0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.03/0.24 % Computer : n124.star.cs.uiowa.edu 0.03/0.24 % Model : x86_64 x86_64 0.03/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.24 % Memory : 32218.625MB 0.03/0.24 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.24 % CPULimit : 300 0.03/0.24 % DateTime : Sat Jul 14 05:55:24 CDT 2018 0.03/0.24 % CPUTime : 8.81/8.80 ============================== Prover9 =============================== 8.81/8.80 Prover9 (32) version 2009-11A, November 2009. 8.81/8.80 Process 16185 was started by sandbox2 on n124.star.cs.uiowa.edu, 8.81/8.80 Sat Jul 14 05:55:32 2018 8.81/8.80 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_16045_n124.star.cs.uiowa.edu". 8.81/8.80 ============================== end of head =========================== 8.81/8.80 8.81/8.80 ============================== INPUT ================================= 8.81/8.80 8.81/8.80 % Reading from file /tmp/Prover9_16045_n124.star.cs.uiowa.edu 8.81/8.80 8.81/8.80 set(prolog_style_variables). 8.81/8.80 set(auto2). 8.81/8.80 % set(auto2) -> set(auto). 8.81/8.80 % set(auto) -> set(auto_inference). 8.81/8.80 % set(auto) -> set(auto_setup). 8.81/8.80 % set(auto_setup) -> set(predicate_elim). 8.81/8.80 % set(auto_setup) -> assign(eq_defs, unfold). 8.81/8.80 % set(auto) -> set(auto_limits). 8.81/8.80 % set(auto_limits) -> assign(max_weight, "100.000"). 8.81/8.80 % set(auto_limits) -> assign(sos_limit, 20000). 8.81/8.80 % set(auto) -> set(auto_denials). 8.81/8.80 % set(auto) -> set(auto_process). 8.81/8.80 % set(auto2) -> assign(new_constants, 1). 8.81/8.80 % set(auto2) -> assign(fold_denial_max, 3). 8.81/8.80 % set(auto2) -> assign(max_weight, "200.000"). 8.81/8.80 % set(auto2) -> assign(max_hours, 1). 8.81/8.80 % assign(max_hours, 1) -> assign(max_seconds, 3600). 8.81/8.80 % set(auto2) -> assign(max_seconds, 0). 8.81/8.80 % set(auto2) -> assign(max_minutes, 5). 8.81/8.80 % assign(max_minutes, 5) -> assign(max_seconds, 300). 8.81/8.80 % set(auto2) -> set(sort_initial_sos). 8.81/8.80 % set(auto2) -> assign(sos_limit, -1). 8.81/8.80 % set(auto2) -> assign(lrs_ticks, 3000). 8.81/8.80 % set(auto2) -> assign(max_megs, 400). 8.81/8.80 % set(auto2) -> assign(stats, some). 8.81/8.80 % set(auto2) -> clear(echo_input). 8.81/8.80 % set(auto2) -> set(quiet). 8.81/8.80 % set(auto2) -> clear(print_initial_clauses). 8.81/8.80 % set(auto2) -> clear(print_given). 8.81/8.80 assign(lrs_ticks,-1). 8.81/8.80 assign(sos_limit,10000). 8.81/8.80 assign(order,kbo). 8.81/8.80 set(lex_order_vars). 8.81/8.80 clear(print_given). 8.81/8.80 8.81/8.80 % formulas(sos). % not echoed (5233 formulas) 8.81/8.80 8.81/8.80 ============================== end of input ========================== 8.81/8.80 8.81/8.80 % From the command line: assign(max_seconds, 300). 8.81/8.80 8.81/8.80 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 8.81/8.80 8.81/8.80 % Formulas that are not ordinary clauses: 8.81/8.80 1 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMax(T_c,V_N_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_Max_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 2 (all V_d_2 c_Int_Oint__ge__less__than2(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))))) # label(fact_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 3 (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]. 8.81/8.80 4 (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]. 8.81/8.80 5 (all V_c_2 all V_b_2 all V_a_2 all V_R_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_fun(T_e,tc_HOL_Obool)),V_R_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),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]. 8.81/8.80 6 (all V_n_2 all V_x_2 all T_c (class_Fields_Ofield(T_c) -> (c_Groups_Oone__class_Oone(T_c) != V_x_2 -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_x_2),V_n_2)),c_Groups_Oone__class_Oone(T_c))),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_x_2),c_Groups_Oone__class_Oone(T_c))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(c_Power_Opower__class_Opower(T_c),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]. 8.81/8.80 7 (all V_s_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> (c_Wellfounded_Owf(T_c,V_s_2) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_c),V_r_2)),hAPP(c_Relation_ORange(T_c,T_c),V_s_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)))))) # label(fact_wf__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 8 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) | V_x_2 = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_order__le__less) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 9 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)))))) # label(fact_neg__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 10 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,hAPP(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),V_A_2)),V_B_2)),V_x_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_c,V_B_2),V_x_2)) & hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_x_2)))) # label(fact_listsp__conj__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 11 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (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_c),c_Groups_Oone__class_Oone(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2)))))) # label(fact_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 12 (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]. 8.81/8.80 13 (all T_d all T_c c_Fun_Oid(tc_prod(T_c,T_d)) = c_Product__Type_Omap__pair(T_c,T_c,T_d,T_d,c_COMBI(T_c),c_COMBI(T_d))) # label(fact_map__pair_Oidentity) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 14 (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]. 8.81/8.80 15 (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]. 8.81/8.80 16 (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]. 8.81/8.80 17 (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]. 8.81/8.80 18 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_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)))) # label(fact_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 19 (all V_n 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_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 20 (all V_A_2 all T_c V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Un__empty__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 21 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_a_2),V_A_2)))) # label(fact_min__max_OInf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.80 22 (all V_f_2 all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),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_c,V_f_2,V_xs_2)))))) # label(fact_list__size__estimation) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 23 (all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2)))) # label(fact_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 24 (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]. 8.81/8.81 25 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls)),V_a))) # label(fact_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 26 (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_Oabs__class_Oabs(T_a),V_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_Ouminus__class_Ouminus(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 27 (all V_bs_2 all V_as_2 all V_f_2 all T_e all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_e,T_d,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),V_Pa_2),V_f_2),V_as_2,V_bs_2) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Omap(T_e,T_c),V_f_2),V_as_2),V_bs_2))) # label(fact_list__all2__map1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 28 (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)) -> hAPP(hAPP(c_List_Otake(tc_Nat_Onat),V_m),c_List_Oupt(V_i,V_n)) = c_List_Oupt(V_i,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m)))) # label(fact_take__upt) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 29 (all V_n_2 all T_c (class_Rings_Osemiring__1(T_c) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,T_c),c_Groups_Oplus__class_Oplus(T_c)),c_Groups_Oone__class_Oone(T_c)),V_n_2,c_Groups_Ozero__class_Ozero(T_c)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2))) # label(fact_semiring__1__class_Oof__nat__code) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 30 (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]. 8.81/8.81 31 (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]. 8.81/8.81 32 (all T_a c_List_Olist_ONil(tc_List_Olist(T_a)) = c_List_Otranspose(T_a,c_List_Olist_ONil(tc_List_Olist(T_a)))) # label(fact_transpose_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 33 (all V_B_2 all V_C_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2))) # label(fact_Int__insert__left__if1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 34 (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]. 8.81/8.81 35 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2)))))) # label(fact_zero__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 36 (all V_B_2 all V_l_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_l_2)) -> (hBOOL(hAPP(c_List_Olistsp(T_c,V_B_2),V_l_2)) -> hBOOL(hAPP(c_List_Olistsp(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_l_2))))) # label(fact_listsp__infI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 37 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__idem_Osubset__comp__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 38 (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]. 8.81/8.81 39 (all V_a_2 all V_xs_2 all V_f_2 all T_e all V_g_2 all T_c all T_d c_List_Ofoldr(T_e,T_c,hAPP(c_Fun_Ocomp(T_d,tc_fun(T_c,T_c),T_e,V_g_2),V_f_2),V_xs_2,V_a_2) = c_List_Ofoldr(T_d,T_c,V_g_2,hAPP(hAPP(c_List_Omap(T_e,T_d),V_f_2),V_xs_2),V_a_2)) # label(fact_foldr__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 40 (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]. 8.81/8.81 41 (all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> c_Relation_Orefl__on(tc_List_Olist(T_c),c_List_Olists(T_c,V_A_2),c_List_Olistrel(T_c,V_r_2)))) # label(fact_listrel__refl__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 42 (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]. 8.81/8.81 43 (all V_B_2 all V_a_2 all V_A_2 all T_c 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_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),V_B_2)) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 44 (all V_r_2 all T_d all V_a_2 all T_c ((exists B_y hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_y),V_a_2)),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_ORange(T_d,T_c),V_r_2))))) # label(fact_Range__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 45 (all V_x_2 all V_f_2 all V_A_2 all T_e all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2),hAPP(hAPP(c_COMBC(T_d,T_e,T_c),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,T_c)),V_A_2),V_f_2),V_x_2))) # label(fact_SUPR__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 46 (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]. 8.81/8.81 47 (all V_A_2 all T_c c_Fun_Oinj__on(T_c,T_c,c_Fun_Oid(T_c),V_A_2)) # label(fact_inj__on__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 48 (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]. 8.81/8.81 49 (all V_B_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_Set_OPow(T_c),V_B_2)))) # label(fact_Pow__bottom) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 50 (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]. 8.81/8.81 51 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_setprod_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 52 (all V_u_2 all V_l_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OatLeastAtMost(T_c,V_less__eq_2,V_l_2,V_u_2))) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)))) # label(fact_ord_OatLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 53 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c))) <-> c_Relation_Ototal__on(T_c,V_A_2,V_r_2))) # label(fact_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 54 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 55 (all V_B_2 all V_A_2 all V_R_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,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(c_Relation_OImage(T_d,T_c,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_B_2))) # label(fact_Image__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 56 (all V_y_2 all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> (V_y_2 = V_x_2 <-> hAPP(V_Rep_2,V_x_2) = hAPP(V_Rep_2,V_y_2)))) # label(fact_type__definition_ORep__inject) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 57 (all V_y_2 all V_f_2 all T_d all T_c all V_x_2 all V_z_2 ((V_z_2 != V_x_2 -> hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_z_2) = hAPP(V_f_2,V_z_2)) & (V_x_2 = V_z_2 -> V_y_2 = hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_z_2)))) # label(fact_fun__upd__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 58 (all V_r_2 all V_A_2 all T_c hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),T_c),c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),c_Relation_OImage(T_c,T_c,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)))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))))) # label(fact_quotient__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 59 (all V_r_2 all T_c (c_Predicate_Otransp(T_c,V_r_2) <-> c_Relation_Otrans(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))))) # label(fact_transp__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 60 (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),V_a),V_b) = 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))))) # label(fact_mult__divide__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 61 (all V_A_2 all V_f_2 all T_d all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_c,T_d,c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,V_f_2),V_B_2))) # label(fact_inj__on__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 62 (all V_f_2 all V_R_2 all T_d all T_c hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),hAPP(c_Relation_Oconverse(T_d,T_d),V_R_2)),V_f_2) = hAPP(c_Relation_Oconverse(T_c,T_c),hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),V_R_2),V_f_2))) # label(fact_converse__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 63 (all V_a (hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = 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]. 8.81/8.81 64 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) <-> -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 65 (all V_x_2 all V_f_2 all V_A_2 all T_e all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),V_A_2),hAPP(hAPP(c_COMBC(T_d,T_e,T_c),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_e,T_c)),V_A_2),V_f_2),V_x_2))) # label(fact_INFI__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 66 (all V_xs all V_n all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_tl__drop) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 67 (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]. 8.81/8.81 68 (all T_c all V_g_2 all V_f_2 all V_Pa_2 ((all B_x (hBOOL(hAPP(V_Pa_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_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),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),V_Pa_2))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),V_g_2))))))) # label(fact_wf__if__measure) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 69 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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),hAPP(c_Nat_OSuc,B_n))) = V_z_2))) # label(fact_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 70 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 71 (all V_l_2 all V_k_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 72 (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]. 8.81/8.81 73 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_b = 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)))) # label(fact_minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 74 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> -(c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (c_Relation_Osym(T_c,V_r_2) -> -c_Relation_Otrans(T_c,V_r_2))))) # label(fact_equivE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 75 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))))) # label(fact_INT__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 76 (all V_B_2 all T_d all V_d_2 all V_A_2 all V_c_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_d_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,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_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_B_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_d_2),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,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_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_d_2),V_A_2)) -> V_B_2 = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,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_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_d_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,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_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_vimage__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 77 (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]. 8.81/8.81 78 (all V_xss_2 all T_d all V_xs_2 all T_c hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),c_List_Ofoldr(tc_List_Olist(T_d),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_d)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_d),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_d)),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_d)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_List_Ofilter(tc_List_Olist(T_d),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_d)),c_fNot),hAPP(hAPP(c_COMBC(tc_List_Olist(T_d),tc_List_Olist(T_d),tc_HOL_Obool),c_fequal),c_List_Olist_ONil(T_d)))),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_c)),V_xs_2))),c_List_Ofoldr(tc_List_Olist(T_d),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_d)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_d))),V_xss_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_transpose__aux__max) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 79 (all V_A_2 all V_I_2 all T_d all T_c exists B_f (c_Fun_Oinj__on(T_c,tc_prod(T_d,T_c),B_f,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,tc_prod(T_d,T_c),B_f),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2))),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_I_2),V_A_2))))) # label(fact_Ex__inj__on__UNION__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 80 (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_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2) <-> V_B_2 = V_A_2)))) # label(fact_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 81 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_card__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 82 (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]. 8.81/8.81 83 (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_Orotate1(T_a),V_xs))) # label(fact_length__rotate1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 84 (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]. 8.81/8.81 85 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (V_a_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))) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)))) # label(fact_atLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 86 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(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_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 87 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 88 (all V_A_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,V_A_2))) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_ys_2),c_List_Olists(T_c,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)),c_List_Olists(T_c,V_A_2))))) # label(fact_append__in__lists__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 89 (all V_f_2 all V_T_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))) -> c_Groups_Oone__class_Oone(T_d) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_S_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))))))) # label(fact_setprod__Un__one) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 90 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__un_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 91 (all V_f_2 all V_B_2 all T_e all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),T_c,T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,T_c),V_B_2)),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,T_c),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_c),T_c,T_e),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),V_A_2)),hAPP(c_COMBC(T_d,T_e,T_c),V_f_2))))) # label(fact_INF__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 92 (all V_xs_2 all T_c ((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_c)),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_c)),V_xs_2))) -> (B_j != B_i -> hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i) != hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_j)))))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 93 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d c_Nitpick_Opair__box_Opair__box__case(T_d,T_e,T_c,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_d,T_e,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_pair__box_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 94 (all V_y_2 all V_x_2 all T_c (class_Orderings_Opreorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 95 (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]. 8.81/8.81 96 (all V_z_2 all T_c (class_Int_Oring__char__0(T_c) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_z_2 <-> hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_of__int__0__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 97 (all V_D_2 all V_C_2 all V_R_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_D_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,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))),c_Wellfounded_Omax__ext(T_c,V_R_2)))))) # label(fact_max__ext__additive) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 98 (all V_A_2 all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Set_Ovimage(T_c,T_d,V_f_2))) # label(fact_vimage__Union) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 99 (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]. 8.81/8.81 100 (all V_B_2 all V_C_2 all T_d all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d),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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_UN__extend__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 101 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__no__zero__divisors(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 | c_Groups_Ozero__class_Ozero(T_c) = V_b_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 102 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 103 (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]. 8.81/8.81 104 (all V_n all V_m (c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m),V_n) -> V_m = c_Groups_Oone__class_Oone(tc_Int_Oint) | V_m = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin))) # label(fact_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 105 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> V_r_2 = c_Transitive__Closure_Otrancl(T_c,V_r_2))) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 106 (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]. 8.81/8.81 107 (all V_ys_2 all V_Pa_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)))) # label(fact_takeWhile__append1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 108 (all V_A_2 all V_f_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (hAPP(V_f_2,B_xa) = hAPP(V_f_2,B_x) -> B_xa = B_x))))) <-> c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 109 (all T_c (class_Lattices_Osemilattice__inf(T_c) -> c_Finite__Set_Ofun__left__comm__idem(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)))) # label(fact_fun__left__comm__idem__inf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 110 (all V_X_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_C_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),V_X_2))))) # label(fact_InterE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 111 (all V_Pa_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> c_List_Olinorder__class_Osorted(T_c,c_List_OdropWhile(T_c,V_Pa_2,V_xs_2))))) # label(fact_sorted__dropWhile) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 112 (all V_a_2 all T_c all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 113 (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]. 8.81/8.81 114 (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]. 8.81/8.81 115 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_y_2 != V_x_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)) -> 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]. 8.81/8.81 116 (all V_B_2 all V_A_2 all T_c all T_d (c_Orderings_Obot__class_Obot(tc_fun(T_c,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) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(V_B_2,B_x))))) # label(fact_UNION__empty__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 117 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))))) # label(fact_pos__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 118 (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]. 8.81/8.81 119 (all V_size_2 all V_nrandom_2 all T_c c_Random__Sequence_Oempty(T_c,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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_c))) # label(fact_Random__Sequence_Oempty__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 120 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2))) # label(fact_bij__betw__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 121 (all V_n_2 all T_c (class_Rings_Oring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),c_Int_Oring__1__class_OInts(T_c))))) # label(fact_Ints__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 122 (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]. 8.81/8.81 123 (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]. 8.81/8.81 124 (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]. 8.81/8.81 125 (all V_A_2 all T_c V_A_2 = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_OPow(T_c),V_A_2))) # label(fact_Union__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 126 (all T_c hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_finite_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 127 (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]. 8.81/8.81 128 (all V_f_2 all T_d all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> hAPP(V_f_2,c_List_Olast(T_c,V_xs_2)) = c_List_Olast(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)))) # label(fact_last__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 129 (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]. 8.81/8.81 130 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 131 (all V_xs all V_x all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_listsum__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 132 (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)))) -> (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]. 8.81/8.81 133 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_d all T_c (c_Big__Operators_Ocomm__monoid__big(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),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]. 8.81/8.81 134 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremdups(T_a,c_List_Oremove1(T_a,V_x,V_xs)) = c_List_Oremove1(T_a,V_x,c_List_Oremdups(T_a,V_xs)))) # label(fact_remove1__remdups) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 135 (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_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)))) # label(fact_minus__add) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 136 (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(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)) = 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)))) # label(fact_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 137 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a))) # label(fact_mult__1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 138 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_d,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_d,T_c,V_f_2),V_A_2)))))) # label(fact_vimage__subsetD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 139 (all V_r_2 all V_a_2 all V_b_2 all T_c (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_a_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Wellfounded_Oacc(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Wellfounded_Oacc(T_c,V_r_2)))))) # label(fact_acc__downwards__aux) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 140 (all V_a_2 all V_times_2 all V_one_2 all T_c V_one_2 = hAPP(hAPP(c_Power_Opower_Opower(T_c,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 141 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))),V_xs_2) = c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)) # label(fact_takeWhile__eq__take) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 142 (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]. 8.81/8.81 143 (all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_c))) # label(fact_setsum__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 144 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oord(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 145 (all T_c all V_as_2 (V_as_2 != c_List_Olist_ONil(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),c_List_Olast(T_c,V_as_2)),hAPP(c_List_Oset(T_c),V_as_2))))) # label(fact_last__in__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 146 (all V_xs_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,c_Groups_Ouminus__class_Ouminus(T_c)),V_f_2)),V_xs_2)))) # label(fact_uminus__listsum__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 147 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),V_A_2)))))) # label(fact_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 148 (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]. 8.81/8.81 149 (all V_r_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel1(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel1__rtrancl__subset__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 150 (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]. 8.81/8.81 151 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,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)))))) = hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_d,T_c),V_A_2),V_x_2))) # label(fact_Inf__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 152 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__partition) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 153 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (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_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))))) # label(fact_rtrancl__trans) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 154 (all V_B_2 all V_J_2 all T_e all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),T_d),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e)),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_d),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2))),V_B_2)))) # label(fact_Int__UN__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 155 (all V_x_2 all T_c c_HOL_OThe(T_c,hAPP(c_fequal,V_x_2)) = V_x_2) # label(fact_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 156 (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]. 8.81/8.81 157 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,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__set__diff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 158 (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]. 8.81/8.81 159 (all V_B_2 all V_A_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_r_2),V_A_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2))))))) # label(fact_Image__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 160 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),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_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))))))) # label(fact_folding__image__simple_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 161 (all V_b_2 all V_r_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Wellfounded_Oacc(T_c,V_r_2))) -> (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_a_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Wellfounded_Oacc(T_c,V_r_2)))))) # label(fact_acc__downwards) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 162 (all V_B_2 all T_e all V_A_2 all V_r_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_e,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Relation_OImage(T_d,T_c,V_r_2)),V_B_2)) = hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Image__INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 163 (all V_B_2 all V_g_2 all T_f all T_e all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_e,T_f,V_g_2,V_B_2) -> c_Fun_Oinj__on(tc_prod(T_c,T_e),tc_prod(T_d,T_f),c_Product__Type_Omap__pair(T_c,T_d,T_e,T_f,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__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 164 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,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(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_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_fun__left__comm_Ofold__rec) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 165 (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]. 8.81/8.81 166 (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]. 8.81/8.81 167 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = 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))) # label(fact_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 168 (all T_c c_Fun_Oid(tc_List_Olist(T_c)) = hAPP(c_List_Omap(T_c,T_c),c_COMBI(T_c))) # label(fact_List_Omap_Oidentity) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 169 (all V_xs_2 all T_c hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_finite__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.81 170 (all V_d_2 all V_b_2 all V_c_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_a_2 -> (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_c_2),V_d_2)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_d_2) <-> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_d_2),V_c_2) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_b_2),V_a_2)))))))) # label(fact_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 171 (all T_c c_Finite__Set_Ofun__left__comm__idem(T_c,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_c),hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)))),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_fun__left__comm__idem__remove) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 172 (all V_xs_2 all V_Pa_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_filter__is__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 173 (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]. 8.81/8.82 174 (all V_r_2 all T_c ((all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))) -> c_Wellfounded_Oacyclic(T_c,V_r_2))) # label(fact_acyclicI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 175 (all V_b_2 all V_B_2 all V_A_2 all T_d all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_SUP1__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 176 (all V_Pa_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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))) -> hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_d),V_Pa_2),c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2))) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)))) # label(fact_bij__image__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 177 (all V_b_2 all V_a_2 all V_r_2 all T_c (c_Relation_Oantisym(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (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_a_2)),V_r_2)) -> V_b_2 = V_a_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 178 (all V_xs_2 all V_Pa_2 all T_c 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_c)),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_c),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_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)))))) # label(fact_dropWhile__nth) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 179 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> V_x_2 = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_min__max_OSup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 180 (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]. 