))). [resolve(5451,a,5470,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))). [resolve(5451,a,5471,a)]. 82.75/82.91 5473 class_Rings_Olinordered__semidom(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semidom) # label(axiom). [assumption]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5473,a,5452,a)]. 82.75/82.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5473,a,5453,a)]. 82.75/82.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)))). [resolve(5473,a,5454,a)]. 82.75/82.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5473,a,5455,a)]. 82.75/82.92 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))). [resolve(5473,a,5456,a)]. 82.75/82.92 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5473,a,5457,a)]. 82.75/82.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5473,a,5458,a)]. 82.75/82.92 Derived: hBOOL(hAPP(hAPP(c_member(A),f156(B,C,A,tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5473,a,5459,a)]. 82.75/82.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(A,f156(A,B,C,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5473,a,5460,a)]. 82.75/82.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5473,a,5461,a)]. 82.75/82.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5473,a,5462,a)]. 82.75/82.92 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | A = B. [resolve(5473,a,5463,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5473,a,5464,a)]. 82.75/82.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5473,a,5465,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5473,a,5466,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5473,a,5467,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5473,a,5468,a)]. 82.75/82.93 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5473,a,5469,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5473,a,5470,a)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))). [resolve(5473,a,5471,a)]. 82.75/82.93 5474 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | C != D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(2298)]. 82.75/82.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C). [resolve(5474,a,5451,a)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C). [resolve(5474,a,5473,a)]. 82.89/83.10 5475 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | C = D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(2298)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C). [resolve(5475,a,5451,a)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C). [resolve(5475,a,5473,a)]. 82.89/83.10 5476 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_less__1__mult) # label(axiom). [clausify(2486)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))). [resolve(5476,a,5451,a)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5476,a,5473,a)]. 82.89/83.10 5477 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_zero__less__power) # label(axiom). [clausify(2554)]. 82.89/83.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5477,a,5473,a)]. 82.89/83.10 5478 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B))) # label(fact_of__nat__0__le__iff) # label(axiom). [clausify(2677)]. 82.89/83.10 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A))). [resolve(5478,a,5451,a)]. 82.89/83.10 5479 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Nat_OSuc,C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | B = D # label(fact_power__inject__base) # label(axiom). [clausify(2790)]. 83.13/83.28 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | A = C. [resolve(5479,a,5451,a)]. 83.13/83.28 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | A = C. [resolve(5479,a,5473,a)]. 83.13/83.28 5480 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_power__Suc__less) # label(axiom). [clausify(2809)]. 83.13/83.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5480,a,5451,a)]. 83.13/83.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5480,a,5473,a)]. 83.13/83.28 5481 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A)))) # label(fact_zero__less__two) # label(axiom). [clausify(2915)]. 83.13/83.28 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5481,a,5451,a)]. 83.13/83.28 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)))). [resolve(5481,a,5473,a)]. 83.13/83.28 5482 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) # label(fact_power__increasing__iff) # label(axiom). [clausify(3034)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))). [resolve(5482,a,5451,a)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5482,a,5473,a)]. 83.44/83.58 5483 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) # label(fact_power__increasing__iff) # label(axiom). [clausify(3034)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))). [resolve(5483,a,5451,a)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5483,a,5473,a)]. 83.44/83.58 5484 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) # label(fact_pos__add__strict) # label(axiom). [clausify(3408)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5484,a,5473,a)]. 83.44/83.58 5485 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_of__nat__less__0__iff) # label(axiom). [clausify(3477)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5485,a,5473,a)]. 83.44/83.58 5486 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(3528)]. 83.44/83.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5486,a,5451,a)]. 83.95/84.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5486,a,5473,a)]. 83.95/84.12 5487 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(3528)]. 83.95/84.12 5488 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A)))) # label(fact_less__add__one) # label(axiom). [clausify(3696)]. 83.95/84.12 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5488,a,5451,a)]. 83.95/84.12 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)))). [resolve(5488,a,5473,a)]. 83.95/84.12 5489 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_of__nat__less__iff) # label(axiom). [clausify(3839)]. 83.95/84.12 5490 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_of__nat__less__iff) # label(axiom). [clausify(3839)]. 83.95/84.12 5491 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_zero__le__power) # label(axiom). [clausify(3964)]. 83.95/84.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5491,a,5451,a)]. 83.95/84.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5491,a,5473,a)]. 83.95/84.12 5492 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)))) # label(fact_power__less__power__Suc) # label(axiom). [clausify(4241)]. 83.95/84.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)))). [resolve(5492,a,5451,a)]. 83.95/84.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)))). [resolve(5492,a,5473,a)]. 83.95/84.14 5493 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Oone__class_Oone(A))) # label(fact_zero__less__one) # label(axiom). [clausify(4254)]. 83.95/84.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(5493,a,5451,a)]. 83.95/84.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))). [resolve(5493,a,5473,a)]. 83.95/84.14 5494 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C))),c_Groups_Oone__class_Oone(A))) # label(fact_power__Suc__less__one) # label(axiom). [clausify(4257)]. 83.95/84.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_Nat_Onat))). [resolve(5494,a,5451,a)]. 83.95/84.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))). [resolve(5494,a,5473,a)]. 83.95/84.14 5495 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C))) # label(fact_one__less__power) # label(axiom). [clausify(4258)]. 83.95/84.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5495,a,5451,a)]. 83.95/84.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5495,a,5473,a)]. 84.16/84.33 5496 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Oone__class_Oone(A)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__one__le__zero) # label(axiom). [clausify(4344)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5496,a,5451,a)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5496,a,5473,a)]. 84.16/84.33 5497 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f535(C,D,B,A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),D))) # label(fact_setprod__pos) # label(axiom). [clausify(4413)]. 84.16/84.33 Derived: hBOOL(hAPP(hAPP(c_member(A),f535(B,C,A,tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C))). [resolve(5497,a,5451,a)]. 84.16/84.33 Derived: hBOOL(hAPP(hAPP(c_member(A),f535(B,C,A,tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5497,a,5473,a)]. 84.16/84.33 5498 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f535(B,C,D,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(D,A),B),C))) # label(fact_setprod__pos) # label(axiom). [clausify(4413)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(A,f535(A,B,C,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,tc_Nat_Onat),A),B))). [resolve(5498,a,5451,a)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(A,f535(A,B,C,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5498,a,5473,a)]. 84.16/84.33 5499 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A))). [resolve(5499,a,5452,a)]. 84.16/84.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A))). [resolve(5499,a,5453,a)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)))). [resolve(5499,a,5454,a)]. 84.16/84.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5499,a,5455,a)]. 84.16/84.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5499,a,5456,a)]. 84.16/84.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A))). [resolve(5499,a,5457,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B))). [resolve(5499,a,5458,a)]. 84.16/84.34 Derived: hBOOL(hAPP(hAPP(c_member(A),f156(B,C,A,tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),C))). [resolve(5499,a,5459,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f156(A,B,C,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,tc_Int_Oint),A),B))). [resolve(5499,a,5460,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5499,a,5461,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5499,a,5462,a)]. 84.16/84.34 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | A = B. [resolve(5499,a,5463,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5499,a,5464,a)]. 84.16/84.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5499,a,5465,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C))). [resolve(5499,a,5466,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C))). [resolve(5499,a,5467,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5499,a,5468,a)]. 84.16/84.34 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5499,a,5469,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B))). [resolve(5499,a,5470,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B))). [resolve(5499,a,5471,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C). [resolve(5499,a,5474,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C). [resolve(5499,a,5475,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5499,a,5476,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5499,a,5477,a)]. 