8.81/8.82 181 (all V_b_2 all V_a_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,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_Relation_OImage(T_c,T_c,V_r_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_equiv__class__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 182 (all V_r_2 all V_x_2 all V_y_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))))) # label(fact_rtrancl__converseI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 183 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (V_d_2 = V_b_2 & V_c_2 = V_a_2 | V_d_2 = V_a_2 & V_c_2 = V_b_2 <-> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_d_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 184 (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),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),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_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 185 (all 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))) & (-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),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]. 8.81/8.82 186 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 187 (all V_y_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),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]. 8.81/8.82 188 (all V_n_2 all V_f_2 all T_c all V_A_2 (hAPP(c_Set_Oimage(tc_Nat_Onat,T_c,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_c),V_A_2)))) # label(fact_nat__seg__image__imp__finite) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 189 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,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(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 190 (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]. 8.81/8.82 191 (all V_Qa_2 all V_Pa_2 (hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)) | hBOOL(V_Pa_2))) # label(help_c__fimplies__1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 192 (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)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_j_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_greaterThanLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 193 (all V_com2 all V_com1 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com1)),c_Com_Ocom_Ocom__size(V_com2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Com_Ocom_Ocom__size(c_Com_Ocom_OSemi(V_com1,V_com2))) # label(fact_com_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 194 (all V_n all V_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k),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_k),V_n))))) # label(fact_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 195 (all V_s_2 all V_g_2 all V_h_2 all V_f_2 all T_d all T_e all V_xs_2 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(c_Fun_Ocomp(T_d,T_e,T_d,V_h_2),hAPP(V_g_2,B_x)) = hAPP(c_Fun_Ocomp(T_e,T_e,T_d,hAPP(V_f_2,B_x)),V_h_2))) -> hAPP(V_h_2,hAPP(c_List_Ofoldl(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_d),V_g_2),V_s_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_e,T_c,hAPP(c_COMBC(T_c,T_e,T_e),V_f_2),hAPP(V_h_2,V_s_2)),V_xs_2))) # label(fact_foldl__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 196 (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]. 8.81/8.82 197 (all V_xss all T_a c_List_Otranspose(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),V_xss)) = c_List_Otranspose(T_a,V_xss)) # label(fact_transpose_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 198 (all V_A_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2))) # label(fact_image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 199 (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]. 8.81/8.82 200 (all V_B_2 all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_B_2)) # label(fact_Un__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 201 (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]. 8.81/8.82 202 (all T_c (class_Groups_Omonoid__add(T_c) -> c_List_Omonoid__add__class_Olistsum(T_c) = c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_listsum__foldl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 203 (all V_B_H_2 all V_A_H_2 all V_A_2 all V_f_2 all T_c all T_d (V_A_H_2 = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_H_2),V_A_H_2)) -> hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,V_f_2)),V_B_H_2)) = V_B_H_2))) # label(fact_image__inv__into__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 204 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_B_2))))))) # label(fact_setprod_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 205 (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]. 8.81/8.82 206 (all V_k c_Int_Osucc(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_succ__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 207 (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]. 8.81/8.82 208 (all V_ys_2 all V_Pa_2 all T_d all T_c (V_ys_2 = c_List_Olist_ONil(T_d) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,c_List_Olist_ONil(T_c),V_ys_2))) # label(fact_list__all2__Nil) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 209 (all V_n_2 all V_m_2 all V_k_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | 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__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 210 (all V_f_2 all T_e all V_g_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),T_d,T_e),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Set_OCollect(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),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_S_2)))),hAPP(c_COMBC(T_c,T_e,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_e,tc_HOL_Obool),T_c),c_fequal),V_f_2)))))),hAPP(c_Set_Oimage(T_c,T_e,V_f_2),V_S_2))))) # label(fact_setsum__image__gen) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 211 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> c_Groups_Ozero__class_Ozero(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)))) # label(fact_divide__Numeral0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 212 (all V_Pa_2 all V_n_2 ((exists B_m (hBOOL(hAPP(V_Pa_2,B_m)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2)))) <-> (exists 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_Pa_2,B_x)))))) # label(fact_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 213 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z))) # label(fact_max__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 214 (all V_n_2 all V_m_2 all V_k_2 (V_m_2 = V_n_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))) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 215 (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_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_k)),V_j)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 216 (all V_xa_2 all V_x_2 all V_R_2 all T_c (c_Wellfounded_Omax__extp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_R_2),V_x_2,V_xa_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_x_2),V_xa_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))))) # label(fact_max__extp__max__ext__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 217 (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]. 8.81/8.82 218 (all T_c all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) -> hBOOL(hAPP(V_Pa_2,c_Hilbert__Choice_OEps(T_c,V_Pa_2))))) # label(fact_someI__ex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 219 (all V_b_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) <-> (exists B_x ((all B_y (V_b_2 = hAPP(V_f_2,B_y) -> B_y = B_x)) & hAPP(V_f_2,B_x) = V_b_2))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 220 (all V_xss_2 all V_xs_2 all T_c hAPP(c_List_Ofoldl(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Oappend(T_c),V_xs_2),V_xss_2) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),c_List_Oconcat(T_c,V_xss_2))) # label(fact_foldl__conv__concat) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 221 (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]. 8.81/8.82 222 (all T_c c_fequal = c_Predicate_Oconversep(T_c,T_c,c_fequal)) # label(fact_conversep__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 223 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),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_d),V_b_2),hAPP(V_B_2,V_a_2))))) # label(fact_mem__Sigma__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 224 (all V_w_2 all V_m_2 (V_m_2 = hAPP(c_Int_Onat,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)) -> V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (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))) # label(fact_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 225 (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]. 8.81/8.82 226 (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]. 8.81/8.82 227 (all V_B_2 all V_A_2 all V_a_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_B_2,V_a_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_INT__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 228 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OAll(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),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(c_COMBC(T_c,T_c,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_c)))))) = c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2))) # label(fact_Sup__Inf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 229 (all V_Pa_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_Pa_2,B_x)) <-> hBOOL(hAPP(V_Pa_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_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_Pa_2,B_x)))) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))))) # label(fact_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 230 (all V_B_2 all V_f_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (c_Fun_Oinj__on(T_c,T_d,V_f_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_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2)))) # label(fact_setsum__reindex__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 231 (all V_i_2 all V_y_2 all V_x_2 all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(V_Pa_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_d)),V_ys_2))) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2),hAPP(hAPP(c_List_Olist__update(T_d,V_ys_2),V_i_2),V_y_2)))))) # label(fact_list__all2__update__cong2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 232 (all V_com2_H all V_com1_H c_Com_Ocom_OSemi(V_com1_H,V_com2_H) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 233 (all V_xs_2 all V_n_2 all V_ys_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_ys_2),hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = V_n_2)) # label(fact_length__n__lists__elem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 234 (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]. 8.81/8.82 235 (all V_z_2 all 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__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)))))) # label(fact_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 236 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Int_Oring__1__class_OInts(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_c))))))) # label(fact_Ints__mult) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 237 (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]. 8.81/8.82 238 (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)) -> (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) <-> V_n_2 = V_m_2))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 239 (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]. 8.81/8.82 240 (all V_a_2 all V_c_2 all V_b_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) -> ((V_c_2 = c_Groups_Ozero__class_Ozero(T_c) -> c_Groups_Ozero__class_Ozero(T_c) = V_a_2) & (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2) = V_b_2) <-> V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))) # label(fact_divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 241 (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]. 8.81/8.82 242 (all V_A_2 all T_c V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 243 (all V_R_2 all V_p_2 all T_c ((exists B_n (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),B_n),V_R_2))) & 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_c,T_c)),V_p_2),c_Transitive__Closure_Otrancl(T_c,V_R_2))))) # label(fact_trancl__power) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 244 (all V_k_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_k_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanAtMost__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 245 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Ocancel__semigroup__add(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_c_2) <-> V_b_2 = V_c_2))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 246 (all V_k c_Int_OBit0(V_k) != c_Int_OMin) # label(fact_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 247 (all V_B_2 all V_a_2 all T_d all V_A_2 all V_u_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_u_2),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_UN__insert__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 248 (all V_z all V_y all V_x (-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_y),V_x)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_y)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_y),V_z)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_z),V_y)) -> -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_z),V_x)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_z))))) # label(fact_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 249 (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]. 8.81/8.82 250 (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]. 8.81/8.82 251 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_d),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_f_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__sup__class_Osup(T_d)),V_f_2)),V_B_2),V_A_2)))) # label(fact_sup__SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 252 (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]. 8.81/8.82 253 (all V_x_2 all V_f_2 all V_xs_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Olinorder__class_Oinsort__insert__key(T_c,T_d,V_f_2,V_x_2,V_xs_2)))))) # label(fact_distinct__insort__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 254 (all V_Qa_2 all V_Pa_2 (hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)) | -hBOOL(V_Qa_2) | -hBOOL(V_Pa_2))) # label(help_c__fconj__1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 255 (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_a) = V_a)) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 256 (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]. 8.81/8.82 257 (all V_T_2 all V_S_2 all V_R_2 all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_R_2),V_S_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_R_2),V_T_2))) # label(fact_rel__comp__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 258 (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]. 8.81/8.82 259 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Oone__class_Oone(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)))) # label(fact_setprod_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 260 (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]. 8.81/8.82 261 (all V_z_2 all V_g_2 all V_h_2 all T_d all V_A_2 all T_c all T_e (class_Groups_Oab__semigroup__mult(T_e) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Fun_Oinj__on(T_c,T_d,V_h_2,V_A_2) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_d,c_Groups_Otimes__class_Otimes(T_e)),V_g_2),V_z_2),hAPP(c_Set_Oimage(T_c,T_d,V_h_2),V_A_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_c,c_Groups_Otimes__class_Otimes(T_e)),hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_g_2),V_h_2)),V_z_2),V_A_2))))) # label(fact_fold__image__reindex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 262 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord__class_Omax(T_c) = c_Orderings_Oord_Omax(T_c,c_Orderings_Oord__class_Oless__eq(T_c)))) # label(fact_max__ord__max) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 263 (all V_g_2 all V_f_2 all V_A_2 all T_c all T_d (class_Fields_Ofield__inverse__zero(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Rings_Oinverse__class_Odivide(T_d)),V_f_2)),V_g_2)),V_A_2)))) # label(fact_setprod__dividef) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 264 (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]. 8.81/8.82 265 (all V_d_2 all V_c_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_a_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),V_d_2))) # label(fact_Int__atLeastAtMostR2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 266 (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]. 8.81/8.82 267 (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]. 8.81/8.82 268 (all T_d all V_A_2 all V_a_2 all V_f_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_a_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),V_a_2),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2))))) # label(fact_vimageI2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 269 (all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) <-> (all B_x hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_f_2),B_x)) = B_x))) # label(fact_surj__iff__all) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 270 (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]. 8.81/8.82 271 (all V_Pa_2 all T_c c_List_Olist__all(T_c,V_Pa_2,c_List_Olist_ONil(T_c))) # label(fact_list__all__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 272 (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]. 8.81/8.82 273 (all V_A_2 all T_d all T_c all V_B_2 ((V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_d,c_Product__Type_Ofst(T_d,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2))) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) & (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_A_2 = hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_d,c_Product__Type_Ofst(T_d,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2)))))) # label(fact_fst__image__times) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 274 (all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (c_Relation_Osym(T_c,V_r_2) -> (c_Relation_Otrans(T_c,V_r_2) -> c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2))))) # label(fact_equivI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 275 (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]. 8.81/8.82 276 (all V_Ga_2 all V_ts_2 all V_G_H_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_G_H_2,V_ts_2) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_G_H_2) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2)))) # label(fact_cut) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 277 (all V_z_2 all V_w_2 all T_c (class_Int_Onumber__ring(T_c) -> (-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_c) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_c),c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_Groups_Otimes__class_Otimes(T_c),V_z_2))),c_COMBI(T_c))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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]. 8.81/8.82 278 (all V_A_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))),hAPP(c_Finite__Set_Ocard(T_d),V_A_2))) -> -c_Fun_Oinj__on(T_d,T_c,V_f_2,V_A_2))) # label(fact_pigeonhole) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 279 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Int_Oring__1__class_OInts(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_c))))))) # label(fact_Ints__diff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 280 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_B_2 = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) -> (V_g_2 = hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e),V_h_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_e),V_g_2),V_A_2)))))) # label(fact_setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 281 (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),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_a))) # label(fact_min__max_Oinf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 282 (all V_f_2 all V_A_2 all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_c,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]. 8.81/8.82 283 (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]. 8.81/8.82 284 (all V_xs_2 all V_f_2 all T_c all V_x_2 all V_Pa_2 all T_d (class_Orderings_Olinorder(T_d) -> (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))) # label(fact_filter__insort__triv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 285 (all V_xs_2 all T_c c_Random_Oselect(T_c,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_c,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_c)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_Code__Numeral_Ocode__numeral),c_Product__Type_OPair(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Onth(T_c),V_xs_2)),c_Code__Numeral_Onat__of)))) # label(fact_select__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 286 (all V_x_2 all V_g_2 all V_f_2 all T_e all T_c all T_d hAPP(hAPP(c_Fun_Ocomp(T_d,T_c,T_e,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]. 8.81/8.82 287 (all V_f_2 all V_a_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)))))) # label(fact_setprod__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 288 (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]. 8.81/8.82 289 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_c),V_B_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_PowD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 290 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Oappend(T_d),V_xs_2),V_ys_2))) # label(fact_map__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 291 (all V_f_2 all V_A_2 all T_c (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)),hAPP(c_Set_Oimage(T_c,T_c,V_f_2),V_A_2)),V_A_2)) -> (c_Fun_Oinj__on(T_c,T_c,V_f_2,V_A_2) -> hAPP(c_Set_Oimage(T_c,T_c,V_f_2),V_A_2) = V_A_2)))) # label(fact_endo__inj__surj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 292 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Relation_OImage(T_c,T_c,c_Relation_OId__on(T_c,V_A_2)),V_B_2)) # label(fact_Image__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 293 (all V_xs_2 all V_Pa_2 all T_c (V_xs_2 = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_filter__id__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 294 (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_n = V_m))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 295 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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),V_B_2) = V_A_2)) # label(fact_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 296 (all V_b all V_a all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> (V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) -> 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]. 8.81/8.82 297 (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]. 8.81/8.82 298 (all V_Pa_2 all V_xs_2 all T_c ((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_c)),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i))))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_all__set__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 299 (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_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)) = hAPP(c_Nat_OSuc,hAPP(c_Int_Onat,V_z)))) # label(fact_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 300 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Int__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 301 (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_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))) = 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)))) # label(fact_atLeastLessThan__add__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 302 (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]. 8.81/8.82 303 (all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2))) -> -(all B_X (hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),B_X)) -> -hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_X),V_C_2)))))) # label(fact_UnionE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 304 (all V_a_2 all V_w_2 all V_b_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_a_2)) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_b_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))))) # label(fact_divide__le__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 305 (all V_B_2 all V_A_2 all T_c (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) & V_B_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> hAPP(hAPP(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__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 306 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = hAPP(c_List_Orotate1(T_c),V_xs_2) <-> V_xs_2 = c_List_Olist_ONil(T_c))) # label(fact_rotate1__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 307 (all V_Pa_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> c_List_Olinorder__class_Osorted(T_c,c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))))) # label(fact_sorted__takeWhile) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 308 (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]. 8.81/8.82 309 (all V_xs_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Oplus__class_Oplus(T_c)),V_f_2)),V_g_2)),V_xs_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_xs_2))))) # label(fact_listsum__addf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 310 (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]. 8.81/8.82 311 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_a -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_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]. 8.81/8.82 312 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))))) # label(fact_neg__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 313 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),hAPP(c_List_Ofilter(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2)),V_xs_2))) # label(fact_length__filter__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 314 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oord(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))))) # label(fact_le__funD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 315 (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]. 8.81/8.82 316 (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]. 8.81/8.82 317 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 8.81/8.82 318 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),c_HOL_OEx(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d)),c_fequal)),V_B_2)))))) # label(fact_INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 319 (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]. 8.81/8.82 320 (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]. 8.81/8.82 321 (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]. 8.81/8.82 322 (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]. 8.81/8.82 323 (all V_i_2 all V_f_2 all V_x_2 all T_c all T_d c_Lazy__Sequence_Ohb__bind(T_d,T_c,hAPP(V_x_2,V_i_2),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c))),V_f_2),V_i_2)) = hAPP(c_New__DSequence_Oneg__bind(T_d,T_c,V_x_2,V_f_2),V_i_2)) # label(fact_New__DSequence_Oneg__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 324 (all V_i_2 all V_yq_2 all V_xq_2 all T_c c_Lazy__Sequence_Oappend(T_c,hAPP(V_xq_2,V_i_2),hAPP(V_yq_2,V_i_2)) = hAPP(c_New__DSequence_Opos__union(T_c,V_xq_2,V_yq_2),V_i_2)) # label(fact_New__DSequence_Opos__union__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 325 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,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]. 8.81/8.82 326 (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]. 8.81/8.82 327 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)))) # label(fact_DiffD2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.82 328 (all V_i_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,hAPP(c_Int_Onat,V_i_2))) <-> (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_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (all B_n (hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) = V_i_2 -> hBOOL(hAPP(V_Pa_2,B_n)))))) # label(fact_split__nat) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 329 (all V_f_2 all V_x_2 all T_c all T_d 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_d,T_c,V_x_2,V_f_2),V_i_2) = hAPP(c_Lazy__Sequence_Obind(T_d,T_c,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_d,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_c)),V_f_2),V_i_2))) & (V_i_2 = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> c_Lazy__Sequence_Oempty(T_c) = hAPP(c_New__DSequence_Opos__decr__bind(T_d,T_c,V_x_2,V_f_2),V_i_2)))) # label(fact_New__DSequence_Opos__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 330 (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]. 8.81/8.83 331 (all V_r_2 all T_c (c_Relation_Oirrefl(T_c,V_r_2) <-> (all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2))))) # label(fact_irrefl__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 332 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),V_b_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_c_2)))) # label(fact_Int__atLeastAtMostL2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 333 (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]. 8.81/8.83 334 (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_Osmall__lazy_H(V_d_2,V_i_2) = c_Lazy__Sequence_Oempty(tc_Int_Oint)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d_2),V_i_2)) -> c_Lazy__Sequence_Osmall__lazy_H(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))))))) # label(fact_small__lazy_H_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 335 (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)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Int_Oint) <-> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m_2 & c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 336 (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_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> (hAPP(hAPP(c_List_Oappend(T_c),V_us_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_vs_2)) = V_xs_2 -> (hAPP(hAPP(c_List_Oappend(T_c),V_us_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_vs_2)) = V_ys_2 -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))))))) # label(fact_listrel1I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 337 (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]. 8.81/8.83 338 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 339 (all V_n V_n = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) # label(fact_max__0L) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 340 (all V_y all V_x (hAPP(c_Nat_OSuc,V_y) = hAPP(c_Nat_OSuc,V_x) -> V_x = V_y)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 341 (all V_n_2 all V_bs_2 all V_as_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_as_2,V_bs_2) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_as_2),hAPP(hAPP(c_List_Odrop(T_d),V_n_2),V_bs_2)))) # label(fact_list__all2__dropI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 342 (all 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))))) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)) # label(fact_int__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 343 (all V_x_2 all V_A_2 all V_L_2 all T_e all T_c all T_d c_Partial__Function_Ofun__lub(T_d,T_c,T_e,V_L_2,V_A_2,V_x_2) = hAPP(V_L_2,hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(tc_fun(T_e,T_d))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool),tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool),T_d),hAPP(c_COMBS(tc_fun(T_e,T_d),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_e,T_d)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_e,T_d),tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_e,T_d))),V_A_2)))),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_fun(T_e,T_d),T_d),tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_e,T_d),T_d),tc_fun(tc_fun(T_e,T_d),tc_HOL_Obool)),T_d),c_COMBB(T_d,tc_HOL_Obool,tc_fun(T_e,T_d))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_e,T_d),T_e,T_d),c_COMBI(tc_fun(T_e,T_d))),V_x_2))))))) # label(fact_fun__lub__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 344 (all V_B_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_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)),V_A_2),V_B_2)),V_C_2))))) # label(fact_Un__least) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 345 (all V_B_2 all V_A_2 all T_c ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_X),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),B_X),V_B_2)))) -> (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2))))) # label(fact_Inter__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 346 (all T_c all V_x_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,V_x_2)) -> hBOOL(hAPP(V_Pa_2,c_Hilbert__Choice_OEps(T_c,V_Pa_2))))) # label(fact_someI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 347 (all V_x all V_y ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y) = c_Nat__Transfer_Otsub(V_x,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) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_tsub__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 348 (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]. 8.81/8.83 349 (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]. 8.81/8.83 350 (all V_z c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),V_z)) # label(fact_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 351 (all V_B_2 all V_X2_2 all V_X1_2 all V_f_2 all T_e all V_r2_2 all V_A2_2 all T_d all V_r1_2 all V_A1_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_d,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_c,T_d,tc_fun(T_e,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X1_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A1_2),V_r1_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_d,tc_HOL_Obool)),V_X2_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_d),V_A2_2),V_r2_2))) -> ((all B_x1 (hBOOL(hAPP(hAPP(c_member(T_c),B_x1),V_A1_2)) -> (all B_x2 (hBOOL(hAPP(hAPP(c_member(T_d),B_x2),V_A2_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(V_f_2,B_x1),B_x2)),V_B_2)))))) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_e,tc_HOL_Obool)),V_X1_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,tc_HOL_Obool)),V_X2_2)),V_f_2))),V_B_2))))))))) # label(fact_UN__equiv__class__type2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 352 (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]. 8.81/8.83 353 (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]. 8.81/8.83 354 (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]. 8.81/8.83 355 (all V_B_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_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),V_B_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)))) # label(fact_insert__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 356 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),V_A_2))))))) # label(fact_Inf__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 357 (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]. 8.81/8.83 358 (all V_x_2 all V_A_2 all T_d all T_c (class_Groups_Ouminus(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(V_A_2,V_x_2)) = hAPP(hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,T_c)),V_A_2),V_x_2))) # label(fact_fun__Compl__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 359 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> c_Fun_Oinj__on(T_c,T_d,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_inj__on__diff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 360 (all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) <-> c_Groups_Ozero__class_Ozero(T_c) = c_Divides_Odiv__class_Omod(T_c,V_b_2,V_a_2)))) # label(fact_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 361 (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]. 8.81/8.83 362 (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]. 8.81/8.83 363 (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]. 8.81/8.83 364 (all V_fs_2 all T_c c_Wellfounded_Owf(T_c,c_List_Omeasures(T_c,V_fs_2))) # label(fact_wf__measures) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 365 (all V_Y_2 all V_X_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> V_Y_2 = V_X_2 | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_Y_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_quotient__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 366 (all V_b all V_a 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_Odivmod__int(V_a,V_b))) # label(fact_div__int__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 367 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)))))) # label(fact_Max__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 368 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),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)),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)))))) # label(fact_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 369 (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]. 8.81/8.83 370 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2)))) # label(fact_atLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 371 (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]. 8.81/8.83 372 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__int_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 373 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 374 (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]. 