84.16/84.34 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | A = C. [resolve(5499,a,5479,a)]. 84.16/84.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5499,a,5480,a)]. 84.16/84.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))). [resolve(5499,a,5481,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))). [resolve(5499,a,5482,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))). [resolve(5499,a,5483,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))). [resolve(5499,a,5484,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5499,a,5485,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5499,a,5486,a)]. 84.16/84.35 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)))). [resolve(5499,a,5488,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5499,a,5491,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)))). [resolve(5499,a,5492,a)]. 84.16/84.35 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5499,a,5493,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B))),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5499,a,5494,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5499,a,5495,a)]. 84.16/84.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5499,a,5496,a)]. 84.16/84.35 Derived: hBOOL(hAPP(hAPP(c_member(A),f535(B,C,A,tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),C))). [resolve(5499,a,5497,a)]. 84.25/84.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f535(A,B,C,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,tc_Int_Oint),A),B))). [resolve(5499,a,5498,a)]. 84.25/84.45 5500 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),D)) # label(fact_power__less__imp__less__exp) # label(axiom). [clausify(4489)]. 84.25/84.45 5501 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_power2__le__imp__le) # label(axiom). [clausify(4505)]. 84.25/84.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5501,a,5451,a)]. 84.25/84.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5501,a,5473,a)]. 84.25/84.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5501,a,5499,a)]. 84.25/84.45 5502 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B))) # label(fact_power__strict__decreasing) # label(axiom). [clausify(4560)]. 84.25/84.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A))). [resolve(5502,a,5451,a)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(5502,a,5473,a)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A))). [resolve(5502,a,5499,a)]. 84.36/84.58 5503 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C))),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Nat_OSuc,C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_power__le__imp__le__base) # label(axiom). [clausify(4577)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5503,a,5451,a)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5503,a,5473,a)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Nat_OSuc,B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5503,a,5499,a)]. 84.36/84.58 5504 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),D),B))) # label(fact_power__decreasing) # label(axiom). [clausify(4698)]. 84.36/84.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),A))). [resolve(5504,a,5451,a)]. 85.72/85.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(5504,a,5473,a)]. 85.72/85.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),A))). [resolve(5504,a,5499,a)]. 85.72/85.95 5505 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) # label(fact_power__le__imp__le__exp) # label(axiom). [clausify(4754)]. 85.72/85.95 5506 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)))) # label(fact_power__gt1) # label(axiom). [clausify(4755)]. 85.72/85.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)))). [resolve(5506,a,5451,a)]. 85.72/85.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)))). [resolve(5506,a,5473,a)]. 85.72/85.95 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)))). [resolve(5506,a,5499,a)]. 85.72/85.95 5507 c_Typedef_Otype__definition(tc_Int_Oint,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Int_ORep__Integ,c_Int_OAbs__Integ,c_Int_OInteg) # label(fact_type__definition__Integ) # label(axiom). [assumption]. 85.72/85.95 5508 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(c_Set_Oimage(A,B,C),c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool))) = E # label(fact_type__definition_ORep__range) # label(axiom). [clausify(224)]. 85.72/85.95 5509 -c_Typedef_Otype__definition(A,B,C,D,E) | -hBOOL(hAPP(hAPP(c_member(B),F),E)) | -hBOOL(hAPP(hAPP(c_member(B),V6),E)) | V6 != F | hAPP(D,V6) = hAPP(D,F) # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(310)]. 85.72/85.95 5510 -c_Typedef_Otype__definition(A,B,C,D,E) | -hBOOL(hAPP(hAPP(c_member(B),F),E)) | -hBOOL(hAPP(hAPP(c_member(B),V6),E)) | V6 = F | hAPP(D,V6) != hAPP(D,F) # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(310)]. 85.72/85.95 Derived: hAPP(c_Set_Oimage(tc_Int_Oint,tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),c_Int_ORep__Integ),c_Orderings_Otop__class_Otop(tc_fun(tc_Int_Oint,tc_HOL_Obool))) = c_Int_OInteg. [resolve(5507,a,5508,a)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),A),c_Int_OInteg)) | -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B),c_Int_OInteg)) | B != A | hAPP(c_Int_OAbs__Integ,B) = hAPP(c_Int_OAbs__Integ,A). [resolve(5507,a,5509,a)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),A),c_Int_OInteg)) | -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B),c_Int_OInteg)) | B = A | hAPP(c_Int_OAbs__Integ,B) != hAPP(c_Int_OAbs__Integ,A). [resolve(5507,a,5510,a)]. 