8.81/8.83 375 (all V_r_2 all T_d all T_c hAPP(c_Relation_ORange(T_d,T_c),V_r_2) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) # label(fact_Domain__converse) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 376 (all V_M_2 all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_M_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_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_M_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_M_2))) # label(fact_INT__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 377 (all V_xs_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_A_2,B_x)))) <-> hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_xs_2)))) # label(fact_in__listsp__conv__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 378 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,V_d_2))) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_d_2))))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 379 (all 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))) = c_Nitpick_Ozero__frac(T_a)) # label(fact_Nitpick_Ozero__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 380 (all V_a_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 381 (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))) & 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__conj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 382 (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 = hAPP(c_Nat_OSuc,V_n_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]. 8.81/8.83 383 (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(c_List_Orev(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),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)))) # label(fact_zip__rev) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 384 (all V_A_2 all T_c (class_Finite__Set_Ofinite(T_c) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_finite__code) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 385 (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]. 8.81/8.83 386 (all V_r_2 all V_xs_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),c_List_Olist_ONil(T_c))),c_List_Olex(T_c,V_r_2)))) # label(fact_Nil2__notin__lex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 387 (all V_a_2 all V_S_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) & class_Orderings_Olinorder(T_c) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),B_x),V_a_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Complete__Lattice_OInf__class_OInf(T_c,V_S_2)),V_a_2))))) # label(fact_Inf__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 388 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> V_y_2 = V_x_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 389 (all V_A_2 all V_l_2 all V_x_2 all T_c c_List_Osublist(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_l_2),V_A_2) = hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(hAPP(c_If(tc_List_Olist(T_c)),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_c),V_x_2),c_List_Olist_ONil(T_c))),c_List_Olist_ONil(T_c))),c_List_Osublist(T_c,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))))) # label(fact_sublist__Cons) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 390 (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]. 8.81/8.83 391 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_atLeastatMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 392 (all V_vs_2 all V_us_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (V_vs_2 = V_us_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_us_2),V_vs_2)),c_List_Olistrel1(T_c,V_r_2))) & V_ys_2 = V_xs_2 -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_us_2)),hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_vs_2))),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_append__listrel1I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 393 (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,V_a,c_Groups_Oone__class_Oone(T_a)))) # label(fact_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 394 (all V_x_2 all T_c c_Hilbert__Choice_OEps(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)) = V_x_2) # label(fact_some__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 395 (all V_com_H_2 all V_fun_H_2 all V_loc_H_2 c_Com_Ocom_OSKIP != c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 396 (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_2 = V_z_H_2 <-> hAPP(c_Int_Onat,V_z_H_2) = hAPP(c_Int_Onat,V_z_2))))) # label(fact_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 397 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setprod__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 398 (all V_a_2 all V_b_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_b_2),V_A_2))))) # label(fact_min__max_Ofold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 399 (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_OBit1(V_k_2)),c_Int_OBit1(V_l_2))))) # label(fact_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 400 (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]. 8.81/8.83 401 (all V_Pa_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))))) # label(fact_distinct__filter) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 402 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_y_2)))))) # label(fact_atLeast__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 403 (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]. 8.81/8.83 404 (all V_y_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> V_y_2 = hAPP(V_f_2,hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),V_y_2))))) # label(fact_f__the__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 405 (all V_xs all T_a V_xs = hAPP(c_List_Orev(T_a),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__rev__ident) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 406 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2)))) # label(fact_ivl__disj__un_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 407 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_INT__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 408 (all V_a_2 all V_b_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_b_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_min__max_Ofold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 409 (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]. 8.81/8.83 410 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 411 (all V_I_2 all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2)))))) # label(fact_notin__set__sublistI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 412 (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]. 8.81/8.83 413 (all V_b all V_a 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,V_a,V_b)),V_b)) # label(fact_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 414 (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]. 8.81/8.83 415 (all V_a all V_b all T_a (class_Rings_Odivision__ring(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> 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]. 8.81/8.83 416 (all V_y_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_y_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2) = V_y_2))) # label(fact_fun__left__comm_Ofold__equality) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 417 (all V_c_2 all V_b_2 all V_A_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (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)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_b_2),V_c_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_c_2),V_A_2))))))) # label(fact_inf__le__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 418 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2),V_z_2),V_A_2) = hAPP(hAPP(V_F_2,V_g_2),V_A_2)))) # label(fact_folding__image_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 419 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_folding_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 420 (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]. 8.81/8.83 421 (all V_xs_2 all V_Pa_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_length__dropWhile__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 422 (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]. 8.81/8.83 423 (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]. 8.81/8.83 424 (all V_r_2 all V_x_2 all V_y_2 all T_c (-hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) & c_Wellfounded_Oacyclic(T_c,V_r_2) <-> c_Wellfounded_Oacyclic(T_c,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),V_r_2)))) # label(fact_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 425 (all V_r_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_r_2),c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_r_2)) # label(fact_r__comp__rtrancl__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 426 (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]. 8.81/8.83 427 (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]. 8.81/8.83 428 (all V_ys_2 all V_xs_2 all V_a_2 all V_f_2 all T_d all T_c hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),hAPP(hAPP(c_List_Oappend(T_d),V_xs_2),V_ys_2)) = hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),V_xs_2)),V_ys_2)) # label(fact_foldl__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 429 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)),V_k) = 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))) # label(fact_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 430 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (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_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))))) # label(fact_rtrancl_Ortrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 431 (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]. 8.81/8.83 432 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> c_List_Oremove1(T_c,V_x_2,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2)) = V_xs_2)) # label(fact_remove1__insort) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 433 (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),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_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))))) # label(fact_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 434 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 435 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_m_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 436 (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]. 8.81/8.83 437 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_c,V_r_2) -> c_Wellfounded_Owf(T_c,V_r_2)))) # label(fact_finite__acyclic__wf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 438 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_u_2)))) # label(fact_ivl__disj__int_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 439 (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]. 8.81/8.83 440 (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]. 8.81/8.83 441 (all V_r_2 all T_c c_Relation_Oirrefl(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))) # label(fact_irrefl__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 442 (all V_x_2 all T_c hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_iso__tuple__UNIV__I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 443 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) -> V_y_2 = V_x_2))) # label(fact_injD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 444 (all V_ys all V_xs all T_a c_List_Oconcat(T_a,hAPP(hAPP(c_List_Oappend(tc_List_Olist(T_a)),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oconcat(T_a,V_xs)),c_List_Oconcat(T_a,V_ys))) # label(fact_concat__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 445 (all V_g_2 all T_e all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_d,T_e,V_g_2,hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) -> c_Fun_Oinj__on(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_g_2),V_f_2),V_A_2)))) # label(fact_comp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 446 (all V_a_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_2)),V_r_2)))) # label(fact_wf__not__refl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 447 (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]. 8.81/8.83 448 (all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_a_2)) # label(fact_Sup__singleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 449 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))))) # label(fact_Inf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 450 (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]. 8.81/8.83 451 (all V_d_2 all V_m_2 (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> (exists B_q hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_d_2),B_q) = V_m_2))) # label(fact_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 452 (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]. 8.81/8.83 453 (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_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = 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)))) # label(fact_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 454 (all V_A_2 all V_I_2 all T_d all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2)),V_A_2))) # label(fact_Int__UN__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 455 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2)))))) # label(fact_finite__Plus) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 456 (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,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))),V_m) = 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)))) # label(fact_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 457 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(T_c,c_Transitive__Closure_Otrancl(T_c,V_r_2)))) # label(fact_wf__trancl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 458 (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]. 8.81/8.83 459 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),c_Transitive__Closure_Ortrancl(T_c,V_s_2))))) # label(fact_rtrancl__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 460 (all V_B_2 all T_c 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_c,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1I1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 461 (all V_R_2 all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),V_R_2) = hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_c,c_Product__Type_Ofst(T_c,T_d)),V_R_2)) # label(fact_fst__eq__Domain) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 462 (all V_x_2 all V_r_2 all T_c (c_Predicate_Oreflp(T_c,V_r_2) -> hBOOL(hAPP(hAPP(V_r_2,V_x_2),V_x_2)))) # label(fact_reflpE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 463 (all V_r_2 all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)))),hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),c_List_Olist_ONil(T_c))))),c_List_Olistrel1(T_c,V_r_2))) <-> V_y_2 = V_x_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) | V_xs_2 = V_ys_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)))) # label(fact_snoc__listrel1__snoc__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 464 (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]. 8.81/8.83 465 (all V_nat_2 all V_f2_2 all V_f1_2 all T_c hAPP(V_f2_2,V_nat_2) = c_Nat_Onat_Onat__case(T_c,V_f1_2,V_f2_2,hAPP(c_Nat_OSuc,V_nat_2))) # label(fact_nat__case__Suc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 466 (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_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)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 467 (all V_s_2 all V_r_2 all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_s_2))) # label(fact_converse__Int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 468 (all V_f_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((exists B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_x != B_xa -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,B_xa)))) & c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(V_f_2,B_x) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_setsum__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 469 (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]. 8.81/8.83 470 (all V_f_2 all V_r_2 all T_c all T_d hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_d,T_d),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_d,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_d,T_d)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_d,T_d),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_d,T_d)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_d),tc_fun(T_c,tc_prod(T_d,T_d))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_d,T_d)),tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,tc_prod(T_d,T_d))),T_c),c_COMBB(T_d,tc_prod(T_d,T_d),T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_prod(T_d,T_d)),T_c),c_Product__Type_OPair(T_d,T_d)),V_f_2))),V_f_2)))),V_r_2))) = hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),V_r_2),V_f_2)) # label(fact_inv__image__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 471 (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),V_b) = 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)))) # label(fact_inf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 472 (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]. 8.81/8.83 473 (all V_A_2 all T_c c_Relation_Oantisym(T_c,c_Relation_OId__on(T_c,V_A_2))) # label(fact_antisym__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 474 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2) = c_Groups_Oone__class_Oone(T_d)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(V_b_2,V_a_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2))))) # label(fact_setprod__delta_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 475 (all V_A_2 all V_B_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_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_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2)))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 476 (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(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.83 477 (all V_Pa_2 all V_B_2 all V_A_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),hAPP(V_B_2,B_x))) -> hBOOL(hAPP(V_Pa_2,B_xa)))))))) # label(fact_ball__UN) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 478 (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]. 8.81/8.84 479 (all V_y_2 all V_x_2 all T_c (class_Rings_Ocomm__ring__1(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),V_y_2))))) # label(fact_minus__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 480 (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),c_Divides_Odivmod__int(V_a,V_b))))) # label(fact_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 481 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))) # label(fact_divide__minus1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 482 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oordered__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_b_2),V_a_2)),V_e_2)),V_d_2)))))) # label(fact_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 483 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 <-> hAPP(c_List_Oset(T_c),V_xs_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_set__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 484 (all V_y_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_f_2),V_y_2)) = V_y_2)) # label(fact_surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 485 (all V_x all V_m 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_m,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m),V_x)) # label(fact_replicate__add) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 486 (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)) -> 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),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]. 8.81/8.84 487 (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_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))) & (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),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))))) # label(fact_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 488 (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]. 8.81/8.84 489 (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]. 8.81/8.84 490 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),c_List_Oset(T_c)),V_A_2))) -> c_Fun_Oinj__on(tc_List_Olist(T_c),tc_List_Olist(T_d),hAPP(c_List_Omap(T_c,T_d),V_f_2),V_A_2))) # label(fact_inj__on__mapI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 491 (all V_B_2 all V_a_2 all T_d all V_A_2 all V_u_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_u_2),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_INT__insert__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 492 (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]. 8.81/8.84 493 (all V_B_2 all V_A_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_B_2)))))))) # label(fact_fold1__Un2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 494 (all V_f_2 all T_e all V_A_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_e),hAPP(hAPP(V_f_2,V_a_2),V_b_2)),hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_e,hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_f_2)),V_A_2))))) # label(fact_pair__imageI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 495 (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]. 8.81/8.84 496 (all V_xs_2 all V_x_2 all V_y_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) -> V_y_2 = V_x_2 | hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__ConsD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 497 (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]. 8.81/8.84 498 (all V_s_2 all V_r_2 all T_c hAPP(c_Relation_OField(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OField(T_c),V_r_2)),hAPP(c_Relation_OField(T_c),V_s_2))) # label(fact_Field__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 499 (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_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 500 (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) | V_b = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 501 (all V_g_2 all V_x_2 all T_e all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),V_g_2))))) # label(fact_folding_Ocommute__left__comp_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 502 (all V_b_2 all V_a_2 all T_c (class_Rings_Osemiring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Nat_Osemiring__1__class_ONats(T_c))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Nat_Osemiring__1__class_ONats(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_c))))))) # label(fact_Nats__mult) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 503 (all V_a_2 all V_b_2 all T_c (class_Rings_Odivision__ring(T_c) -> (V_b_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2) = c_Groups_Oone__class_Oone(T_c) <-> V_b_2 = V_a_2)))) # label(fact_right__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 504 (all V_com2_2 all V_com1_2 all V_fun_2 c_Com_Ocom_Ocom__size(c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com1_2)),c_Com_Ocom_Ocom__size(V_com2_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_com_Osize_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 505 (all V_r_2 all V_b_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(c_Relation_OImage(T_c,T_d,V_r_2),V_A_2)))))) # label(fact_rev__ImageI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 506 (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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),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_diff__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 507 (all V_A_2 all T_c 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_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_A_2)) # label(fact_Compl__disjoint2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 508 (all V_b_2 all V_a_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_c,T_d,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)) -> hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))))) # label(fact_curryD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 509 (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]. 8.81/8.84 510 (all V_x_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_x_2)) # label(fact_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 511 (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]. 8.81/8.84 512 (all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_fequal,V_y_2))) <-> V_x_2 = V_y_2)) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 513 (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]. 8.81/8.84 514 (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]. 8.81/8.84 515 (all T_e all T_d all T_c 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_c,T_d,T_e,V_r_2,V_s_2),V_a_2),V_c_2))))) # label(fact_pred__comp_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 516 (all V_Qa_2 all V_Pa_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_Set_OCollect(T_c),V_Pa_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_OCollect(T_d),V_Qa_2))) = hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),V_Pa_2))),V_Qa_2)))) # label(fact_Collect__split) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 517 (all V_n V_n = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 518 (all V_l_2 all 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)) <-> -(-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2))))) # label(fact_dvd_OgreaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 519 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) <-> V_b_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))) # label(fact_equation__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 520 (all V_M_2 all V_A_2 all T_d all V_u_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_u_2),hAPP(V_M_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_u_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),V_A_2),V_M_2)))))) # label(fact_le__INF__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 521 (all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),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))))) # label(fact_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 522 (all V_Pa_2 all V_A_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_A_2)) -> (hBOOL(hAPP(V_Pa_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> ((all B_a all B_F (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_F)) -> (hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),B_a),B_F)) -> (hBOOL(hAPP(V_Pa_2,B_F)) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_F)))))))) -> hBOOL(hAPP(V_Pa_2,V_F_2))))))) # label(fact_finite__subset__induct) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 523 (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]. 8.81/8.84 524 (all V_g_2 all V_f_2 all T_d all T_c all T_f all T_e hAPP(c_Fun_Ocomp(tc_prod(T_e,T_f),T_e,tc_prod(T_c,T_d),c_Product__Type_Ofst(T_e,T_f)),c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,V_f_2,V_g_2)) = hAPP(c_Fun_Ocomp(T_c,T_e,tc_prod(T_c,T_d),V_f_2),c_Product__Type_Ofst(T_c,T_d))) # label(fact_fst__comp__map__pair) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 525 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,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_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) & (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_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_UN__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 526 (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]. 8.81/8.84 527 (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]. 8.81/8.84 528 (all V_y_2 all V_x_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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)))) # label(fact_less__than__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 529 (all V_s_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2)),hAPP(c_Relation_ODomain(T_c,T_d),V_s_2))))) # label(fact_Domain__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 530 (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]. 8.81/8.84 531 (all V_b_2 all V_a_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_d,V_r_2),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(V_r_2,V_b_2),V_a_2)))) # label(fact_conversep__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 532 (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]. 8.81/8.84 533 (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]. 8.81/8.84 534 (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]. 8.81/8.84 535 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> ((exists B_f c_Fun_Obij__betw(T_c,T_d,B_f,V_A_2,V_B_2)) <-> hAPP(c_Finite__Set_Ocard(T_d),V_B_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_BIJ) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 536 (all V_B_2 all V_A_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_d,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(c_Relation_ORange(T_d,T_c),V_A_2)),hAPP(c_Relation_ORange(T_d,T_c),V_B_2))))) # label(fact_Range__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 537 (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]. 8.81/8.84 538 (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]. 8.81/8.84 539 (all V_x_2 all V_g_2 all V_f_2 all T_f all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),hAPP(c_Fun_Ocomp(T_f,tc_fun(T_e,T_c),T_d,V_f_2),V_g_2)),V_x_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_x_2))),hAPP(c_Product__Type_Osnd(T_d,T_e),V_x_2))) # label(fact_split__comp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 540 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)) <-> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 541 (all V_Qa_2 all T_d all T_c all V_y_2 all V_x_2 all V_Pa_2 (hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_Pa_2),V_Qa_2)) -> hBOOL(hAPP(hAPP(V_Qa_2,V_x_2),V_y_2))))) # label(fact_rev__predicate2D) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 542 (all V_ys_2 all V_xs_2 all V_x_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_ys_2) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_c,T_d)),T_d,c_List_Olist_ONil(tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d)))),T_d),c_COMBB(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(T_d))),hAPP(hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),T_d),c_List_Olist_OCons(tc_prod(T_c,T_d))),hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2)))),hAPP(c_List_Ozip(T_c,T_d),V_xs_2))),V_ys_2)) # label(fact_zip__Cons1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 543 (all V_A_2 all V_f_2 all T_d all V_z_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) <-> (exists B_x (V_z_2 = hAPP(V_f_2,B_x) & hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)))))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 544 (all V_B_2 all V_A_2 all T_c all T_d (hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(V_B_2,B_x))))) # label(fact_INTER__UNIV__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 545 (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)) -> 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)) <-> V_m_2 = hAPP(c_Int_Onat,V_w_2))) # label(fact_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 546 (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]. 8.81/8.84 547 (all V_y_2 all V_x_2 all V_b_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Oone__class_Oone(T_c)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_b_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)))))) # label(fact_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 548 (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]. 8.81/8.84 549 (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]. 8.81/8.84 550 (all V_x_2 all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_R_2),V_S_2))) <-> hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2)),V_x_2)))) # label(fact_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 551 (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]. 8.81/8.84 552 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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_2),V_B_2) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_B_2))))))))) # label(fact_Max_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 553 (all V_com_H_2 all V_fun_H_2 all V_com_2 all V_fun_2 (V_fun_H_2 = V_fun_2 & V_com_H_2 = V_com_2 <-> c_Com_Ocom_OWhile(V_fun_2,V_com_2) = c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2))) # label(fact_com_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 554 (all V_Pa_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)) = hAPP(c_Set_OCollect(T_c),hAPP(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_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2)))),V_Pa_2))) # label(fact_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 555 (all V_xs_2 all V_x_2 all V_r_2 all T_c hAPP(c_Relation_OImage(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Olistrel(T_c,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))) = c_List_Oset__Cons(T_c,hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),hAPP(c_Relation_OImage(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Olistrel(T_c,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),V_xs_2),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))))) # label(fact_listrel__Cons) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 556 (all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> hAPP(c_Set_Oimage(T_c,T_d,V_Rep_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = V_A_2)) # label(fact_type__definition_ORep__range) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 557 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),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_C_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))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))) # label(fact_Times__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 558 (all V_k all V_l ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l -> c_Divides_Opdivmod(V_k,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))) & (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]. 8.81/8.84 559 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) & class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),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]. 8.81/8.84 560 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d 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_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_UN__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 561 (all V_xs_2 all T_c c_Random_Oselect__weight(T_c,V_xs_2) = c_Random_Oselect__weight(T_c,hAPP(c_List_Ofilter(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Code__Numeral_Ocode__numeral,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_COMBK(tc_HOL_Obool,T_c)),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]. 8.81/8.84 562 (all V_xs_2 all V_f_2 all T_c all T_d c_List_Obutlast(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Obutlast(T_d,V_xs_2))) # label(fact_map__butlast) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 563 (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]. 8.81/8.84 564 (all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 565 (all V_n all V_m hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)) # label(fact_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 566 (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]. 8.81/8.84 567 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (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_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_rtrancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 568 (all V_B_2 all V_a_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_a_2),V_B_2)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 569 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(c_Fun_Ofun__upd(T_d,T_c,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]. 8.81/8.84 570 (all T_c c_Fun_Obij__betw(T_c,T_c,c_Fun_Oid(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_bij__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 571 (all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_a_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_a_2),V_B_2))) # label(fact_Union__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 572 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_A_2),V_C_2))),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_C_2))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBS(T_d,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_d),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Int__Inter__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 573 (all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)) <-> c_Wellfounded_Owf(T_c,V_r_2))) # label(fact_wfP__wf__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 574 (all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2) = V_xs_2)) # label(fact_removeAll__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 575 (all V_B_2 all V_A_2 all V_a_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_B_2,V_a_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_UN__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 576 (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)) -> (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_j_2),V_i_2)),V_u_2)),V_n_2) = V_m_2))) # label(fact_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 577 (all V_x_2 all V_A_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)))))) # label(fact_fold1__insert__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 578 (all V_B_2 all V_A_2 all V_f_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_c,tc_HOL_Obool),T_e),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_e,T_d,V_f_2),V_A_2)),V_B_2)) # label(fact_UN__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 579 (all V_X_2 all V_r_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_X_2)))))) # label(fact_finite__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 580 (all V_n all V_ys all V_xs all T_a hAPP(hAPP(c_List_Onth(T_a),V_ys),V_n) = 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))) # label(fact_nth__append__length__plus) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 581 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_a) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) # label(fact_min__max_Osup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 582 (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]. 8.81/8.84 583 (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]. 8.81/8.84 584 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__less__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 585 (all V_l c_Int_OMin != c_Int_OBit0(V_l)) # label(fact_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 586 (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]. 8.81/8.84 587 (all V_X_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) != 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__strict)))) # label(fact_smin__emptyI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 588 (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_d all T_c (c_Finite__Set_Ofolding__image(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (c_Fun_Oinj__on(T_d,T_d,V_h_2,V_A_2) -> hAPP(hAPP(V_F_2,V_g_2),hAPP(c_Set_Oimage(T_d,T_d,V_h_2),V_A_2)) = hAPP(hAPP(V_F_2,hAPP(c_Fun_Ocomp(T_d,T_c,T_d,V_g_2),V_h_2)),V_A_2))))) # label(fact_folding__image_Oreindex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 589 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) & c_List_Olist__all(T_c,V_Pa_2,V_ys_2) <-> c_List_Olist__all(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)))) # label(fact_list__all__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 590 (all V_d_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_d_2)))) # label(fact_Int__atLeastAtMostL1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 591 (all V_r_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,V_r_2)),c_Relation_OId(T_c)) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_reflcl__trancl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 592 (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]. 8.81/8.84 593 (all V_xs_2 all V_f_2 all T_e all V_a_2 all V_g_2 all T_d all T_c hAPP(c_List_Ofoldl(T_c,T_d,V_g_2,V_a_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_f_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_e,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_e,T_d),tc_fun(T_e,T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_c),tc_fun(tc_fun(T_e,T_d),tc_fun(T_e,T_c)),T_c),c_COMBB(T_d,T_c,T_e)),V_g_2)),V_f_2),V_a_2),V_xs_2)) # label(fact_foldl__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 594 (all V_xs all T_a 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)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_Osublists(T_a,V_xs))) # label(fact_length__sublists) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 595 (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]. 8.81/8.84 596 (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]. 8.81/8.84 597 (all T_c c_Nat_Ocompow(T_c,T_c) = c_Nat_Ofunpow(T_c)) # label(fact_funpow__code__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 598 (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]. 8.81/8.84 599 (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]. 8.81/8.84 600 (all V_w all V_v 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)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))) # label(fact_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 601 (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]. 8.81/8.84 602 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) <-> c_List_Olist__ex(T_c,V_Pa_2,V_ys_2) | c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 603 (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))) -> 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),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_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 604 (all V_xs_2 all T_c all V_Pa_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_Pa_2,hAPP(hAPP(c_List_Onth(T_c),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_c)),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_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))))))) # label(fact_length__takeWhile__less__P__nth) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 605 (all V_B_2 all V_b_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(V_B_2,V_a_2))) -> hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_UN__I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 606 (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]. 8.81/8.84 607 (all V_x_2 all V_f_2 all T_c all T_d all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_k_2 -> hAPP(c_Random_Oiterate(T_d,T_c,V_k_2,V_f_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_c,tc_prod(T_d,T_c)),hAPP(V_f_2,V_x_2)),c_Random_Oiterate(T_d,T_c,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))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(c_Product__Type_OPair(T_d,T_c),V_x_2) = hAPP(c_Random_Oiterate(T_d,T_c,V_k_2,V_f_2),V_x_2)))) # label(fact_iterate_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 608 (all V_k V_k = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k)) # label(fact_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 609 (all V_B_2 all V_A_2 all T_c (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(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 610 (all V_R_2 all T_d all T_c V_R_2 = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_d),V_R_2),c_Relation_OId(T_d))) # label(fact_R__O__Id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 611 (all V_y_2 all V_x_2 all V_Pa_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),V_x_2)) -> hBOOL(hAPP(V_Pa_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_Pa_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) <-> hBOOL(hAPP(V_Pa_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)))))) # label(fact_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 612 (all V_B_2 all V_A_2 all T_d all T_c c_Nitpick_Oprod(T_c,T_d,V_A_2,V_B_2) = hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),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(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))))) # label(fact_Nitpick_Oprod__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 613 (all V_f_2 all V_xs_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))))) # label(fact_listsum__distinct__conv__setsum__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 614 (all V_f_2 all T_d all V_Pa_2 all T_c (c_FunDef_Oreduction__pair(T_c,V_Pa_2) -> c_FunDef_Oreduction__pair(T_d,hAPP(hAPP(c_FunDef_Orp__inv__image(T_c,T_d),V_Pa_2),V_f_2)))) # label(fact_rp__inv__image__rp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 615 (all V_com_2 all V_fun_2 all V_loc_2 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2))) # label(fact_com_Osize_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 616 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_minus__le__self__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 617 (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)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)))) # label(fact_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 618 (all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> V_x_2 = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Inf__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 619 (all V_B_2 all V_k_2 all V_A_2 all T_c ((all B_n hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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_c,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_c,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_c,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]. 8.81/8.84 620 (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]. 8.81/8.84 621 (all V_xs_2 all V_n_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__takeD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 622 (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]. 8.81/8.84 623 (all V_Pa_2 all T_c ((all B_x -hBOOL(hAPP(V_Pa_2,B_x))) <-> hAPP(c_Set_OCollect(T_c),V_Pa_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 624 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_B_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2)))))) # label(fact_INF__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 625 (all V_b_2 all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_b_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_a_2),V_b_2))))))) # label(fact_min__max_Ofold__inf__le__inf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 626 (all V_f2_2 all V_f1_2 all T_c hAPP(c_Nat_Onat_Onat__rec(T_c,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]. 8.81/8.84 627 (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]. 8.81/8.84 628 (all V_r_2 all V_xs_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_xs_2)),c_List_Olistrel1(T_c,V_r_2)))) # label(fact_not__Nil__listrel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 629 (all V_R_2 all V_n_2 all T_c hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2)) # label(fact_relpow_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.84 630 (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]. 8.81/8.84 631 (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]. 8.81/8.85 632 (all 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)) = V_A_2) # label(fact_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 633 (all V_x_2 all V_A_2 all T_c (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(c_Finite__Set_Ocard(T_c),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))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),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_card__Diff__singleton__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 634 (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]. 8.81/8.85 635 (all V_R_2 all T_c hAPP(c_Relation_ORange(T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_R_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Range__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 636 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),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_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(V_h_2,c_Big__Operators_Olattice__class_OInf__fin(T_c,V_N_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_Inf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 637 (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]. 8.81/8.85 638 (all V_ys_2 all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2))))))) # label(fact_rtrancl__listrel1__ConsI2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 639 (all V_A_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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)))) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 640 (all V_s_2 all V_s_H_2 all T_e all V_r_2 all V_r_H_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_H_2),V_r_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool)),V_s_H_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_H_2),V_s_H_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_2),V_s_2)))))) # label(fact_rel__comp__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 641 (all V_a_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,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_d,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))),hAPP(hAPP(c_Set_Oinsert(T_c),c_HOL_OThe(T_c,hAPP(hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_a_2))),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_inj__vimage__singleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 642 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> hAPP(c_Finite__Set_Ocard(T_d),V_B_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2))) # label(fact_bij__betw__same__card) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 643 (all V_X_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_C_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),V_X_2))))) # label(fact_InterD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 644 (all V_c all V_b all V_a 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) = 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))) # label(fact_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 645 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,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_Max_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 646 (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]. 8.81/8.85 647 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2))))) # label(fact_minus__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 648 (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]. 8.81/8.85 649 (all V_k c_Int_OBit0(V_k) = c_Int_Opred(c_Int_OBit1(V_k))) # label(fact_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 650 (all V_A_2 all T_c hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),V_A_2)) = c_List_Olists(T_c,V_A_2)) # label(fact_lists__eq__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 651 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 652 (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]. 8.81/8.85 653 (all V_S_2 all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_S_2)),V_R_2)) -> c_FunDef_Oreduction__pair(T_c,hAPP(hAPP(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)),V_R_2),V_S_2))))) # label(fact_reduction__pairI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 654 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2))) # label(fact_Compl__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 655 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_B_2)) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Min__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 656 (all V_f_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,T_c),c_Int_Onat),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Int_Oint),V_f_2),V_A_2))) # label(fact_transfer__int__nat__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 657 (all V_z_2 (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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 658 (all V_v_2 all V_n_2 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)) = 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))))) # label(fact_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 659 (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]. 8.81/8.85 660 (all V_z_2 all V_f_2 all T_d all T_c hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_z_2))) # label(fact_fold__graph_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 661 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)) = hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_x_2)))) # label(fact_image__uminus__greaterThan) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 662 (all T_c c_Wellfounded_Owf(T_c,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))) # label(fact_wf__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 663 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Relation_OField(T_c),V_r_2))))) # label(fact_finite__Field) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 664 (all V_A_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)))) # label(fact_Sup__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 665 (all V_a_2 all T_c (class_Int_Oring__char__0(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2) <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2)))) # label(fact_Ints__double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 666 (all V_s2_2 all V_s1_2 all T_d all T_c hAPP(c_Lazy__Sequence_Obind(T_c,tc_prod(T_c,T_d),V_s1_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_c,T_d))),tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_c,T_d)),T_c),c_Lazy__Sequence_Obind(T_d,tc_prod(T_c,T_d),V_s2_2)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_c,T_d))),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_Lazy__Sequence_Olazy__sequence(tc_prod(T_c,T_d)),T_d),c_Lazy__Sequence_Osingle(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))) = c_Lazy__Sequence_Oproduct(T_c,T_d,V_s1_2,V_s2_2)) # label(fact_) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 667 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_x = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)))) # label(fact_min__max_Oinf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 668 (all V_x_2 all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),V_x_2)) = hAPP(c_Product__Type_Ofst(T_c,T_e),V_x_2)) # label(fact_fst__apsnd) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 669 (all T_c hAPP(c_Fun_Ocomp(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,T_c,c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint)),c_Nitpick_ORep__Frac(T_c)) = c_Nitpick_Onum(T_c)) # label(fact_Nitpick_Onum__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 670 (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]. 8.81/8.85 671 (all V_r_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_List_Olist_ONil(T_c))),c_List_Olistrel(T_c,V_r_2)))) # label(fact_listrel_ONil) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 672 (all V_xs_2 all V_x_2 all T_c hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_List_Oset_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 673 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Predicate_OPowp(T_c,V_A_2),V_B_2)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_B_2)) -> hBOOL(hAPP(V_A_2,B_x)))))) # label(fact_Powp__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 674 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> c_Finite__Set_Ofun__left__comm(T_e,T_d,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,T_d),T_e),V_f_2),V_g_2)))) # label(fact_fun__left__comm_Ofun__left__comm__apply) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 675 (all V_y_2 all V_x_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2) = c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),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]. 8.81/8.85 676 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Oab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_c_2),V_d_2) -> (V_d_2 = V_c_2 <-> V_a_2 = V_b_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 677 (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)) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a))) # label(fact_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 678 (all V_com2_2 all V_com1_2 all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OSemi(V_com1_2,V_com2_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I47_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 679 (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_x_2 = V_y_2)) # label(fact_transfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 680 (all V_t_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(c_List_Orev(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) -> c_List_OtakeWhile(T_d,hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_d),hAPP(c_Orderings_Oord__class_Oless(T_c),V_t_2)),V_f_2),V_xs_2) = hAPP(c_List_Ofilter(T_d,hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_d),hAPP(c_Orderings_Oord__class_Oless(T_c),V_t_2)),V_f_2)),V_xs_2)))) # label(fact_filter__equals__takeWhile__sorted__rev) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 681 (all V_S_2 all V_R_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_R_2)),c_Transitive__Closure_Ortrancl(T_c,V_S_2))),c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_rtrancl__Un__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 682 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_d,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_Ovimage(T_c,T_d,V_f_2),V_B_2)),V_A_2))))) # label(fact_vimage__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 683 (all V_Qa_2 all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Qa_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(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_fdisj),V_Pa_2)),V_Qa_2)))))) # label(fact_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 684 (all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_setprod__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 685 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2))))))) # label(fact_min__max_OSup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 686 (all V_a (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_a -> 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]. 8.81/8.85 687 (all V_d_2 all V_c_2 all V_b_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)))) # label(fact_Int__atLeastAtMostR1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 688 (all T_c (class_Lattices_Osemilattice__sup(T_c) & class_Orderings_Olinorder(T_c) -> c_Lattices_Osemilattice__sup__class_Osup(T_c) = c_Orderings_Oord__class_Omax(T_c))) # label(fact_sup__max) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 689 (all V_ts_2 all V_t_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),V_t_2),V_ts_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),V_t_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) & c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2))) # label(fact_derivs__insertD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 690 (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]. 8.81/8.85 691 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_divide__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 692 (all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> (hAPP(c_Set_Oimage(T_c,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_c,tc_HOL_Obool)) -> c_Fun_Oinj__on(T_c,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_finite__UNIV__surj__inj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 693 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBS(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_c),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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal))),c_Orderings_Oord__class_Omin(T_c)))))),hAPP(hAPP(c_COMBC(T_c,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_c),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_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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_B_2)))))))) # label(fact_min__max_Oinf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 694 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_R_2 c_New__Random__Sequence_Opos__not__random__dseq(V_R_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__not__seq(hAPP(hAPP(hAPP(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]. 8.81/8.85 695 (all V_l_2 (V_l_2 = c_Int_OPls <-> c_Int_OBit0(V_l_2) = c_Int_OPls)) # label(fact_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 696 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)))) # label(fact_min__max_OInf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 697 (all V_Pa_2 all T_d all T_c c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,T_d),V_Pa_2)),c_Product__Type_OPair(T_c,T_d)))) = c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),V_Pa_2)) # label(fact_split__paired__Eps) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 698 (all V_y_2 all V_r_2 all T_c ((all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_y_2),V_y_2)),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__irreflexive) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 699 (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]. 8.81/8.85 700 (all V_Y_2 all T_c V_Y_2 = hAPP(c_Set_Oimage(T_c,T_c,c_COMBI(T_c)),V_Y_2)) # label(fact_image__ident) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 701 (all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 702 (all V_x_2 all V_k_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_k_2),V_x_2) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_list__update__nonempty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 703 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Set_Oimage(T_d,T_c,V_f_2)),V_B_2))) # label(fact_image__UN) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 704 (all V_w_2 all V_c_2 all V_b_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))) # label(fact_divide__less__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 705 (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]. 8.81/8.85 706 (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]. 8.81/8.85 707 (all V_B_2 all V_A_2 all T_d all T_c (hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hAPP(V_B_2,B_x) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_INTER__UNIV__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 708 (all V_m all V_n all 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_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__self3) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 709 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Inter__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 710 (all V_g_2 all V_f_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),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_c,V_f_2,V_xs_2)),c_List_Olist_Olist__size(T_c,V_g_2,V_xs_2))))) # label(fact_list__size__pointwise) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 711 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,V_n),V_m)) # label(fact_minus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 712 (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]. 8.81/8.85 713 (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(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))) & (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))))) # label(fact_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 714 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 715 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_y_2)),hAPP(V_f_2,V_x_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)))) # label(fact_fun__left__comm_Ofun__comp__comm) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 716 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_b),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,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))) # label(fact_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 717 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 | (exists B_j (V_m_2 = hAPP(c_Nat_OSuc,B_j) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),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__0__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 718 (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]. 8.81/8.85 719 (all V_B_2 all V_A_2 all V_C_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_C_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(c_Set_Oimage(T_c,T_d,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_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)))))) # label(fact_inj__on__image__Int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 720 (all V_B_2 all V_a_2 all V_A_2 all T_c 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_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_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_Diff__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 721 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),c_Set_Oimage(T_d,T_c,V_f_2)),hAPP(c_Set_OPow(T_d),V_A_2))),hAPP(c_Set_OPow(T_c),V_B_2))))) # label(fact_image__Pow__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 722 (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),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z))) # label(fact_inf__sup__aci_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 723 (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]. 8.81/8.85 724 (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]. 8.81/8.85 725 (all V_f_2 all V_a_2 all V_A_2 all T_c all T_d (class_Groups_Oab__group__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_setsum__diff1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 726 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),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)) = c_List_Oupt(V_i,V_j))) # label(fact_upt__conv__Cons) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 727 (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]. 8.81/8.85 728 (all V_A_2 all V_C_2 all T_c ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_X),V_C_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),B_X)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_C_2))))) # label(fact_InterI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 729 (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_b) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 730 (all V_a_2 all V_B_2 all V_A_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_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))) -> (hBOOL(hAPP(hAPP(c_member(T_d),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_INT__D) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 731 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I19_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 732 (all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),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)))) = V_A_2)) # label(fact_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 733 (all V_g_2 all T_e all T_f all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) -> (hAPP(c_Set_Oimage(T_f,T_e,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_f,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_prod(T_d,T_f),tc_prod(T_c,T_e),c_Product__Type_Omap__pair(T_d,T_c,T_f,T_e,V_f_2,V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_d,T_f),tc_HOL_Obool)))))) # label(fact_map__pair__surj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 734 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_a_2),V_A_2) = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_min__max_OSup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 735 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orotate1(T_a),V_xs)) = hAPP(c_List_Orotate1(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate1__rotate__swap) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 736 (all V_i_2 all V_xq_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))) = hAPP(c_New__DSequence_Opos__not__seq(V_xq_2),V_i_2)) # label(fact_New__DSequence_Opos__not__seq__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 737 (all V_N_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),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)))))) # label(fact_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 738 (all V_ys all V_xs all T_b all T_a 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)) = 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))) # label(fact_length__zip) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 739 (all V_Pa_2 all V_f_2 all T_c all T_d c_Random__Sequence_Omap(T_d,T_c,V_f_2,V_Pa_2) = c_Random__Sequence_Obind(T_d,T_c,V_Pa_2,hAPP(c_Fun_Ocomp(T_c,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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))))),T_d,c_Random__Sequence_Osingle(T_c)),V_f_2))) # label(fact_Random__Sequence_Omap__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 740 (all V_ys all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Otl(T_a),V_xs)),V_ys) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_tl__append2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 741 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_A_2 = hAPP(c_Set_Oimage(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_image__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 742 (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]. 8.81/8.85 743 (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]. 8.81/8.85 744 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Oplus__class_Oplus(T_d)),V_g_2),c_Groups_Ozero__class_Ozero(T_d)),V_A_2)))) # label(fact_setsum_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 745 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2) # label(fact_Int__absorb) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 746 (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]. 8.81/8.85 747 (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_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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_y),V_z))))) # label(fact_add__divide__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 748 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Ominus__class_Ominus(T_c)),V_f_2)),V_g_2)),V_A_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_g_2),V_A_2)))) # label(fact_setsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 749 (all V_f_2 all T_c c_FunDef_Ois__measure(T_c,V_f_2)) # label(fact_is__measure_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 750 (all V_p_2 all V_g_2 all T_f all V_f_2 all T_c all T_d all T_e hAPP(c_Product__Type_Oapfst(T_f,T_c,T_d,V_g_2),hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_f,V_f_2),V_p_2)) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),hAPP(c_Product__Type_Oapfst(T_f,T_c,T_e,V_g_2),V_p_2))) # label(fact_apsnd__apfst__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 751 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 752 (all V_f_2 all T_e all T_d all T_c c_Product__Type_Ocurry(T_c,T_d,T_e,hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_f_2)) = V_f_2) # label(fact_curry__split) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 753 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,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]. 8.81/8.85 754 (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(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),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_OMin),V_b))) # label(fact_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 755 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(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_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 756 (all T_c c_Transitive__Closure_Otrancl(T_c,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) # label(fact_trancl__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 757 (all V_N_2 all V_h_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),B_x),B_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(V_h_2,hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_N_2)) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_hom__fold1__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 758 (all V_p_2 all V_c_2 all T_e all T_d all V_z_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,tc_fun(T_c,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_d,T_e),B_x),B_y) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(hAPP(V_c_2,B_x),B_y))))))) # label(fact_mem__splitE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 759 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oring(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),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]. 8.81/8.85 760 (all V_y_2 all V_n_2 all V_x_2 all T_c (V_y_2 = V_x_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_y_2)))))) # label(fact_in__set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 761 (all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) & hBOOL(hAPP(c_List_Odistinct(T_c),V_ys_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_List_Oset(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_ys_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))) # label(fact_distinct__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 762 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> 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)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_c))) # label(fact_diff__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 763 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OAll(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),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)))),c_Orderings_Oord__class_Oless__eq(T_c))))) = c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2))) # label(fact_Inf__Sup) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 764 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),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))))) # label(fact_of__int__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 765 (all T_d all V_b_2 all V_B_2 all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2))))) # label(fact_SUP1__I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 766 (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]. 