86.74/86.91 5511 -c_Typedef_Otype__definition(A,B,C,D,E) | c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)) = hAPP(c_Set_Oimage(B,A,D),E) # label(fact_type__definition_OAbs__image) # label(axiom). [clausify(558)]. 86.74/86.91 Derived: c_Orderings_Otop__class_Otop(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_Int_Oint,c_Int_OAbs__Integ),c_Int_OInteg). [resolve(5511,a,5507,a)]. 86.74/86.91 5512 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(C,F) != hAPP(C,V6) | F = V6 # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(1830)]. 86.74/86.91 5513 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(C,F) = hAPP(C,V6) | F != V6 # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(1830)]. 86.74/86.91 5514 -c_Typedef_Otype__definition(A,B,C,D,E) | hBOOL(hAPP(hAPP(c_member(B),hAPP(C,F)),E)) # label(fact_type__definition_ORep) # label(axiom). [clausify(2354)]. 86.74/86.91 5515 -c_Typedef_Otype__definition(A,B,C,D,E) | -hBOOL(hAPP(hAPP(c_member(B),F),E)) | hAPP(C,hAPP(D,F)) = F # label(fact_type__definition_OAbs__inverse) # label(axiom). [clausify(2847)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),A),c_Int_OInteg)) | hAPP(c_Int_ORep__Integ,hAPP(c_Int_OAbs__Integ,A)) = A. [resolve(5515,a,5507,a)]. 86.74/86.91 5516 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(D,hAPP(C,F)) = F # label(fact_type__definition_ORep__inverse) # label(axiom). [clausify(3683)]. 86.74/86.91 5517 c_Typedef_Otype__definition(tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat,c_Code__Numeral_Onat__of,c_Code__Numeral_Oof__nat,c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_type__definition__code__numeral) # label(axiom). [assumption]. 86.74/86.91 Derived: hAPP(c_Set_Oimage(tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat,c_Code__Numeral_Onat__of),c_Orderings_Otop__class_Otop(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)). [resolve(5517,a,5508,a)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | B != A | hAPP(c_Code__Numeral_Oof__nat,B) = hAPP(c_Code__Numeral_Oof__nat,A). [resolve(5517,a,5509,a)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | B = A | hAPP(c_Code__Numeral_Oof__nat,B) != hAPP(c_Code__Numeral_Oof__nat,A). [resolve(5517,a,5510,a)]. 86.74/86.91 Derived: c_Orderings_Otop__class_Otop(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,c_Code__Numeral_Oof__nat),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))). [resolve(5517,a,5511,a)]. 86.74/86.91 Derived: hAPP(c_Code__Numeral_Onat__of,A) != hAPP(c_Code__Numeral_Onat__of,B) | A = B. [resolve(5517,a,5512,a)]. 86.74/86.91 Derived: hAPP(c_Code__Numeral_Onat__of,A) = hAPP(c_Code__Numeral_Onat__of,B) | A != B. [resolve(5517,a,5513,a)]. 86.74/86.91 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,A)) = A. [resolve(5517,a,5515,a)]. 90.02/90.26 Derived: hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,A)) = A. [resolve(5517,a,5516,a)]. 90.02/90.26 5518 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 90.02/90.26 5519 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C) # label(fact_min__diff__distrib__left) # label(axiom). [clausify(227)]. 90.02/90.26 5520 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B)) # label(fact_minus__le__iff) # label(axiom). [clausify(245)]. 90.02/90.26 5521 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B)) # label(fact_minus__le__iff) # label(axiom). [clausify(245)]. 90.02/90.26 5522 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(495)]. 90.02/90.26 5523 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(495)]. 90.02/90.26 5524 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_neg__le__iff__le) # label(axiom). [clausify(670)]. 90.02/90.26 5525 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_neg__le__iff__le) # label(axiom). [clausify(670)]. 90.02/90.26 5526 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C) # label(fact_max__diff__distrib__left) # label(axiom). [clausify(880)]. 90.02/90.26 5527 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)) = hAPP(c_SetInterval_Oord__class_OlessThan(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_image__uminus__greaterThan) # label(axiom). [clausify(1076)]. 90.02/90.26 5528 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(1188)]. 90.02/90.26 5529 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(1188)]. 90.02/90.26 5530 -class_Groups_Oordered__ab__group__add(A) | c_Fun_Oinj__on(A,A,c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_inj__uminus) # label(axiom). [clausify(1233)]. 90.02/90.26 5531 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),hAPP(c_SetInterval_Oord__class_OatMost(A),B)) = hAPP(c_SetInterval_Oord__class_OatLeast(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_image__uminus__atMost) # label(axiom). [clausify(1258)]. 90.02/90.26 5532 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f165(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),F),C) != E | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(F,D)),E)) # label(fact_setsum__nonneg__leq__bound) # label(axiom). [clausify(1344)]. 90.02/90.26 5533 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(D,f165(E,F,D,C,B,A)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) != F | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(D,E)),F)) # label(fact_setsum__nonneg__leq__bound) # label(axiom). [clausify(1344)]. 90.02/90.26 5534 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f194(D,E,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(E,D) # label(fact_setsum__nonneg__0) # label(axiom). [clausify(1566)]. 