8.81/8.85 767 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_List_Oset(T_c),c_List_Oremdups(T_c,V_xs_2))) # label(fact_set__remdups) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 768 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__le__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 769 (all V_x_2 all V_g_2 all V_f_2 all T_e all T_c all T_d hAPP(V_f_2,hAPP(V_g_2,V_x_2)) = hAPP(hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2),V_x_2)) # label(fact_o__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 770 (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]. 8.81/8.85 771 (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_Odrop(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)))) # label(fact_nth__drop_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 772 (all V_is_2 all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_prod(T_c,tc_Nat_Onat)),V_Pa_2),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_prod(T_c,tc_Nat_Onat)),c_Nat_OSuc),c_Product__Type_Osnd(T_c,tc_Nat_Onat)))),hAPP(hAPP(c_List_Ozip(T_c,tc_Nat_Onat),V_xs_2),V_is_2))) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_prod(T_c,tc_Nat_Onat)),V_Pa_2),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),hAPP(hAPP(c_List_Ozip(T_c,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]. 8.81/8.85 773 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_List_Omap(tc_List_Olist(T_d),tc_List_Olist(T_c)),hAPP(c_List_Omap(T_d,T_c),V_f_2)),c_List_Otranspose(T_d,V_xs_2)) = c_List_Otranspose(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_d),tc_List_Olist(T_c)),hAPP(c_List_Omap(T_d,T_c),V_f_2)),V_xs_2))) # label(fact_transpose__map__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 774 (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)),V_b),V_c))) # label(fact_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 775 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> (exists B_n exists B_f hAPP(c_Set_Oimage(tc_Nat_Onat,T_c,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))) = V_A_2))) # label(fact_finite__conv__nat__seg__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 776 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 777 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> V_i = hAPP(c_List_Ohd(tc_Nat_Onat),c_List_Oupt(V_i,V_j)))) # label(fact_hd__upt) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 778 (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]. 8.81/8.85 779 (all V_xs_2 all V_x_2 all T_c c_Random_Oselect__weight(T_c,V_xs_2) = c_Random_Oselect__weight(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_c),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),V_x_2)),V_xs_2))) # label(fact_select__weight__cons__zero) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 780 (all V_C_2 all V_B_2 all V_A_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,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_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_e),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_UN__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.85 781 (all V_B_2 all V_f_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_d),c_Fun_Oid(T_d)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_setprod__reindex__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 782 (all V_a2_2 all V_a1_2 all V_f_2 all T_e all V_r2_2 all V_A2_2 all T_d all V_r1_2 all V_A1_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_d,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_c,T_d,tc_fun(T_e,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a1_2),V_A1_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_a2_2),V_A2_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_e,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r1_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a1_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_d,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a2_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,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]. 8.81/8.86 783 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_divide__zero) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 784 (all V_x_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_2,V_F_2) -> V_x_2 = hAPP(hAPP(V_f_2,V_x_2),V_x_2))) # label(fact_folding__one__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 785 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_y_2)))))) # label(fact_lessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 786 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omax(T_c),V_x_2),V_xs_2))) # label(fact_min__max_OSup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 787 (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]. 8.81/8.86 788 (all V_r_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_r_2) = c_Transitive__Closure_Otrancl(T_c,V_r_2)) # label(fact_trancl__unfold__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 789 (all V_y_2 all V_x_2 all T_c (class_Lattices_Oboolean__algebra(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2) <-> V_y_2 = V_x_2))) # label(fact_compl__eq__compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 790 (all V_Pa_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2))))) # label(fact_distinct__dropWhile) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 791 (all V_z_2 all V_f_2 all T_d all T_c c_Nitpick_Ofold__graph_H(T_c,T_d,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),V_z_2)) # label(fact_fold__graph_H_Ointros_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 792 (all V_m all V_n all V_a all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> 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)) = 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)))))) # label(fact_power__diff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 793 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 794 (all V_a_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,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_c),V_a_2),V_A_2))))) # label(fact_INF1__E) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 795 (all V_B_2 all V_A_2 all T_c hAPP(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_B_2),V_A_2)) # label(fact_Un__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 796 (all V_ys_2 all V_xs_2 all T_c hAPP(c_List_Otl(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(T_c),T_c,hAPP(c_List_Otl(T_c),V_ys_2),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Oappend(T_c)),V_ys_2))),V_xs_2)) # label(fact_tl__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 797 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_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)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_B_2)) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))))) # label(fact_Min_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 798 (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]. 8.81/8.86 799 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_List_Orev(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Orev(T_d),V_xs_2))) # label(fact_rev__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 800 (all V_n_2 all V_Pa_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_Pa_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)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),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_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(hAPP(V_Pa_2,B_i),B_j))))))) # label(fact_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 801 (all V_i c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_upt__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 802 (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]. 8.81/8.86 803 (all V_xs_2 all T_c 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_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Odrop(T_c),V_m_2),V_xs_2))),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2)))))) # label(fact_set__drop__subset__set__drop) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 804 (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]. 8.81/8.86 805 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oring(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_b_2),V_a_2)),V_e_2)),V_d_2) = V_c_2))) # label(fact_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 806 (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]. 8.81/8.86 807 (all V_x_2 all V_A_2 all T_c (class_Lattices_Odistrib__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2))))),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_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)))))) # label(fact_sup__Inf1__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 808 (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)))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 809 (all V_n_2 all V_m_2 all V_k_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) <-> V_n_2 = V_m_2 | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2)) # label(fact_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 810 (all V_s_2 all V_Z_2 all V_p_2 all V_Pa_2 all T_c (hBOOL(hAPP(hAPP(c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,V_p_2),V_Z_2),V_s_2)) <-> hBOOL(hAPP(V_p_2,V_s_2)) & hBOOL(hAPP(hAPP(V_Pa_2,V_Z_2),V_s_2)))) # label(fact_peek__and__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 811 (all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_c,T_d,c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_surj__imp__inj__inv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 812 (all V_Ga_2 all T_c c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) # label(fact_empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 813 (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]. 8.81/8.86 814 (all V_r_2 all T_c hAPP(c_Relation_Oconverse(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olistrel1(T_c,V_r_2)) = c_List_Olistrel1(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2))) # label(fact_listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 815 (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]. 8.81/8.86 816 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_x_2)) # label(fact_Min_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 817 (all V_xs_2 all T_c all V_b_2 all V_a_2 (V_b_2 != V_a_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),c_List_Oremove1(T_c,V_b_2,V_xs_2))))))) # label(fact_in__set__remove1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 818 (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_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))))) = 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__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 819 (all V_k_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OlessThan(T_c,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]. 8.81/8.86 820 (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_Int_Oint),V_w_2),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)))))) # label(fact_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 821 (all V_u_2 all V_l_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThanLessThan(T_c,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]. 8.81/8.86 822 (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),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)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))) # label(fact_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 823 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_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_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))))) # label(fact_min__max_OInf__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 824 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),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)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2))))) # label(fact_Union__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 825 (all V_xs_2 all V_Pa_2 all T_c ((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_c)),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n))))) <-> c_List_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_list__all__length) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 826 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_c),V_A_2)))) # label(fact_Pow__top) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 827 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_UN__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 828 (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]. 8.81/8.86 829 (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]. 8.81/8.86 830 (all V_n all V_a all T_a (class_Rings_Oring__1__no__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 831 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,V_r_2))) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_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_c)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n)),hAPP(hAPP(c_List_Onth(T_c),V_ys_2),B_n))),V_r_2)))))) # label(fact_listrel__iff__nth) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 832 (all V_r_2 all V_A_2 all V_f_2 all T_d all T_c (class_Fields_Ofield(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Rings_Oinverse__class_Odivide(T_c)),V_f_2)),V_r_2)),V_A_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),V_r_2))) # label(fact_setsum__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 833 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 834 (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]. 8.81/8.86 835 (all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__one) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 836 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 837 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n))) # label(fact_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 838 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Olist__ex(tc_Int_Oint,V_Pa_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_Pa_2),V_i_2,V_j_2))) # label(fact_list__ex__iff__not__all__inverval__int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 839 (all V_r_2 all V_q_2 all T_c (c_Nitpick_Oless__frac(T_c,V_q_2,V_r_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),c_Nitpick_Oplus__frac(T_c,V_q_2,c_Nitpick_Ouminus__frac(T_c,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_Nitpick_Oless__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 840 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_A_2 = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_inj__vimage__image__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 841 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_a_2)))))) # label(fact_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 842 (all V_list_2 all V_a_2 all V_f2_2 all V_f1_2 all T_d all T_c hAPP(hAPP(V_f2_2,V_a_2),V_list_2) = hAPP(c_List_Olist_Olist__case(T_c,T_d,V_f1_2,V_f2_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_a_2),V_list_2))) # label(fact_list_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 843 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_bij__betw__empty2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 844 (all V_k_2 (c_Int_OMin = c_Int_OBit1(V_k_2) <-> V_k_2 = c_Int_OMin)) # label(fact_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 845 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olinorder__class_OMin(T_c,V_N_2))))))) # label(fact_Min_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 846 (all V_b_2 all V_a_2 all T_c (class_Groups_Oab__group__add(T_c) -> (V_b_2 = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)))) # label(fact_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 847 (all V_n all V_m (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) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_m = V_n))) # label(fact_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 848 (all V_Pa_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_Pa_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_Pa_2,B_i),B_j))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 849 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),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),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2))))) # label(fact_Union__upper) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 850 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_A_2),V_C_2))),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_C_2))) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBS(T_d,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_d),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Un__Union__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 851 (all V_a_2 all V_r_2 all T_c all T_d hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool),hAPP(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_c),c_member(tc_prod(T_d,T_c))),hAPP(c_Product__Type_OPair(T_d,T_c),V_a_2))),V_r_2)) = hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_Image__singleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 852 (all V_x_2 all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,V_r_2) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),V_x_2)))) # label(fact_accp__wfPD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 853 (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]. 8.81/8.86 854 (all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_finite__Domain) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 855 (all V_h_2 all V_B_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2)),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_h_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_setsum__reindex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 856 (all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 all V_com_2 all V_fun_2 all V_loc_2 c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2)) # label(fact_com_Osimps_I36_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 857 (all V_n all V_m ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_n) & (V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 8.81/8.86 858 (all V_xs_2 all V_Pa_2 all T_c (c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) = c_List_Olist_ONil(T_c) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_dropWhile__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 859 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_x_2),V_A_2))))) # label(fact_Sup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 860 (all V_c all V_b all V_a 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) = 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))) # label(fact_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 861 (all V_Pa_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),hAPP(c_Set_OCollect(tc_prod(T_d,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,tc_HOL_Obool),V_Pa_2))) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d)),hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Pa_2)))) # label(fact_Range__Collect__split) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 862 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olexord(T_c,V_r_2))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_y_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olex(T_c,V_r_2))))) # label(fact_lexord__lex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 863 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_A_2))) # label(fact_subset__refl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 864 (all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> V_x_2 = c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 865 (all T_c c_Relation_Otrans(T_c,c_Relation_OId(T_c))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 866 (all V_ys_2 all V_f_2 all T_e all V_xs_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_f_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_e),tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_e,tc_prod(T_c,T_d)),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_f_2))),hAPP(hAPP(c_List_Ozip(T_c,T_e),V_xs_2),V_ys_2))) # label(fact_zip__map2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 867 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)))) # label(fact_IntD2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 868 (all V_Pa_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2))) -> (hBOOL(hAPP(hAPP(V_Pa_2,V_ax_2),V_ay_2)) -> ((all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2))) -> (all B_aa all B_ba (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_aa),B_ba))),V_r_2)) -> (hBOOL(hAPP(hAPP(V_Pa_2,B_a),B_b)) -> hBOOL(hAPP(hAPP(V_Pa_2,B_aa),B_ba))))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_bx_2),V_by_2)))))) # label(fact_rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 869 (all V_R_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_R_2) -> c_Wellfounded_Owf(T_c,V_R_2))) # label(fact_wf__no__loop) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 870 (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]. 8.81/8.86 871 (all V_xs all T_a all 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__appendL) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 872 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_B_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2))))))) # label(fact_min__max_OSup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 873 (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]. 8.81/8.86 874 (all V_n all V_m c_Divides_Odivmod__nat(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__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 875 (all V_n all V_m all V_k ((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))) & (V_k = 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)))) # label(fact_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 876 (all V_y_2 all V_x_2 all V_g_2 all V_f_2 all T_c all T_d all T_e c_Fun_Ofun__upd(T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_2),V_g_2),V_x_2,hAPP(V_f_2,V_y_2)) = hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_2),c_Fun_Ofun__upd(T_c,T_e,V_g_2,V_x_2,V_y_2))) # label(fact_fun__upd__comp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 877 (all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,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]. 8.81/8.86 878 (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),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),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__assoc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 879 (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]. 8.81/8.86 880 (all V_f_2 all T_d all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> c_Relation_Otrans(T_d,hAPP(hAPP(c_Relation_Oinv__image(T_c,T_d),V_r_2),V_f_2)))) # label(fact_trans__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 881 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_list__ex__iff__raw) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 882 (all V_B_2 all V_C_2 all T_d all V_A_2 all T_c 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_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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_d),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_INT__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 883 (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]. 8.81/8.86 884 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> V_xs = 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))))) # label(fact_append__butlast__last__id) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 885 (all V_s_2 all V_r_2 all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_s_2))) # label(fact_converse__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 886 (all V_A_2 all T_c c_Relation_Otrans(T_c,c_Relation_OId__on(T_c,V_A_2))) # label(fact_trans__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 887 (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]. 8.81/8.86 888 (all V_f_2 all V_B_2 all T_e all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_e,V_f_2),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(T_e,tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2),V_f_2))) # label(fact_INT__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 889 (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]. 8.81/8.86 890 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_d),V_A_2)),hAPP(c_Relation_ODomain(T_c,T_d),V_B_2)) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Domain__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 891 (all V_xs_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Orderings_Obot__class_Obot(T_c)),V_xs_2))) # label(fact_Sup__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 892 (all V_z_2 all V_R_2 all V_n_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_z_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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]. 8.81/8.86 893 (all V_A_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) -> (c_Fun_Oinj__on(T_d,T_c,V_f_2,V_A_2) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2))))) # label(fact_finite__imageD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 894 (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]. 8.81/8.86 895 (all V_B_2 all V_A_2 all T_c 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)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_2))),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Inter__Un__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 896 (all V_A_2 all T_d all V_I_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_A_2,B_x))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_finite__INT) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 897 (all V_xs all V_x all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs) = 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))) # label(fact_drop__Suc__Cons) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 898 (all V_x_2 all T_c (class_Groups_Oone(T_c) -> (c_Groups_Oone__class_Oone(T_c) = V_x_2 <-> V_x_2 = c_Groups_Oone__class_Oone(T_c)))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 899 (all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_Nil__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 900 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 ((-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))))) # label(fact_filter_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 901 (all V_A_2 all V_xs_2 all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_c,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_c,tc_Nat_Onat)),c_member(tc_Nat_Onat)),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_c,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_c)),V_xs_2))))) = c_List_Osublist(T_c,V_xs_2,V_A_2)) # label(fact_sublist__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 902 (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]. 8.81/8.86 903 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__comm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_y_2)) -> (c_Groups_Ozero__class_Ozero(T_c) = V_x_2 & V_y_2 = c_Groups_Ozero__class_Ozero(T_c) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),V_y_2) = c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 904 (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]. 8.81/8.86 905 (all V_xs_2 all V_f_2 all T_c 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_c,tc_Nat_Onat),V_f_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) = c_List_Olist_Olist__size(T_c,V_f_2,V_xs_2)) # label(fact_list__size__conv__listsum) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 906 (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)) = c_List_Obutlast(T_a,V_xs)) & (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))))) # label(fact_butlast__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 907 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_B_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)))))) # label(fact_Max_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 908 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> 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)))) # label(fact_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 909 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_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_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__left__absorb) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 910 (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)))) -> (V_x_2 = V_y_2 <-> hAPP(c_Code__Numeral_Oof__nat,V_x_2) = hAPP(c_Code__Numeral_Oof__nat,V_y_2))))) # label(fact_code__numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 911 (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]. 8.81/8.86 912 (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]. 8.81/8.86 913 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)))))))) # label(fact_le__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 914 (all V_c_2 all V_a_2 all V_b_2 all T_c (class_Groups_Ocancel__semigroup__add(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_a_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_a_2) <-> V_c_2 = V_b_2))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 915 (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]. 8.81/8.86 916 (all V_x_2 all V_n_2 all T_c all V_f_2 hAPP(V_f_2,hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),hAPP(V_f_2,V_x_2))) # label(fact_funpow__swap1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 917 (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]. 8.81/8.86 918 (all V_x_2 all V_A_2 all T_c (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(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 919 (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]. 8.81/8.86 920 (all V_Pa_2 all V_Qa_2 all V_c_2 all V_P_Ha_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_P_Ha_2),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> ((all B_Z all B_s (hBOOL(hAPP(hAPP(V_Pa_2,B_Z),B_s)) -> hBOOL(hAPP(hAPP(V_P_Ha_2,B_Z),B_s)))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))))) # label(fact_conseq1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 921 (all V_x_2 all T_c V_x_2 = c_Hilbert__Choice_OEps(T_c,hAPP(c_fequal,V_x_2))) # label(fact_some__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 922 (all V_r_2 all V_s_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Relation_OId(T_c)),V_s_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,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_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_s_2))))) # label(fact_rtrancl__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 923 (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_j)),V_k) = 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))) # label(fact_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 924 (all V_A_2 all V_S_2 all V_R_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_R_2),V_S_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_d,T_c,V_S_2),V_A_2))) # label(fact_Un__Image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 925 (all V_v_2 all V_f_2 all V_a_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,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)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),V_a_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Int_Oint),V_f_2),c_Int_Onat))) = c_Nat_Onat_Onat__case(T_c,V_a_2,V_f_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_nat__case__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 926 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,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]. 8.81/8.86 927 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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 (V_n_2 = hAPP(c_Nat_OSuc,B_m) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),V_R_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),B_m),V_R_2))))))))) # label(fact_rel__pow__E2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 928 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2))))))) # label(fact_Sup__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 929 (all V_xs_2 all V_x_2 all V_zs_2 all V_ys_2 all T_c (c_List_Olist_ONil(T_c) = V_ys_2 & hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = V_zs_2 | (exists B_ys_H (V_ys_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_ys_H) & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),B_ys_H),V_zs_2))) <-> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_zs_2))) # label(fact_append__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 930 (all V_a_2 all V_f_2 all V_A_2 all T_d all T_c (class_Orderings_Olinorder(T_c) & class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),V_A_2),V_f_2)),V_a_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,B_x)),V_a_2)) & hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2))))))) # label(fact_INF__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 931 (all V_s_2 all V_r_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),c_Predicate_Oconversep(T_d,T_c,V_r_2)),c_Predicate_Oconversep(T_d,T_c,V_s_2)) = c_Predicate_Oconversep(T_d,T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool))),V_r_2),V_s_2))) # label(fact_converse__meet) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 932 (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_SetInterval_Oord__class_OatLeastLessThan(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))))) # label(fact_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.86 933 (all V_B_2 all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(T_d)),V_f_2)),V_B_2))) # label(fact_vimage__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 934 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(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__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2)))))))) # label(fact_Inf__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 935 (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]. 8.81/8.87 936 (all V_n_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),hAPP(c_List_Olexn(T_c,V_r_2),V_n_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = V_n_2 & V_n_2 = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_lexn__length) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 937 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),T_d),c_Set_OPow(T_c)),V_B_2)) = hAPP(c_Set_OPow(T_c),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_Pow__INT__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 938 (all V_B_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Set_Ovimage(T_c,T_d,V_f_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),c_Set_Oinsert(T_d)),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)) # label(fact_vimage__eq__UN) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 939 (all V_i_2 all V_h_2 all V_g_2 all V_f_2 all T_d all T_c all T_f all T_e all T_h all T_g hAPP(c_Fun_Ocomp(tc_prod(T_g,T_h),tc_prod(T_e,T_f),tc_prod(T_c,T_d),c_Product__Type_Omap__pair(T_g,T_e,T_h,T_f,V_f_2,V_g_2)),c_Product__Type_Omap__pair(T_c,T_g,T_d,T_h,V_h_2,V_i_2)) = c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,hAPP(c_Fun_Ocomp(T_g,T_e,T_c,V_f_2),V_h_2),hAPP(c_Fun_Ocomp(T_h,T_f,T_d,V_g_2),V_i_2))) # label(fact_map__pair_Ocomp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 940 (all V_P_Ha_2 all V_Pa_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_Ha_2,B_x)) <-> hBOOL(hAPP(V_Pa_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_Ha_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_Pa_2)))) # label(fact_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 941 (all V_d_2 all V_m_2 (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m_2,V_d_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> (exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_d_2),B_q)))) # label(fact_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 942 (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_n_2 = V_m_2)) # label(fact_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 943 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> (hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = V_A_2))) # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 944 (all V_v_2 all V_i_2 all V_xs_2 all V_x_2 all T_c c_Nat_Onat_Onat__case(tc_List_Olist(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_v_2),V_xs_2),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_c),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,T_c,tc_List_Olist(T_c)),c_List_Olist__update(T_c,V_xs_2)),V_v_2)),V_i_2) = hAPP(hAPP(c_List_Olist__update(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_i_2),V_v_2)) # label(fact_list__update_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 945 (all V_xs_2 all V_Pa_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_length__takeWhile__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 946 (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]. 8.81/8.87 947 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,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(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_B_2))))) # label(fact_image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 948 (all V_C_2 all V_B_2 all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),V_B_2) -> (hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) -> hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_B_2),V_C_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))))) # label(fact_union__comp__emptyR) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 949 (all V_A_2 all T_c (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_c),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_card__ge__0__finite) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 950 (all V_B_2 all V_A_2 all T_c c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2))) # label(fact_Union__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 951 (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]. 8.81/8.87 952 (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(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_OnegDivAlg(V_a,V_b))))) # label(fact_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 953 (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]. 8.81/8.87 954 (all V_R_2 all T_c 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_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_c,V_f_2,V_R_2)))))) # label(fact_mlex__leq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 955 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(tc_fun(T_c,tc_HOL_Obool),c_Wellfounded_Omin__ext(T_c,V_r_2)))) # label(fact_min__ext__wf) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 956 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2))))))) # label(fact_min__max_OInf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 957 (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(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_z_H))))) # label(fact_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 958 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_b_2),V_a_2)),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2))))) # label(fact_converseI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 959 (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]. 8.81/8.87 960 (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]. 8.81/8.87 961 (all V_n_2 all V_k_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),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),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_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 962 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),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_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel1(T_c,V_r_2)),c_List_Olistrel1(T_c,V_s_2))))) # label(fact_listrel1__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 963 (all V_x_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,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]. 8.81/8.87 964 (all T_c c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)) = c_List_Olists(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_lists__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 965 (all V_xss_2 all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_c)),c_List_Otranspose__rel(T_c)),hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_xss_2))) -> c_List_Otranspose(T_c,V_xss_2) = c_List_Otranspose(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_xss_2)))) # label(fact_transpose_Opsimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 966 (all V_x_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_ys_2))),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))))) # label(fact_rtrancl__listrel1__ConsI1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 967 (all V_A_2 all V_x_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)),V_x_2))))) # label(fact_Inf__lower) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 968 (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]. 8.81/8.87 969 (all V_y all V_x 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)))) = c_Nitpick_Oint__lcm(V_x,V_y)) # label(fact_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 970 (all V_A_2 (c_Nat__Transfer_Onat__set(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))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),V_A_2))))) # label(fact_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 971 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_B_2),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))))) # label(fact_Un__eq__Union) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 972 (all V_xs_2 all V_f_2 all T_d all V_n_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Orotate(T_d,V_n_2),V_xs_2)) = hAPP(c_List_Orotate(T_c,V_n_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_rotate__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 973 (all V_n_2 all V_a_2 all V_times_2 all V_one_2 all T_c hAPP(hAPP(V_times_2,V_a_2),hAPP(hAPP(c_Power_Opower_Opower(T_c,V_one_2,V_times_2),V_a_2),V_n_2)) = hAPP(hAPP(c_Power_Opower_Opower(T_c,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]. 8.81/8.87 974 (all V_ys_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) & c_List_Olinorder__class_Osorted(T_c,V_ys_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),hAPP(c_List_Oset(T_c),V_ys_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_x),B_xa)))))) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))) # label(fact_sorted__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 975 (all V_p_2 all V_g_2 all V_f_2 all T_d all T_c 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_c,T_d),V_p_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_c,T_d),V_p_2)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Product__Type_Oprod_Oprod__size(T_c,T_d,V_f_2,V_g_2,V_p_2)) # label(fact_prod__size__simp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 976 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2)) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__rev) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 977 (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]. 8.81/8.87 978 (all V_n_2 all V_k_2 all V_m_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))) <-> (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__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 979 (all V_X_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_X_2)) = V_X_2)) # label(fact_inv__image__comp) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 980 (all 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)))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))))) # label(fact_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 981 (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]. 8.81/8.87 982 (all V_g_2 all V_f_2 all V_B_2 all V_A_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) & class_Groups_Ocomm__monoid__mult(T_c) -> (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_f_2,B_x) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_g_2),V_B_2)))))) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 983 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c) != V_a_2))) # label(fact_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 984 (all V_B_2 all V_A_2 all V_C_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 985 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),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)))))),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))))) # label(fact_card__Diff1__less) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 986 (all T_c hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))) = c_List_Olists(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_lists__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 987 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) = hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) # label(fact_rtrancl__converse) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 988 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 989 (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)) -> V_a = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 990 (all V_B_2 all V_a_2 all V_A_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(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_Set_Oinsert(T_c),V_a_2),V_B_2))) # label(fact_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 991 (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_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 992 (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]. 8.81/8.87 993 (all V_k_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2)) # label(fact_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 994 (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]. 8.81/8.87 995 (all V_r_2 all T_d all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))) -> -(all B_y -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),B_y)),V_r_2))))) # label(fact_DomainE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 996 (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]. 8.81/8.87 997 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_ys_2) -> (c_Fun_Oinj__on(T_d,T_c,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_List_Oset(T_d),V_xs_2)),hAPP(c_List_Oset(T_d),V_ys_2))) -> V_xs_2 = V_ys_2))) # label(fact_map__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 998 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_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]. 8.81/8.87 999 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((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_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = V_B_2))) # label(fact_UN__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1000 (all V_list_2 all V_g_2 all T_e all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_g_2),V_list_2)) = hAPP(hAPP(c_List_Omap(T_e,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_list_2)) # label(fact_List_Omap_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1001 (all V_a all T_a (class_Groups_Omonoid__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__1__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1002 (all V_ys_2 all V_xs_2 all V_f_2 all T_e all T_d all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_e,T_d),tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_d,tc_prod(T_c,T_d)),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_f_2))),hAPP(hAPP(c_List_Ozip(T_e,T_d),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Omap(T_e,T_c),V_f_2),V_xs_2)),V_ys_2)) # label(fact_zip__map1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1003 (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]. 8.81/8.87 1004 (all V_y_2 all V_x_2 all T_c (class_Lattices_Oboolean__algebra(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2)))))) # label(fact_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1005 (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]. 8.81/8.87 1006 (all V_xs all V_n all V_m all T_a hAPP(c_List_Orotate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_xs) = hAPP(c_List_Orotate(T_a,V_m),hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__rotate) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1007 (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]. 8.81/8.87 1008 (all V_g_2 all V_f_2 all T_f all T_e all T_d all T_c hAPP(hAPP(c_COMBS(tc_prod(T_c,T_d),T_d,T_e),hAPP(hAPP(c_COMBB(T_f,tc_fun(T_d,T_e),tc_prod(T_c,T_d)),V_f_2),hAPP(hAPP(c_COMBB(T_c,T_f,tc_prod(T_c,T_d)),V_g_2),c_Product__Type_Ofst(T_c,T_d)))),c_Product__Type_Osnd(T_c,T_d)) = hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBB(T_f,tc_fun(T_d,T_e),T_c),V_f_2),V_g_2))) # label(fact_split__comp__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1009 (all V_m all 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)) = 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))) # label(fact_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1010 (all V_a_2 all V_xs_2 all V_f_2 all T_c all T_d hAPP(c_List_Ofoldl(T_c,T_d,hAPP(c_COMBC(T_d,T_c,T_c),V_f_2),V_a_2),hAPP(c_List_Orev(T_d),V_xs_2)) = c_List_Ofoldr(T_d,T_c,V_f_2,V_xs_2,V_a_2)) # label(fact_foldr__foldl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1011 (all V_xs_2 all V_g_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Ofilter(T_d,hAPP(hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,tc_HOL_Obool),T_d),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]. 8.81/8.87 1012 (all V_n_2 all V_z_2 all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_R_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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]. 8.81/8.87 1013 (all V_nat_H c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1014 (all V_k c_Int_Opred(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OMin)) # label(fact_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1015 (all T_c c_Wellfounded_Owf(tc_fun(T_c,tc_HOL_Obool),c_Wellfounded_Ofinite__psubset(T_c))) # label(fact_wf__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1016 (all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_vimage__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1017 (all V_B_2 all V_a_2 all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_B_2))) # label(fact_image__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1018 (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]. 8.81/8.87 1019 (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]. 8.81/8.87 1020 (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]. 8.81/8.87 1021 (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]. 8.81/8.87 1022 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_c (c_Predicate_Otransp(T_c,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]. 8.81/8.87 1023 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2))))) # label(fact_sorted__insort) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1024 (all V_x_2 all V_Qa_2 all V_Pa_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_Pa_2),V_Qa_2)) -> (hBOOL(hAPP(V_Pa_2,V_x_2)) -> hBOOL(hAPP(V_Qa_2,V_x_2))))) # label(fact_predicate1D) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1025 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (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_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_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_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_B_2)))))) # label(fact_setsum__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1026 (all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_A_2)) # label(fact_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1027 (all V_B_2 all T_c V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_B_2)) # label(fact_Int__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1028 (all V_B_2 all V_A_2 all T_c (class_Lattices_Odistrib__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBS(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_c),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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal))),c_Lattices_Osemilattice__inf__class_Oinf(T_c)))))),hAPP(hAPP(c_COMBC(T_c,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_c),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_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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))))))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)))))))) # label(fact_inf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1029 (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]. 8.81/8.87 1030 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Sup__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1031 (all V_x_2 all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1032 (all V_x_2 all T_c hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_UNIV__I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1033 (all V_y_2 all V_x_2 all V_S_2 all V_R_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),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_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)) <-> 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_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_sup__Un__eq2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1034 (all T_c 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_c,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)),V_B_2),V_A_2)))) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1035 (all V_xs all T_a all V_ys ((V_ys != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_ys) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),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__append) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1036 (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]. 8.81/8.87 1037 (all V_x_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_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)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_setsum_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1038 (all V_x_2 all V_A_2 all T_c (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(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1039 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))))) # label(fact_refl__onD1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1040 (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]. 8.81/8.87 1041 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Oacyclic(T_c,V_r_2))) # label(fact_wf__acyclic) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1042 (all V_xs_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = V_xs_2))) # label(fact_List_Oinsert__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1043 (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(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))) & (-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))) -> c_Divides_OposDivAlg(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))))))) # label(fact_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1044 (all V_code__numeral_H_2 all V_code__numeral_2 (V_code__numeral_2 = V_code__numeral_H_2 <-> c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2) = c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_2))) # label(fact_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1045 (all V_a3_2 all V_a1_2 all V_s_2 all V_r_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(c_Predicate_Opred__comp(T_c,T_d,T_e,V_r_2,V_s_2),V_a1_2),V_a3_2)) <-> (exists B_b (hBOOL(hAPP(hAPP(V_s_2,B_b),V_a3_2)) & hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_b)))))) # label(fact_pred__comp_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1046 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_finite__PlusD_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1047 (all V_f_2 all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) -> c_Wellfounded_Owf(T_c,c_Wellfounded_Omlex__prod(T_c,V_f_2,V_R_2)))) # label(fact_wf__mlex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1048 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))))))) # label(fact_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1049 (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]. 8.81/8.87 1050 (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]. 8.81/8.87 1051 (all V_g_2 all V_f_2 all T_f all T_d all T_e all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_prod(T_e,T_f)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_f),tc_fun(T_d,tc_prod(T_e,T_f))),hAPP(hAPP(c_COMBB(tc_fun(T_f,tc_prod(T_e,T_f)),tc_fun(tc_fun(T_d,T_f),tc_fun(T_d,tc_prod(T_e,T_f))),T_c),c_COMBB(T_f,tc_prod(T_e,T_f),T_d)),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_f,tc_prod(T_e,T_f)),T_c),c_Product__Type_OPair(T_e,T_f)),V_f_2))),V_g_2)) = c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,V_f_2,V_g_2)) # label(fact_map__pair__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1052 (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]. 8.81/8.87 1053 (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]. 8.81/8.87 1054 (all V_I_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2))))) # label(fact_distinct__sublistI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1055 (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_b),V_c)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_c)))))) # label(fact_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1056 (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]. 8.81/8.87 1057 (all V_d all T_b all T_a (class_Lazy__Sequence_Osmall__lazy(T_a) & class_Lazy__Sequence_Osmall__lazy(T_b) -> c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_prod(T_a,T_b),V_d) = 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)))) # label(fact_small__lazy__prod__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1058 (all V_xs_2 all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_xs_2))) # label(fact_Sup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1059 (all V_xs all V_i all T_a 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)) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs))) # label(fact_rev__take) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1060 (all V_xs_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2))))) # label(fact_distinct__sort) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1061 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_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))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1062 (all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),hAPP(c_List_Orev(T_c),c_List_OtakeWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),V_xs_2))) = c_List_OdropWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),hAPP(c_List_Orev(T_c),V_xs_2))))) # label(fact_dropWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1063 (all V_B_H_2 all V_B_2 all V_A_H_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2) = V_B_H_2 -> c_Fun_Obij__betw(T_c,T_d,V_f_2,V_B_2,V_B_H_2))))) # label(fact_bij__betw__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1064 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1065 (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]. 8.81/8.87 1066 (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) = 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_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,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]. 8.81/8.87 1067 (all V_w_2 all V_v_2 all T_c (class_Int_Onumber__ring(T_c) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) = hAPP(hAPP(c_HOL_OLet(T_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),c_Groups_Otimes__class_Otimes(T_c)),c_COMBI(T_c))))) # label(fact_power__number__of__even__number__of) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1068 (all V_A_2 all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OAll(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),T_c),hAPP(c_COMBS(tc_fun(T_c,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_c,tc_HOL_Obool)),c_fimplies),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,tc_HOL_Obool))),V_A_2)))),c_member(T_c))))) # label(fact_Inter__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1069 (all V_Pa_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__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_a),B_b))) -> hBOOL(hAPP(hAPP(V_Pa_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_Pa_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1070 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_x_2,V_y_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1071 (all V_b_2 all V_a_2 (c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,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)) <-> -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2)))) # label(fact_dvd_OatLeastatMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1072 (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(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)) & (-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_a_2 | 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)) & (V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | 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))))) # label(fact_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1073 (all V_Pa_2 all V_b_2 all V_a_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_a_2),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2),B_x)))) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_b_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_a_2) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_a_2,V_b_2)))) # label(fact_list__all2I) # label(axiom) # label(non_clause). [assumption]. 8.81/8.87 1074 (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]. 8.81/8.87 1075 (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]. 8.81/8.87 1076 (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]. 8.81/8.87 1077 (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]. 8.81/8.87 1078 (all V_x_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_ys_2))),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel1I2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1079 (all V_R2_2 all V_R1_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R1_2),V_R2_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Wellfounded_Oacc(T_c,V_R2_2)),c_Wellfounded_Oacc(T_c,V_R1_2))))) # label(fact_acc__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1080 (all 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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1081 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,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_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1082 (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]. 8.81/8.88 1083 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_fun(T_d,tc_fun(T_e,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]. 8.81/8.88 1084 (all V_S_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) & class_Orderings_Olinorder(T_c) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_S_2)))))) # label(fact_less__Sup__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1085 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))))) # label(fact_Compl__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1086 (all T_c (class_Orderings_Olinorder(T_c) -> c_Orderings_Oord_Omin(T_c,hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_c))) = c_Orderings_Oord__class_Omax(T_c))) # label(fact_dual__min) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1087 (all V_xs all V_n all T_a hAPP(c_List_Otl(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),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_tl__take) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1088 (all V_C_2 all V_B_2 all V_A_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_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)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_d,V_r_2),V_C_2)),V_B_2)))) # label(fact_Image__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1089 (all V_b_2 all V_a_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2)))))) # label(fact_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1090 (all T_c c_Nitpick_Odenom(T_c) = hAPP(c_Fun_Ocomp(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,T_c,c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint)),c_Nitpick_ORep__Frac(T_c))) # label(fact_Nitpick_Odenom__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1091 (all V_xs_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Otl(T_d),V_xs_2)) = hAPP(c_List_Otl(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_map__tl) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1092 (all V_B_2 all V_D_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_D_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Diff__mono) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1093 (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]. 8.81/8.88 1094 (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]. 8.81/8.88 1095 (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_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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)))) # label(fact_add__frac__num) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1096 (all V_r_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))))) # label(fact_quotientI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1097 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1098 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_OdropWhile(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2),V_xs_2)) = c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_dropWhile__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1099 (all V_A_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,V_A_2))))) # label(fact_in__lists__conv__set) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1100 (all V_n_2 all V_m_2 all V_k_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m_2),V_n_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]. 8.81/8.88 1101 (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]. 8.81/8.88 1102 (all V_B_2 all V_b_2 all V_A_2 all V_a_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(hAPP(c_Set_Oinsert(T_d),V_b_2),V_B_2))) = hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),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),hAPP(hAPP(c_Set_Oinsert(T_d),V_b_2),V_B_2)))),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2))))) # label(fact_insert__times__insert) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1103 (all V_B_2 all V_C_2 all V_a_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2)))) # label(fact_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1104 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_x_2))))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1105 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2))))) # label(fact_setsum__abs) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1106 (all V_b all V_a all T_a (class_Rings_Odivision__ring(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),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))) # label(fact_minus__divide__left) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1107 (all V_k c_Int_OBit0(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit0(V_k))) # label(fact_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1108 (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]. 8.81/8.88 1109 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_d,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]. 8.81/8.88 1110 (all V_Pa_2 all V_f_2 all T_d all T_c hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(T_d,tc_HOL_Obool,T_c),V_Pa_2),V_f_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(c_Set_OCollect(T_d),V_Pa_2))) # label(fact_vimage__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1111 (all V_B_2 all V_A_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_d,T_c),V_A_2)),hAPP(c_Relation_ORange(T_d,T_c),V_B_2))),hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Range__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1112 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_m_2,V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1113 (all V_b_2 all V_B_2 all V_a_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_B_2)) -> V_a_2 = V_b_2) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2))))) # label(fact_insertCI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1114 (all V_l all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit0(V_k)),V_l) = c_Int_OBit0(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k),V_l))) # label(fact_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1115 (all V_n hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_n)) = V_n) # label(fact_nat__of__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1116 (all V_R_2 all V_f_2 all V_T_2 all T_d all V_S_2 all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_T_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_e),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e)),V_f_2)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),c_Set_OCollect(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_T_2)))),V_R_2)))),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),hAPP(hAPP(c_COMBS(T_d,tc_fun(T_c,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_e),tc_fun(tc_fun(T_c,tc_HOL_Obool),T_e),T_d),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e)),hAPP(c_COMBC(T_c,T_d,T_e),V_f_2))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Set_OCollect(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),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_S_2)))),hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),V_R_2))))),V_T_2))))) # label(fact_setsum__setsum__restrict) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1117 (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]. 8.81/8.88 1118 (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_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))))) # label(fact_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1119 (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))))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),V_D_2) = V_t_2)))) # label(fact_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1120 (all V_Qa_2 all T_d all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> ((all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Qa_2),B_y)))))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),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),V_Pa_2))),V_Qa_2)))))))) # label(fact_finite__Collect__bounded__ex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1121 (all V_y_2 all V_x_2 all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,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]. 8.81/8.88 1122 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_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__commutes) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1123 (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]. 8.81/8.88 1124 (all V_xs_2 all V_f_2 all T_d all V_n_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Otake(T_d),V_n_2),V_xs_2)) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_take__map) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1125 (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]. 8.81/8.88 1126 (all V_Pa_2 all T_c -c_List_Olist__ex1(T_c,V_Pa_2,c_List_Olist_ONil(T_c))) # label(fact_list__ex1__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1127 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_b_2),V_a_2)),V_r_2)))) # label(fact_converse__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1128 (all T_c c_List_Orotate(T_c,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Fun_Oid(tc_List_Olist(T_c))) # label(fact_rotate0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1129 (all V_y_2 all V_x_2 (-hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)) | V_x_2 = V_y_2)) # label(help_c__fequal__1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1130 (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_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))) # label(fact_dvd_OatLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1131 (all V_a_2 all V_w_2 all V_b_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) & class_Int_Onumber(T_c) -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)) <-> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = c_Groups_Ozero__class_Ozero(T_c) -> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)) & (c_Groups_Ozero__class_Ozero(T_c) != hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))) # label(fact_divide__eq__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1132 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) -> c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),c_List_OtakeWhile(T_c,V_Pa_2,V_ys_2)))) # label(fact_takeWhile__append2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1133 (all V_l all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l))) # label(fact_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1134 (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]. 8.81/8.88 1135 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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)) -> (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_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)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2))))))) # label(fact_folding_Ounion) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1136 (all V_l hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l))) # label(fact_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1137 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (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_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_trancl__into__trancl2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1138 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> hAPP(c_Finite__Set_Ocard(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1139 (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]. 8.81/8.88 1140 (all V_a_2 all V_b_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)))) # label(fact_atLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1141 (all V_A_2 all V_F_2 all V_f_2 all T_c (c_Big__Operators_Osemilattice__big(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2) = hAPP(V_F_2,V_A_2)))) # label(fact_semilattice__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1142 (all V_z_2 all T_c (class_Rings_Oring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)),c_Int_Oring__1__class_OInts(T_c))))) # label(fact_Ints__of__int) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1143 (all V_a_2 all V_b_2 all T_c (V_a_2 = V_b_2 <-> hBOOL(hAPP(hAPP(c_member(T_c),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_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1144 (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]. 