90.02/90.26 5535 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(D,f194(E,D,C,B,A)))) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(D,E) # label(fact_setsum__nonneg__0) # label(axiom). [clausify(1566)]. 90.02/90.26 5536 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),hAPP(c_SetInterval_Oord__class_OatLeast(A),B)) = hAPP(c_SetInterval_Oord__class_OatMost(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_image__uminus__atLeast) # label(axiom). [clausify(1858)]. 90.02/90.26 5537 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__less__iff__less) # label(axiom). [clausify(2248)]. 90.02/90.26 5538 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__less__iff__less) # label(axiom). [clausify(2248)]. 90.02/90.26 5539 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__iff) # label(axiom). [clausify(2693)]. 90.02/90.26 5540 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__iff) # label(axiom). [clausify(2693)]. 90.02/90.26 5541 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(2697)]. 90.02/90.26 5542 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(2697)]. 90.02/90.27 5543 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_le__minus__iff) # label(axiom). [clausify(2772)]. 90.02/90.27 5544 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_le__minus__iff) # label(axiom). [clausify(2772)]. 90.02/90.27 5545 -class_Groups_Oordered__ab__group__add(A) | c_SetInterval_Oord__class_OgreaterThanAtMost(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),B)) # label(fact_image__uminus__atLeastLessThan) # label(axiom). [clausify(2917)]. 90.02/90.27 5546 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_le__imp__neg__le) # label(axiom). [clausify(3299)]. 90.02/90.27 5547 -class_Groups_Oordered__ab__group__add(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),c_SetInterval_Oord__class_OatLeastAtMost(A,C,B)) # label(fact_image__uminus__atLeastAtMost) # label(axiom). [clausify(3373)]. 90.02/90.27 5548 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(3440)]. 90.02/90.27 5549 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(3440)]. 90.02/90.27 5550 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) = hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)) # label(fact_image__uminus__lessThan) # label(axiom). [clausify(3752)]. 90.02/90.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B). [resolve(5518,a,5519,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),A)). [resolve(5518,a,5520,a)]. 90.02/90.27 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)). [resolve(5518,a,5522,a)]. 90.02/90.27 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),D)). [resolve(5518,a,5523,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5518,a,5524,a)]. 90.02/90.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5518,a,5525,a)]. 90.02/90.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),B). [resolve(5518,a,5526,a)]. 90.02/90.27 Derived: hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5518,a,5527,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5518,a,5528,a)]. 90.02/90.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5518,a,5529,a)]. 90.02/90.27 Derived: c_Fun_Oinj__on(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5518,a,5530,a)]. 90.02/90.27 Derived: hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5518,a,5531,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f165(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),E),B) != D | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(E,C)),D)). [resolve(5518,a,5532,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(C,f165(D,E,C,B,A,tc_Int_Oint)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B) != E | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(C,D)),E)). [resolve(5518,a,5533,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f194(C,D,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(D,C). [resolve(5518,a,5534,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(C,f194(D,C,B,A,tc_Int_Oint)))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(C,D). [resolve(5518,a,5535,a)]. 90.02/90.27 Derived: hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5518,a,5536,a)]. 90.02/90.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5537,a)]. 90.17/90.37 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5538,a)]. 90.17/90.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5539,a)]. 90.17/90.37 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)). [resolve(5518,a,5541,a)]. 90.17/90.37 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)). [resolve(5518,a,5542,a)]. 90.17/90.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5543,a)]. 90.17/90.37 Derived: c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,B),A)). [resolve(5518,a,5545,a)]. 90.17/90.37 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,B,A)). [resolve(5518,a,5547,a)]. 90.17/90.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5548,a)]. 90.17/90.37 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5518,a,5549,a)]. 90.17/90.37 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)). [resolve(5518,a,5550,a)]. 90.17/90.37 5551 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(4073)]. 90.17/90.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5551,a,5518,a)]. 90.17/90.37 5552 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(4073)]. 90.75/90.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5552,a,5518,a)]. 90.75/90.91 5553 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B)) # label(fact_minus__less__iff) # label(axiom). [clausify(4187)]. 90.75/90.