8.81/8.88 1145 (all V_m 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)))) = 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))))) # label(fact_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1146 (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_Otimes__class_Otimes(tc_Nat_Onat),V_y),V_z)) = 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)) # label(fact_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1147 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_R2_2 all V_R1_2 all T_c c_New__Random__Sequence_Opos__union(T_c,V_R1_2,V_R2_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__union(T_c,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))) # label(fact_pos__union__def) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1148 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_DomainI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1149 (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),V_y) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),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]. 8.81/8.88 1150 (all V_D_2 all V_C_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_C_2,V_D_2) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_D_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> c_Fun_Obij__betw(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_D_2)))))) # label(fact_bij__betw__combine) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1151 (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]. 8.81/8.88 1152 (all V_Pa_2 all T_c (class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),V_Pa_2)) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1)))))) # label(fact_enum__ex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1153 (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]. 8.81/8.88 1154 (all V_b_2 all V_a_2 all T_c (hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) -> V_b_2 = V_a_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1155 (all V_fa_2 all T_c c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_fa_2),c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c))) # label(fact_lazy__sequence_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1156 (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]. 8.81/8.88 1157 (all V_xs_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2)),c_List_Oset(T_c)) = hAPP(c_List_Oset(T_c),c_List_Oconcat(T_c,V_xs_2))) # label(fact_set__concat) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1158 (all V_Pa_2 all V_xs_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_Bex__set__list__ex) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1159 (all V_r_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2)) = hAPP(c_Relation_ODomain(T_c,T_d),V_r_2)) # label(fact_Range__converse) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1160 (all V_xs_2 all T_c c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2))))) # label(fact_sorted__transpose) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1161 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))))) # label(fact_trancl__converseI) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1162 (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]. 8.81/8.88 1163 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(V_h_2,c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_N_2)) = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_min__max_OSup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1164 (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]. 8.81/8.88 1165 (all V_Pa_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),B_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),B_a),B_b)))) -> hBOOL(hAPP(hAPP(V_Pa_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_Pa_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1166 (all V_z_2 all V_Pa_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_Pa_2),V_z_2) = hAPP(hAPP(V_Pa_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_z_2)),hAPP(c_Product__Type_Osnd(T_d,T_e),V_z_2))) # label(fact_split__beta) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1167 (all V_Pa_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> hBOOL(hAPP(V_Pa_2,V_a_2)))) # label(fact_CollectE) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1168 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(V_B_2,V_a_2))))) # label(fact_SigmaD2) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1169 (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]. 8.81/8.88 1170 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_INT__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1171 (all V_x_2 all V_n_2 all T_c hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,hAPP(c_Nat_OSuc,V_n_2),V_x_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_set__replicate__Suc) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1172 (all V_k_2 all V_i_2 all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_k_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2)))))) # label(fact_atLeast__iff) # label(axiom) # label(non_clause). [assumption]. 8.81/8.88 1173 (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]. 8.81/8.88 1174 (all V_xs_2 all V_x_2 all T_c ((exists B_ys (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),B_ys))) & (exists B_zs V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_zs))))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__decomp__first) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1175 (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))) & (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) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_div__if) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1176 (all V_B_2 all V_A_2 all T_d all V_C_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Times__subset__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1177 (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]. 8.91/8.88 1178 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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),c_Groups_Oone__class_Oone(T_d)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)))) # label(fact_setprod_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1179 (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]. 8.91/8.88 1180 (all V_x_2 all V_Pa_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) -> (hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_d,V_Pa_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(c_List_Ofilter(T_d,V_Pa_2),V_xs_2)))))) # label(fact_filter__insort) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1181 (all T_c all V_ys_2 all V_xs_2 (V_ys_2 = V_xs_2 <-> c_List_Olist__all2(T_c,T_c,c_fequal,V_xs_2,V_ys_2))) # label(fact_list__all2__eq) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1182 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_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)))) # label(fact_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1183 (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_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_d)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_d)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1184 (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(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,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))))) # label(fact_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1185 (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]. 8.91/8.88 1186 (all V_n_2 all V_k_2 all V_m_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n_2 = V_m_2 <-> 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__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1187 (all V_M_2 all V_A_2 all V_i_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_M_2)),hAPP(V_M_2,V_i_2)))))) # label(fact_INF__leI) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1188 (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]. 8.91/8.88 1189 (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)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_x))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1190 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) -> V_z_2 = hAPP(V_g_2,B_x))) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,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]. 8.91/8.88 1191 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2))))) # label(fact_finite__surj) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1192 (all V_n_2 all V_m_2 (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) <-> V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2)) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1193 (all V_Qa_2 all V_Pa_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(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_fdisj),V_Pa_2)),V_Qa_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_c),V_Pa_2)),hAPP(c_Set_OCollect(T_c),V_Qa_2))) # label(fact_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1194 (all V_xs all V_x all T_a all V_n ((V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> 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)) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_take__Cons_H) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1195 (all V_M_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_M_2)) -> (exists B_h c_Fun_Obij__betw(tc_Nat_Onat,T_c,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_c),V_M_2)),V_M_2)))) # label(fact_ex__bij__betw__nat__finite) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1196 (all V_x_2 all V_A_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((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_c),V_x_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_A_2)))))))) # label(fact_below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1197 (all V_j_2 all V_i_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),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_c)),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_c)),V_xs_2))) -> (hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_j_2) = hAPP(hAPP(c_List_Onth(T_c),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]. 8.91/8.88 1198 (all V_C_2 all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2)) # label(fact_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1199 (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),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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))))) # label(fact_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1200 (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]. 8.91/8.88 1201 (all V_B_2 all V_A_2 all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_B_2),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__eq__Inter) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1202 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> (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]. 8.91/8.88 1203 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Union__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1204 (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]. 8.91/8.88 1205 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Orev(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__filter) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1206 (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_n = V_m))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1207 (all V_Pa_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_Pa_2,hAPP(c_Int_ORep__Integ,B_x)))) -> hBOOL(hAPP(V_Pa_2,V_y_2))))) # label(fact_Rep__Integ__induct) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1208 (all V_n_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (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_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)))))) # label(fact_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1209 (all V_xss_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_c)),c_List_Otranspose__rel(T_c)),hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_xss_2))) -> c_List_Otranspose(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_xss_2)) = hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_Olist__case(tc_List_Olist(T_c),T_c,c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_c),c_COMBK(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c))))),V_xss_2)))),c_List_Otranspose(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),V_xs_2),c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_c)),T_c,c_List_Olist_ONil(tc_List_Olist(T_c)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_OCons(tc_List_Olist(T_c))),c_List_Olist_ONil(tc_List_Olist(T_c)))))),V_xss_2))))))) # label(fact_transpose_Opsimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1210 (all V_C_2 all V_B_2 all T_e all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_d),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_UN__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1211 (all V_k_2 all T_c (class_Rings_Oring__1(T_c) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_k_2)) & (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_Groups_Ouminus__class_Ouminus(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k_2))) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),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(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(T_c,T_c)),tc_fun(tc_Int_Oint,T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),T_c),tc_fun(tc_fun(tc_Int_Oint,tc_fun(T_c,T_c)),tc_fun(tc_Int_Oint,T_c)),tc_Int_Oint),c_COMBB(tc_fun(T_c,T_c),T_c,tc_Int_Oint)),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_fun(T_c,T_c),T_c),tc_Int_Oint),c_HOL_OLet(T_c,T_c)),c_Int_Oring__1__class_Oof__int(T_c)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(T_c,T_c),tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,T_c)),tc_Int_Oint),c_COMBS(T_c,T_c,T_c)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(T_c,T_c),tc_fun(T_c,tc_fun(T_c,T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_fun(T_c,T_c))),tc_Int_Oint),c_COMBB(T_c,tc_fun(T_c,T_c),T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),tc_Int_Oint),c_If(T_c)),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_c,T_c,T_c),c_Groups_Oplus__class_Oplus(T_c)),c_COMBI(T_c))))),hAPP(hAPP(c_COMBC(T_c,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_c),c_Groups_Oplus__class_Oplus(T_c)),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),c_Groups_Oplus__class_Oplus(T_c)),c_COMBI(T_c)))),c_Groups_Oone__class_Oone(T_c))))),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))))) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_k_2))))) # label(fact_of__int__num) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1212 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Osgn__class_Osgn(T_c,V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_sgn__less) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1213 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_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_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_Diff__partition) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1214 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(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),V_B_2)) # label(fact_Un__left__absorb) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1215 (all V_xs_2 all T_c hAPP(c_Set_Oimage(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),c_List_Oset(T_c)),hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_Osublists(T_c,V_xs_2))) = hAPP(c_Set_OPow(T_c),hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_sublists__powset) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1216 (all T_d all T_c all V_t_2 all V_s_2 (hAPP(c_Product__Type_Ofst(T_c,T_d),V_t_2) = hAPP(c_Product__Type_Ofst(T_c,T_d),V_s_2) & hAPP(c_Product__Type_Osnd(T_c,T_d),V_t_2) = hAPP(c_Product__Type_Osnd(T_c,T_d),V_s_2) <-> V_t_2 = V_s_2)) # label(fact_Pair__fst__snd__eq) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1217 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_d,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_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)))) # label(fact_image__Int) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1218 (all V_xs_2 all V_n_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2))))) # label(fact_distinct__rotate) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1219 (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_OBit0(V_k_2)),c_Int_OBit0(V_l_2))))) # label(fact_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1220 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),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)))))) = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_2)))) # label(fact_min__max_OSup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1221 (all V_zs_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_ys_2),V_zs_2)),c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_zs_2)),c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2))))))) # label(fact_listrel__rtrancl__trans) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1222 (all V_A_2 all V_f_2 all T_d all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_a_2)),V_A_2)))) # label(fact_vimageD) # label(axiom) # label(non_clause). [assumption]. 8.91/8.88 1223 (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]. 8.91/8.88 1224 (all V_I_2 all V_xs_2 all T_c hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2)) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(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)),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)),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(T_c,tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_c),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_c)),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]. 8.91/8.89 1225 (all V_r_2 all V_xs_2 all V_ys_2 all V_y_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)),V_xs_2)),c_List_Olistrel(T_c,V_r_2))) -> -(all B_y all B_ys (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_ys) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),B_y)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_ys_2),B_ys)),c_List_Olistrel(T_c,V_r_2)))))))) # label(fact_listrel__Cons1) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1226 (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]. 8.91/8.89 1227 (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]. 8.91/8.89 1228 (all V_zs_2 all V_ys_2 all V_xs_2 all V_x_2 all T_c ((exists B_ys_H (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_ys_H) = V_ys_2 & hAPP(hAPP(c_List_Oappend(T_c),B_ys_H),V_zs_2) = V_xs_2)) | V_ys_2 = c_List_Olist_ONil(T_c) & V_zs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) <-> hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_zs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_Cons__eq__append__conv) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1229 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)))) # label(fact_Int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1230 (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))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = 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))))))) # label(fact_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1231 (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_Ouminus__class_Ouminus(tc_Int_Oint),V_i) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),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))) # label(fact_zabs__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1232 (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)) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Osgn__class_Osgn(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_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]. 8.91/8.89 1233 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,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)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_psubset__subset__trans) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1234 (all V_r_2 all T_c all T_d c_Predicate_Oconversep(T_d,T_c,c_Predicate_Oconversep(T_c,T_d,V_r_2)) = V_r_2) # label(fact_conversep__conversep) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1235 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2))))) # label(fact_subset__Pow__Union) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1236 (all V_r_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Relation_OId(T_c)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_r_2)) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_rtrancl__unfold) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1237 (all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,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_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__UNIV__card__ge__0) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1238 (all V_f_2 all V_A_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) & class_Groups_Ocomm__monoid__mult(T_c) -> (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_c,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]. 8.91/8.89 1239 (all V_s_2 all V_xs_2 all T_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(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_c)),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Product__Type_Ofst(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Oselect__weight(T_c,V_xs_2),V_s_2))),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c),T_c),c_Product__Type_Osnd(tc_Code__Numeral_Ocode__numeral,T_c)),V_xs_2)))))) # label(fact_select__weight__member) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1240 (all V_A_2 all T_c hAPP(c_Relation_Oconverse(T_c,T_c),c_Relation_OId__on(T_c,V_A_2)) = c_Relation_OId__on(T_c,V_A_2)) # label(fact_converse__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1241 (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]. 8.91/8.89 1242 (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]. 8.91/8.89 1243 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)))) # label(fact_set__insort) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1244 (all V_ys all T_b all T_a c_List_Olist_ONil(tc_prod(T_a,T_b)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),c_List_Olist_ONil(T_a)),V_ys)) # label(fact_zip__Nil) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1245 (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 V_x = 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)))))) # label(fact_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1246 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_d_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_d_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_d_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_a_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_a_2)))))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1247 (all V_Qa_2 all V_Pa_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2),V_Qa_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_Pa_2))) -> (all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_x)),V_Qa_2)) -> B_x = B_y)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_Pa_2)))))) # label(fact_rtrancl__Un__separator__converseE) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1248 (all V_a2_2 all V_a1_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_a1_2),V_a2_2))) <-> hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_a2_2)) & hBOOL(hAPP(V_A_2,V_a1_2)))) # label(fact_listsp_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1249 (all V_ys_2 all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olistrel(T_c,V_r_2)))))) # label(fact_listrel_OCons) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1250 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_le__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1251 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1252 (all V_a_2 all T_c c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_a_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) # label(fact_hoare__derivs_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1253 (all V_m all V_n 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))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_div__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1254 (all V_xs_2 all T_c (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_List_Oset(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_card__distinct) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1255 (all V_f_2 all V_S_2 all V_T_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_T_2),V_S_2))) -> c_Groups_Oone__class_Oone(T_d) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_S_2)))))) # label(fact_setprod__mono__one__right) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1256 (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]. 8.91/8.89 1257 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) & hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2))))) # label(fact_equiv__class__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1258 (all V_n_2 all V_r_2 all T_c hAPP(c_List_Olexn(T_c,V_r_2),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_prod(tc_prod(T_c,tc_List_Olist(T_c)),tc_prod(T_c,tc_List_Olist(T_c))),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_Product__Type_Omap__pair(tc_prod(T_c,tc_List_Olist(T_c)),tc_List_Olist(T_c),tc_prod(T_c,tc_List_Olist(T_c)),tc_List_Olist(T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)))),c_Wellfounded_Olex__prod(T_c,tc_List_Olist(T_c),V_r_2,hAPP(c_List_Olexn(T_c,V_r_2),V_n_2)))),hAPP(c_Set_OCollect(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),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_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),hAPP(c_Nat_OSuc,V_n_2))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),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_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),hAPP(c_Nat_OSuc,V_n_2))))))) # label(fact_lexn_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1259 (all V_xs_2 all T_c hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,tc_prod(T_c,T_c)),hAPP(hAPP(c_COMBS(T_c,T_c,tc_prod(T_c,T_c)),c_Product__Type_OPair(T_c,T_c)),c_COMBI(T_c))),V_xs_2)) # label(fact_zip__same__conv__map) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1260 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1261 (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]. 8.91/8.89 1262 (all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1263 (all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,T_d)),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,T_d),tc_HOL_Obool)))) -> (hAPP(c_Finite__Set_Ocard(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) != hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__fun__UNIVD1) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1264 (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]. 8.91/8.89 1265 (all V_a_2 all T_c c_FunDef_Ois__measure(T_c,V_a_2)) # label(fact_is__measure_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1266 (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(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)))) & (-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)))) -> 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)))) = 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))))) # label(fact_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1267 (all V_f_2 all T_c all T_e all T_d c_Product__Type_Oapsnd(T_d,T_e,T_c,V_f_2) = c_Product__Type_Omap__pair(T_c,T_c,T_d,T_e,c_Fun_Oid(T_c),V_f_2)) # label(fact_apsnd__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1268 (all T_c 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_c,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)),V_B_2),V_A_2))))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1269 (all V_b all V_a 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_a),V_b)) = V_a) # label(fact_fst__conv) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1270 (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]. 8.91/8.89 1271 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))))) # label(fact_min__max_OInf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1272 (all V_size_2 all V_nrandom_2 all V_x_2 all T_c 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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_DSequence_Osingle(T_c),V_x_2)) = hAPP(hAPP(hAPP(c_Random__Sequence_Osingle(T_c),V_x_2),V_nrandom_2),V_size_2)) # label(fact_Random__Sequence_Osingle__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1273 (all V_r_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,c_Wellfounded_Oacc(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))))) # label(fact_lists__accD) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1274 (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]. 8.91/8.89 1275 (all V_f_2 all T_e all T_d all T_c V_f_2 = hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),c_Product__Type_Ocurry(T_c,T_d,T_e,V_f_2))) # label(fact_split__curry) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1276 (all V_ys_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> (c_List_Olinorder__class_Osorted(T_c,V_ys_2) -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_ys_2)) -> (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_sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1277 (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]. 8.91/8.89 1278 (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]. 8.91/8.89 1279 (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_Int_Oint),V_x_2),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))))))) # label(fact_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1280 (all V_b_2 all V_a_2 all T_c (class_Rings_Osemiring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Nat_Osemiring__1__class_ONats(T_c))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Nat_Osemiring__1__class_ONats(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_b_2)),c_Nat_Osemiring__1__class_ONats(T_c))))))) # label(fact_Nats__add) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1281 (all V_z (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z)) # label(fact_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1282 (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]. 8.91/8.89 1283 (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]. 8.91/8.89 1284 (all V_i_2 all V_Pa_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_Pa_2,B_i)) -> hBOOL(hAPP(V_Pa_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,B_i),V_p_2)))))) -> (hBOOL(hAPP(V_Pa_2,V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_p_2)) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))))) # label(fact_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1285 (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_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)) = 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))) # label(fact_zip__update) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1286 (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)) -> V_x = hAPP(hAPP(c_List_Onth(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_i))) # label(fact_nth__replicate) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1287 (all V_list_H_2 all V_a_H_2 all V_list_2 all V_a_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_list_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_H_2),V_list_H_2) <-> V_a_H_2 = V_a_2 & V_list_H_2 = V_list_2)) # label(fact_list_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1288 (all V_A_2 all V_A_H_2 all V_r_2 all V_r_H_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_H_2),V_r_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_H_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_d,V_r_H_2),V_A_H_2)),hAPP(c_Relation_OImage(T_c,T_d,V_r_2),V_A_2)))))) # label(fact_Image__mono) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1289 (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]. 8.91/8.89 1290 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),c_Orderings_Otop__class_Otop(T_c)),V_A_2)))) # label(fact_Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1291 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_subset__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1292 (all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,V_R_2) -> (exists B_d hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,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_c,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_COMBS(T_c,T_c,tc_HOL_Obool),V_R_2),c_COMBI(T_c))))),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c)),c_fequal)),V_R_2)))))))) # label(fact_part__equivp__typedef) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1293 (all T_c all V_a_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,V_a_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_OCollect(T_c),V_Pa_2))))) # label(fact_CollectI) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1294 (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]. 8.91/8.89 1295 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d hAPP(hAPP(V_f1_2,V_a_2),V_b_2) = c_Product__Type_Oprod_Oprod__rec(T_d,T_e,T_c,V_f1_2,hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2))) # label(fact_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1296 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_d,V_f_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)))) # label(fact_bij__betw__imp__surj) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1297 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,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_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_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_inj__image__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1298 (all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> -(c_Predicate_Oreflp(T_c,V_R_2) -> (c_Predicate_Osymp(T_c,V_R_2) -> -c_Predicate_Otransp(T_c,V_R_2))))) # label(fact_equivpE) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1299 (all V_n_2 all V_x_2 (V_x_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> 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))))) # label(fact_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1300 (all V_i_2 all V_n_2 all V_inc_2 all T_c (class_Rings_Osemiring__1(T_c) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_c,V_inc_2,V_n_2,hAPP(V_inc_2,V_i_2)) = c_Nat_Osemiring__1__class_Oof__nat__aux(T_c,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]. 8.91/8.89 1301 (all V_R_2 all T_c hAPP(c_Relation_OField(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),V_R_2)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),c_Relation_OField(T_c)),V_R_2))) # label(fact_Field__Union) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1302 (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]. 8.91/8.89 1303 (all V_A_2 all V_x_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1304 (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]. 8.91/8.89 1305 (all V_i_1 all T_a hAPP(c_New__DSequence_Opos__empty(T_a),V_i_1) = c_Lazy__Sequence_Oempty(T_a)) # label(fact_New__DSequence_Opos__empty__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1306 (all V_x_2 all V_A_2 all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2)) -> (c_Fun_Oinj__on(T_e,T_c,V_g_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2)))) -> hAPP(hAPP(c_Fun_Ocomp(T_c,T_e,T_d,c_Fun_Othe__inv__into(T_e,T_c,V_A_2,V_g_2)),c_Fun_Othe__inv__into(T_c,T_d,hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2),V_f_2)),V_x_2) = hAPP(c_Fun_Othe__inv__into(T_e,T_d,V_A_2,hAPP(c_Fun_Ocomp(T_c,T_d,T_e,V_f_2),V_g_2)),V_x_2))))) # label(fact_the__inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1307 (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]. 8.91/8.89 1308 (all V_n all V_i hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_i)),V_n) = c_Code__Numeral_Onat__of__aux(V_i,V_n)) # label(fact_nat__of__aux__def) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1309 (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]. 8.91/8.89 1310 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),B_x)))) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_sorted__Cons) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1311 (all V_A_2 all T_c V_A_2 = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_double__complement) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1312 (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]. 8.91/8.89 1313 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Set_Ovimage(T_c,T_d,V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__INT) # label(axiom) # label(non_clause). [assumption]. 8.91/8.89 1314 (all V_A_2 all T_c (hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> -hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) | V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1315 (all V_r_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),c_List_Olist_ONil(T_c))),c_List_Olistrel(T_c,V_r_2))) -> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_listrel__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1316 (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]. 