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),A)). [resolve(5553,a,5518,a)]. 90.75/90.91 5554 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B)) # label(fact_minus__less__iff) # label(axiom). [clausify(4187)]. 90.75/90.91 5555 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(4360)]. 90.75/90.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5555,a,5518,a)]. 90.75/90.91 5556 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(4360)]. 90.75/90.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5556,a,5518,a)]. 90.75/90.91 5557 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(4746)]. 90.75/90.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5557,a,5518,a)]. 90.75/90.91 5558 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(4746)]. 90.75/90.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5558,a,5518,a)]. 90.75/90.91 5559 -class_Groups_Oordered__ab__group__add(A) | c_SetInterval_Oord__class_OgreaterThanLessThan(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,C,B)) # label(fact_image__uminus__greaterThanLessThan) # label(axiom). [clausify(4804)]. 101.23/101.42 Derived: c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,B,A)). [resolve(5559,a,5518,a)]. 101.23/101.42 5560 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(4849)]. 101.23/101.42 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5560,a,5518,a)]. 101.23/101.42 5561 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(4849)]. 101.23/101.42 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5561,a,5518,a)]. 101.23/101.42 5562 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_image__uminus__greaterThanAtMost) # label(axiom). [clausify(4866)]. 101.23/101.42 Derived: hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5562,a,5518,a)]. 101.23/101.42 5563 c_List_Onat__list(A) | -c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) # label(fact_nat__list__def) # label(axiom). [clausify(438)]. 101.23/101.42 5564 -c_List_Onat__list(A) | c_List_Oembed__list(c_List_Oreturn__list(A)) = A # label(fact_transfer__nat__int__list__return__embed) # label(axiom). [clausify(229)]. 101.23/101.42 5565 -c_List_Onat__list(A) | c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) # label(fact_nat__list__def) # label(axiom). [clausify(438)]. 101.23/101.42 Derived: -c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | c_List_Oembed__list(c_List_Oreturn__list(A)) = A. [resolve(5563,a,5564,a)]. 101.23/101.42 5566 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),f73(C,B,A)),c_Groups_Ozero__class_Ozero(A)))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(585)]. 101.23/101.42 5567 class_Rings_Odvd(tc_Int_Oint) # label(arity_Int__Oint__Rings_Odvd) # label(axiom). [assumption]. 101.23/101.42 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),f73(B,A,tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))). [resolve(5566,b,5567,a)]. 101.23/101.42 5568 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) | hBOOL(hAPP(B,f73(C,B,A))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(585)]. 101.23/101.42 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))) | hBOOL(hAPP(A,f73(B,A,tc_Int_Oint))). [resolve(5568,b,5567,a)]. 101.74/101.98 5569 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),f74(C,B,A)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),c_Groups_Ozero__class_Ozero(A)))) | -hBOOL(hAPP(B,D)) # label(fact_unity__coeff__ex) # label(axiom). [clausify(585)]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),f74(B,A,tc_Int_Oint)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) | -hBOOL(hAPP(A,C)). [resolve(5569,b,5567,a)]. 101.74/101.98 5570 class_Rings_Odvd(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Odvd) # label(axiom). [assumption]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),f73(B,A,tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))). [resolve(5570,a,5566,b)]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(A,f73(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5570,a,5568,b)]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),f74(B,A,tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(A,C)). [resolve(5570,a,5569,b)]. 101.74/101.98 5571 class_Rings_Odvd(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Odvd) # label(axiom). [assumption]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),f73(B,A,tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))). [resolve(5571,a,5566,b)]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))) | hBOOL(hAPP(A,f73(B,A,tc_Nat_Onat))). [resolve(5571,a,5568,b)]. 101.74/101.98 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),f74(B,A,tc_Nat_Onat)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) | -hBOOL(hAPP(A,C)). [resolve(5571,a,5569,b)]. 101.74/101.98 5572 -class_Rings_Odvd(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) != D | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvdI) # label(axiom). [clausify(2684)]. 101.74/101.98 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5572,a,5567,a)]. 101.74/101.98 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5572,a,5570,a)]. 101.74/101.98 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5572,a,5571,a)]. 101.74/101.98 5573 -class_Rings_Ocomm__ring(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(2769)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),B))),D))). [resolve(5573,b,5567,a)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),E),B))),D))). [resolve(5573,b,5570,a)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),E),B))),D))). [resolve(5573,b,5571,a)]. 102.33/102.52 5574 -class_Rings_Ocomm__ring(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(2769)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),B))),D))). [resolve(5574,b,5567,a)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),E),B))),D))). [resolve(5574,b,5570,a)]. 102.33/102.52 Derived: -class_Rings_Ocomm__ring(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),E),B))),D))). [resolve(5574,b,5571,a)]. 102.33/102.52 5575 -class_Rings_Odvd(A) | -class_Rings_Ocomm__ring(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I4_J) # label(axiom). [clausify(3675)]. 113.43/113.65 5576 -class_Rings_Odvd(A) | -class_Rings_Ocomm__ring(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I4_J) # label(axiom). [clausify(3675)]. 113.43/113.65 5577 class_Groups_Oab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 113.43/113.65 5578 -class_Groups_Oab__semigroup__add(A) | c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),B,C) = hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),C),B) # label(fact_foldr__conv__foldl) # label(axiom). [clausify(327)]. 113.43/113.65 Derived: c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A,B) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5577,a,5578,a)]. 113.43/113.65 5579 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 113.43/113.65 Derived: c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A,B) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A). [resolve(5579,a,5578,a)]. 113.43/113.65 5580 class_Groups_Oab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 113.43/113.65 Derived: c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A,B) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A). [resolve(5580,a,5578,a)]. 113.43/113.65 5581 -class_Groups_Oab__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(4674)]. 113.43/113.65 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(5581,a,5580,a)]. 113.43/113.65 5582 class_Groups_Oab__semigroup__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 113.43/113.65 5583 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f52(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),F),D),C) # label(fact_fold__image__cong) # label(axiom). [clausify(426)]. 113.43/113.65 5584 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f52(E,D,F,C,B,A)) != hAPP(F,f52(E,D,F,C,B,A)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),E),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),F),E),C) # label(fact_fold__image__cong) # label(axiom). [clausify(426)]. 113.43/113.65 5585 -class_Groups_Oab__semigroup__mult(A) | c_Finite__Set_Ofun__left__comm(A,A,c_Groups_Otimes__class_Otimes(A)) # label(fact_fun__left__comm) # label(axiom). [clausify(682)]. 113.43/113.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f52(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),C),B). [resolve(5582,a,5583,a)]. 113.73/113.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f52(D,C,E,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(E,f52(D,C,E,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),D),B). [resolve(5582,a,5584,a)]. 113.73/113.95 Derived: c_Finite__Set_Ofun__left__comm(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)). [resolve(5582,a,5585,a)]. 113.73/113.95 5586 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 113.73/113.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f52(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),C),B). [resolve(5586,a,5583,a)]. 113.73/113.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f52(D,C,E,B,A,tc_Nat_Onat)) != hAPP(E,f52(D,C,E,B,A,tc_Nat_Onat)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),D),B). [resolve(5586,a,5584,a)]. 113.73/113.95 Derived: c_Finite__Set_Ofun__left__comm(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)). [resolve(5586,a,5585,a)]. 113.73/113.95 5587 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),f179(B,A)),f180(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f179(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f180(B,A)),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),B)),B)) # label(fact_fold1__in) # label(axiom). [clausify(1465)]. 113.73/113.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),f179(A,tc_Code__Numeral_Ocode__numeral)),f180(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f179(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f180(A,tc_Code__Numeral_Ocode__numeral)),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),A)),A)). [resolve(5587,a,5582,a)]. 113.73/113.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),f179(A,tc_Nat_Onat)),f180(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f179(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f180(A,tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),A)),A)). [resolve(5587,a,5586,a)]. 113.73/113.95 5588 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),B)),hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),C)) # label(fact_fold1__Un) # label(axiom). [clausify(1911)]. 113.99/114.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = B | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B) | hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),A)),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5588,a,5582,a)]. 113.99/114.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B) | hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),A)),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),B)). [resolve(5588,a,5586,a)]. 113.99/114.20 5589 -class_Groups_Oab__semigroup__mult(A) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),B)) # label(fact_fold1__insert) # label(axiom). [clausify(2400)]. 113.99/114.20 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5589,a,5582,a)]. 113.99/114.20 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),A)). [resolve(5589,a,5586,a)]. 114.63/114.85 5590 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(c_Finite__Set_Ofold1(A,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Groups_Otimes__class_Otimes(A)),C),B) # label(fact_fold1__eq__fold) # label(axiom). [clausify(2506)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5590,a,5582,a)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),B),A). [resolve(5590,a,5586,a)]. 114.63/114.85 5591 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),F),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),F),hAPP(hAPP(c_Set_Oinsert(B),D),C)) # label(fact_fold__image__insert) # label(axiom). [clausify(2876)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5591,a,5582,a)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5591,a,5586,a)]. 114.63/114.85 5592 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofold__graph(A,A,c_Groups_Otimes__class_Otimes(A),B,C),D)) | hBOOL(hAPP(hAPP(c_member(A),B),C)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(A,A,c_Groups_Otimes__class_Otimes(A),E,hAPP(hAPP(c_Set_Oinsert(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),D))) # label(fact_fold__graph__insert__swap) # label(axiom). [clausify(3543)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A,B),C)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),D,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),D),C))). [resolve(5592,a,5582,a)]. 114.63/114.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A,B),C)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),C))). [resolve(5592,a,5586,a)]. 114.93/115.15 5593 -class_Groups_Oab__semigroup__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(3545)]. 114.93/115.15 5594 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f52(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),C),B). [resolve(5594,a,5583,a)]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f52(D,C,E,B,A,tc_Int_Oint)) != hAPP(E,f52(D,C,E,B,A,tc_Int_Oint)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),D),B). [resolve(5594,a,5584,a)]. 114.93/115.15 Derived: c_Finite__Set_Ofun__left__comm(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)). [resolve(5594,a,5585,a)]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),f179(A,tc_Int_Oint)),f180(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f179(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f180(A,tc_Int_Oint)),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),A)),A)). [resolve(5594,a,5587,a)]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B) | hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),A)),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),B)). [resolve(5594,a,5588,a)]. 114.93/115.15 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),A)). [resolve(5594,a,5589,a)]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),B),A). [resolve(5594,a,5590,a)]. 114.93/115.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5594,a,5591,a)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint),A,B),C)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint),D,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),C))). [resolve(5594,a,5592,a)]. 115.40/115.64 5595 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofold__graph(A,A,c_Groups_Otimes__class_Otimes(A),B,C),D)) | -hBOOL(hAPP(hAPP(c_member(A),E),C)) | hBOOL(hAPP(hAPP(c_member(A),B),C)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(A,A,c_Groups_Otimes__class_Otimes(A),E,hAPP(hAPP(c_Set_Oinsert(A),B),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(A),E),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))),D)) # label(fact_fold__graph__permute__diff) # label(axiom). [clausify(4041)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A,B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),D),B)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),D,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),D),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))),C)). [resolve(5595,a,5582,a)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A,B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),D),B)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),D),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))),C)). [resolve(5595,a,5586,a)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint),A,B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),D),B)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | hBOOL(hAPP(c_Finite__Set_Ofold__graph(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint),D,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),D),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))),C)). [resolve(5595,a,5594,a)]. 115.40/115.64 5596 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -c_Fun_Oinj__on(B,D,E,C) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),V6),hAPP(c_Set_Oimage(B,D,E),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(c_Fun_Ocomp(D,A,B,F),E)),V6),C) # label(fact_fold__image__reindex) # label(axiom). [clausify(4534)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,C,D,B) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,E),D)),F),B). [resolve(5596,a,5582,a)]. 115.40/115.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,C,D,B) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_OnatAlarm clock 119.85/120.05 Prover9 interrupted 119.85/120.05 EOF