8.92/8.89 1317 (all V_x_2 all V_g_2 all T_f all V_f_2 all T_c all T_d all T_e hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_f,T_e),V_x_2))),hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_f,T_e),V_x_2))) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),hAPP(c_Product__Type_Oapfst(T_f,T_c,T_e,V_g_2),V_x_2))) # label(fact_apsnd__apfst) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1318 (all V_f_2 all T_d all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_d),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_d),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_c,T_d),tc_fun(T_c,tc_HOL_Obool)),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)),V_f_2)))),V_Pa_2))))))) # label(fact_finite__image__set) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1319 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))) # label(fact_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1320 (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]. 8.92/8.89 1321 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_m)) # label(fact_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1322 (all V_r_2 all V_q_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),c_Nitpick_Oplus__frac(T_c,V_q_2,c_Nitpick_Ouminus__frac(T_c,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> c_Nitpick_Oless__eq__frac(T_c,V_q_2,V_r_2))) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1323 (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]. 8.92/8.89 1324 (all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_List_Ohd(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_hd__in__set) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1325 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> 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__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1326 (all V_r_2 all V_x_2 all T_c hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_r_2) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_singleton__quotient) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1327 (all V_N_2 all V_h_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_2,V_F_2) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y)) = hAPP(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,hAPP(V_F_2,V_N_2)) = hAPP(V_F_2,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_folding__one__idem_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1328 (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),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x))) # label(fact_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1329 (all V_code__numeral_2 all V_f2_2 all V_f1_2 all T_c c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_c,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_c,V_f1_2,V_f2_2,V_code__numeral_2))) # label(fact_code__numeral_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1330 (all V_S_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_Set_Oimage(T_d,T_c,V_f_2)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_S_2))) # label(fact_image__Union) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1331 (all V_n 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),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),V_n)) # label(fact_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1332 (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))) -> 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))) = 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))) & (-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_Olist_ONil(tc_Int_Oint) = 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))))) # label(fact_upto__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1333 (all V_b_2 all V_a_2 all V_c_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Ointernal__split(T_d,T_e,T_c),V_c_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2)) = hAPP(hAPP(V_c_2,V_a_2),V_b_2)) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1334 (all V_w_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),V_w_2)) & 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]. 8.92/8.89 1335 (all V_s_2 all T_e all V_r_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,V_r_2) -> (c_Relation_Osingle__valued(T_d,T_e,V_s_2) -> c_Relation_Osingle__valued(T_c,T_e,hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_2),V_s_2))))) # label(fact_single__valued__rel__comp) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1336 (all V_r all V_q 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),V_q)),V_r) = 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(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]. 8.92/8.89 1337 (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]. 8.92/8.89 1338 (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]. 8.92/8.89 1339 (all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_bij__is__inj) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1340 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (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(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_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)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2))))) # label(fact_setsum_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1341 (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]. 8.92/8.89 1342 (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]. 8.92/8.89 1343 (all V_a_2 all V_b_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_a_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_atLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1344 (all V_a_2 all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),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))))) -> V_b_2 = V_a_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1345 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Qa_2) | -hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)))) # label(help_c__fconj__3) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1346 (all V_z_2 all V_m_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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Int_Onat,V_z_2))))) # label(fact_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1347 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = V_B_2 -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)) -> hAPP(V_h_2,hAPP(V_f_2,B_a)) = hAPP(V_g_2,B_a))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_h_2),V_B_2)))))) # label(fact_setsum__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1348 (all V_Qa_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),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)))),V_Qa_2))))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Qa_2),B_x))))))))) # label(fact_finite__Collect__bex) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1349 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_m_2,V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1350 (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]. 8.92/8.89 1351 (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]. 8.92/8.89 1352 (all V_x V_x = hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_x))) # label(fact_nat__of__inverse) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1353 (all V_com2_2 all V_com1_2 all V_fun_2 hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com1_2)),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com2_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_com_Osize_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1354 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> c_Relation_Osym(tc_List_Olist(T_c),c_List_Olistrel(T_c,V_r_2)))) # label(fact_listrel__sym) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1355 (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]. 8.92/8.89 1356 (all V_f_2 all V_A_2 all T_c (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_A_2),hAPP(c_Set_Oimage(T_c,T_c,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_c,T_c,V_f_2,V_A_2)))) # label(fact_finite__surj__inj) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1357 (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]. 8.92/8.89 1358 (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]. 8.92/8.89 1359 (all V_f_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_F_2)) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(V_f_2,B_x))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,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]. 8.92/8.89 1360 (all V_v_2 (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__eq(tc_Int_Oint),V_v_2),c_Int_OPls)))) # label(fact_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.89 1361 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,T_c),V_A_2),V_x_2) = c_Complete__Lattice_OSup__class_OSup(T_c,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)))))))) # label(fact_Sup__apply) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1362 (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]. 8.92/8.90 1363 (all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> c_Finite__Set_Ofun__left__comm__idem(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)))) # label(fact_fun__left__comm__idem) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1364 (all V_a_2 all V_b_2 all V_A_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_b_2),V_A_2))))) # label(fact_fold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1365 (all V_d_2 all V_c_2 all T_f all V_b_2 all V_a_2 all T_c all T_d all T_e (hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_a_2),V_b_2) = hAPP(c_Fun_Ocomp(T_f,T_d,T_c,V_c_2),V_d_2) -> (all B_v hAPP(V_a_2,hAPP(V_b_2,B_v)) = hAPP(V_c_2,hAPP(V_d_2,B_v))))) # label(fact_o__eq__elim) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1366 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(tc_List_Olist(T_c),tc_List_Olist(T_d),hAPP(c_List_Omap(T_c,T_d),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_inj__mapD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1367 (all V_Pa_2 all T_d all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> V_Pa_2 = c_Quotient_OBabs(T_c,T_d,c_Quotient_ORespects(T_c,V_R_2),V_Pa_2))) # label(fact_babs__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1368 (all V_y_2 all V_z_2 all V_f_2 all T_d all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_y_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_c),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]. 8.92/8.90 1369 (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_b = V_a & V_d != 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)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_r),V_c)))))) # label(fact_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1370 (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]. 8.92/8.90 1371 (all V_k_2 all V_i_2 all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_k_2),V_i_2))))) # label(fact_greaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1372 (all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Olinordered__semidom(T_d) -> ((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_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)))))) # label(fact_setprod__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1373 (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]. 8.92/8.90 1374 (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]. 8.92/8.90 1375 (all V_ys all V_y all V_xs all V_x all T_a 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))) = 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))) # label(fact_splice_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1376 (all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Ocomm__semiring__1(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,B_x))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2))))) # label(fact_setprod__zero) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1377 (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]. 8.92/8.90 1378 (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_x != V_y -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))))) # label(fact_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1379 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> 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)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)))) # label(fact_of__int__add) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1380 (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]. 8.92/8.90 1381 (all V_Pa_2 ((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_Pa_2,hAPP(c_Int_Onat,B_x))))) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))) # label(fact_ex__nat) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1382 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_d)))) # label(fact_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1383 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Max_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1384 (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]. 8.92/8.90 1385 (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]. 8.92/8.90 1386 (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]. 8.92/8.90 1387 (all V_f_2 all V_m_2 all V_n_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (-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_c),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))))) # label(fact_setsum__cl__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1388 (all V_xs_2 all V_Qa_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,hAPP(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),V_Qa_2)),V_Pa_2)),V_xs_2) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2))) # label(fact_filter__filter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1389 (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]. 8.92/8.90 1390 (all V_f_2 all T_d all T_c all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_c) -> hAPP(c_List_Ohd(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) = hAPP(V_f_2,hAPP(c_List_Ohd(T_c),V_xs_2)))) # label(fact_hd__map) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1391 (all V_C_2 all V_B_2 all V_A_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_e),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_INT__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1392 (all V_B_2 all V_A_2 all T_d all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = V_B_2 & V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool)))) # label(fact_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1393 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_IntI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1394 (all T_d all T_c (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(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool)))))) # label(fact_finite__Plus__UNIV__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1395 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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_Groups_Otimes__class_Otimes(T_d),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_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),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)),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),c_Groups_Oone__class_Oone(T_d)),V_B_2)))))) # label(fact_fold__image__Un__Int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1396 (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]. 8.92/8.90 1397 (all V_w_2 hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(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)))) # label(fact_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1398 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,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_c),hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)))),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))))),V_B_2),V_A_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) # label(fact_minus__fold__remove) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1399 (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]. 8.92/8.90 1400 (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]. 8.92/8.90 1401 (all V_Y_2 all T_c V_Y_2 = hAPP(c_Set_Ovimage(T_c,T_c,c_COMBI(T_c)),V_Y_2)) # label(fact_vimage__ident) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1402 (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]. 8.92/8.90 1403 (all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,c_List_Olist_ONil(T_d)) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_list__all2__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1404 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> c_Groups_Ozero__class_Ozero(T_a) = V_a))) # label(fact_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1405 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,T_c,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_COMBK(tc_Nat_Onat,T_c),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_c),V_A_2))) # label(fact_card_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1406 (all V_f_2 all T_c all T_d c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_d)) = c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c)) # label(fact__07) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1407 (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]. 8.92/8.90 1408 (all V_c_2 all V_b_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_x),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_c_2),V_A_2)),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_b_2),V_c_2))))))) # label(fact_min__max_Ofold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1409 (all V_v all V_i all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_v)) # label(fact_list__update_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1410 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),V_A_2))))))) # label(fact_Sup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1411 (all V_xs_2 all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2)) = hAPP(c_List_OremoveAll(T_d,hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)))) # label(fact_map__removeAll__inj) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1412 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Groups_Oordered__ab__semigroup__add__imp__le(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_c_2)))))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1413 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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))))) -> V_A_2 = c_Orderings_Obot__class_Obot(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_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1414 (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]. 8.92/8.90 1415 (all V_n_2 all T_c (class_Rings_Ocomm__semiring__1(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),c_Groups_Oone__class_Oone(T_c)),c_Groups_Oone__class_Oone(T_c))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),c_Nat_Osemiring__1__class_Oof__nat(T_c)),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_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),c_Groups_Oone__class_Oone(T_c))))) # label(fact_gauss__sum) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1416 (all V_x_2 all T_d all T_c (class_Orderings_Obot(T_c) -> hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_d,T_c)),V_x_2) = c_Orderings_Obot__class_Obot(T_c))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1417 (all V_xs_2 all T_c hAPP(c_List_Olist_Olist__case(tc_List_Olist(T_c),T_c,c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_c),hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Oappend(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c)))),V_xs_2) = hAPP(c_List_Orotate1(T_c),V_xs_2)) # label(fact_rotate1__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1418 (all V_xs_2 all T_c c_Random_Opick(T_c,hAPP(c_List_Ofilter(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Code__Numeral_Ocode__numeral,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_COMBK(tc_HOL_Obool,T_c)),hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))))),V_xs_2)) = c_Random_Opick(T_c,V_xs_2)) # label(fact_pick__drop__zero) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1419 (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]. 8.92/8.90 1420 (all V_c_2 all T_d all T_c all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) & (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1421 (all V_a_2 all V_f_2 all V_xs_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,V_a_2)),hAPP(V_f_2,B_x))))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_a_2),V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_xs_2)))) # label(fact_insort__is__Cons) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1422 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Orderings_Otop__class_Otop(T_c) = c_Complete__Lattice_OSup__class_OSup(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Sup__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1423 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> V_A_2 = hAPP(c_List_Oset(T_c),c_List_Olinorder__class_Osorted__list__of__set(T_c,V_A_2)) & c_List_Olinorder__class_Osorted(T_c,c_List_Olinorder__class_Osorted__list__of__set(T_c,V_A_2)) & hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Olinorder__class_Osorted__list__of__set(T_c,V_A_2)))))) # label(fact_sorted__list__of__set) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1424 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (V_y_2 != c_Groups_Ozero__class_Ozero(T_c) | c_Groups_Ozero__class_Ozero(T_c) != V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_y_2),V_y_2))))))) # label(fact_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1425 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(tc_List_Olist(T_c),tc_List_Olist(T_d),hAPP(c_List_Omap(T_c,T_d),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))) <-> c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_inj__map) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1426 (all T_d all V_x_2 all V_f_2 all T_c hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_rangeI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1427 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_b_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) = V_a_2))) # label(fact_minus__equation__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1428 (all V_B_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_B_2,V_a_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UN__upper) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1429 (all T_c (class_Nat_Osize(T_c) -> c_FunDef_Ois__measure(T_c,c_Nat_Osize__class_Osize(T_c)))) # label(fact_measure__size) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1430 (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)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_m))) # label(fact_mod__if) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1431 (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]. 8.92/8.90 1432 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1433 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)))) # label(fact_INT__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1434 (all V_Qa_2 all V_Q_Ha_2 all V_c_2 all V_Pa_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),V_c_2),V_Q_Ha_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> ((all B_Z all B_s (hBOOL(hAPP(hAPP(V_Q_Ha_2,B_Z),B_s)) -> hBOOL(hAPP(hAPP(V_Qa_2,B_Z),B_s)))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))))) # label(fact_conseq2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1435 (all V_p_2 all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_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_xs_2))) -> hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_p_2)),hAPP(hAPP(c_List_Onth(T_d),V_ys_2),V_p_2)))))) # label(fact_list__all2__nthD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1436 (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]. 8.92/8.90 1437 (all V_k_2 all V_C_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_C_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2))) -> ((all B_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c),V_C_2)) -> hAPP(c_Finite__Set_Ocard(T_c),B_c) = V_k_2)) -> ((all B_c1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c1),V_C_2)) -> (all B_c2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c2),V_C_2)) -> (B_c1 != B_c2 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),B_c1),B_c2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_fun(T_c,tc_HOL_Obool)),V_C_2)) = hAPP(c_Finite__Set_Ocard(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2))))))) # label(fact_card__partition) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1438 (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]. 8.92/8.90 1439 (all V_xs_2 all V_g_2 all T_d all V_f_2 all T_c c_List_Olist_Olist__size(T_d,hAPP(c_Fun_Ocomp(T_c,tc_Nat_Onat,T_d,V_f_2),V_g_2),V_xs_2) = c_List_Olist_Olist__size(T_c,V_f_2,hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_xs_2))) # label(fact_list__size__map) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1440 (all V_B_2 all V_A_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),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(c_Relation_ODomain(T_c,T_d),V_A_2)),hAPP(c_Relation_ODomain(T_c,T_d),V_B_2))))) # label(fact_Domain__Int__subset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1441 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_b_2),V_a_2)),V_r_2)))) # label(fact_converseD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1442 (all T_d all T_c c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool)) = c_Sum__Type_OPlus(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1443 (all V_r_2 all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_r__into__trancl_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1444 (all V_m_2 all V_n_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)))) # label(fact_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1445 (all V_r_2 all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 <-> c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) # label(fact_quotient__is__empty) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1446 (all V_x_2 all V_y_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_y_2)),hAPP(V_f_2,V_x_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2)))) # label(fact_folding_Ocommute__comp) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1447 (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]. 8.92/8.90 1448 (all V_a_2 all V_A_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)),V_a_2),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_fold1__eq__fold__idem) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1449 (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]. 8.92/8.90 1450 (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]. 8.92/8.90 1451 (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]. 8.92/8.90 1452 (all V_g_2 all T_e all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_B_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_B_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),V_g_2)))))) # label(fact_folding_Ocommute__left__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1453 (all V_w_2 all V_c_2 all V_b_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Ofield__inverse__zero(T_c) -> ((V_c_2 = c_Groups_Ozero__class_Ozero(T_c) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)) & (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))) # label(fact_divide__eq__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1454 (all V_A_2 all T_c all V_b_2 all V_a_2 (V_a_2 = V_b_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Relation_OId__on(T_c,V_A_2)))))) # label(fact_Id__on__eqI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1455 (all V_h_2 all T_d all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_h_2),V_F_2))))) # label(fact_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1456 (all V_R_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) -> V_b_2 != V_a_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_R_2))) | V_b_2 = V_a_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1457 (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_zmult__int) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1458 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_set__empty2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1459 (all V_m all V_n 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)) = hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m))) # label(fact_plus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1460 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_2) = c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),c_Relation_OId(T_c)))) # label(fact_rtrancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1461 (all V_R_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(T_c,T_c),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_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2)) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)) # label(fact_rtrancl__is__UN__rel__pow) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1462 (all T_c hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_finite_OemptyI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1463 (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]. 8.92/8.90 1464 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)))))) # label(fact_inj__on__strict__subset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1465 (all V_A_2 all T_c V_A_2 = hAPP(c_Set_Ovimage(T_c,T_c,c_Fun_Oid(T_c)),V_A_2)) # label(fact_vimage__id) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1466 (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]. 8.92/8.90 1467 (all V_xs all T_a V_xs = c_List_Osplice(T_a,V_xs,c_List_Olist_ONil(T_a))) # label(fact_splice__Nil2) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1468 (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]. 8.92/8.90 1469 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Ofilter(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2)),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_filter__map) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1470 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(V_k)) = c_Int_OBit1(c_Int_Opred(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)))) # label(fact_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1471 (all V_A_2 all V_f_2 all T_d all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2))) # label(fact_vimage__Compl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1472 (all V_g_2 all V_A_2 all T_d all V_I_2 all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_I_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_A_2,B_x))))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_I_2)) -> (B_xa != B_x -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_A_2,B_x)),hAPP(V_A_2,B_xa))))))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_d,c_Groups_Otimes__class_Otimes(T_e)),V_g_2),c_Groups_Oone__class_Oone(T_e)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_c,c_Groups_Otimes__class_Otimes(T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),T_e,T_c),hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_d,c_Groups_Otimes__class_Otimes(T_e)),V_g_2),c_Groups_Oone__class_Oone(T_e))),V_A_2)),c_Groups_Oone__class_Oone(T_e)),V_I_2)))))) # label(fact_fold__image__UN__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1473 (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]. 8.92/8.90 1474 (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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1475 (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]. 8.92/8.90 1476 (all V_S_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_COMBI(tc_fun(T_c,tc_HOL_Obool))) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_S_2)) # label(fact_Union__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1477 (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]. 8.92/8.90 1478 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> c_Relation_Otrans(tc_List_Olist(T_c),c_List_Olexord(T_c,V_r_2)))) # label(fact_lexord__transI) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1479 (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]. 8.92/8.90 1480 (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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),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_rx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1481 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))))))) # label(fact_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1482 (all V_n all V_m 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)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n)) # label(fact_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1483 (all V_n 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),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)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n))) # label(fact_power__divide) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1484 (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) = c_List_Olist_ONil(tc_Nat_Onat)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),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)) = c_List_Oupt(V_i,V_j)))) # label(fact_upt__rec) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1485 (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)))) -> 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))) = c_Divides_OnegDivAlg(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))) -> 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)) = c_Divides_OnegDivAlg(V_a_2,V_b_2)))) # label(fact_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1486 (all V_y all V_x all T_a (class_Int_Onumber__ring(T_a) -> 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)))) = 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)))) # label(fact_power2__diff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1487 (all T_c hAPP(c_Set_Oimage(T_c,T_c,c_Fun_Oid(T_c)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_surj__id) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1488 (all V_b_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) = V_a_2 | V_b_2 = V_a_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_a_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_b_2)))) # label(fact_square__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1489 (all V_g_2 all V_f_2 all T_c all T_d (class_Enum_Oenum(T_c) & class_Orderings_Oorder(T_d) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_c),hAPP(hAPP(c_COMBS(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Oord__class_Oless__eq(T_d)),V_f_2)),V_g_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2))))) # label(fact_order__fun_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1490 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)),V_a_2))))) # label(fact_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1491 (all V_ys_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),B_x),hAPP(c_List_Oset(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_List_Ozip(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),B_x)))) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_ys_2) -> (c_List_Oconcat(T_c,V_xs_2) = c_List_Oconcat(T_c,V_ys_2) <-> V_ys_2 = V_xs_2)))) # label(fact_concat__eq__concat__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1492 (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_n_2),hAPP(c_Nat_OSuc,V_m_2))) <-> V_n_2 = V_m_2))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1493 (all V_A_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(c_COMBK(T_c,T_d),c_Groups_Ozero__class_Ozero(T_c))),V_A_2) = c_Groups_Ozero__class_Ozero(T_c))) # label(fact_setsum__0) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1494 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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)))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1495 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__Compl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1496 (all V_g_2 all V_f_2 all T_d all T_c V_f_2 = c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_override__on__emptyset) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1497 (all V_Pa_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Ofilter(T_d,V_Pa_2),V_xs_2)))))) # label(fact_sorted__filter) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1498 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),B_x))))))))) # label(fact_strict__below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1499 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_x_2 all T_c c_New__DSequence_Oneg__single(T_c,V_x_2) = hAPP(hAPP(hAPP(hAPP(c_New__Random__Sequence_Oneg__single(T_c),V_x_2),V_nrandom_2),V_size_2),V_seed_2)) # label(fact_neg__single__def) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1500 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> c_Relation_Osym(T_c,c_Transitive__Closure_Otrancl(T_c,V_r_2)))) # label(fact_sym__trancl) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1501 (all V_l_2 all V_k_2 ((V_k_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (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)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_l_2 -> c_Divides_Odivmod__int(V_k_2,V_l_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> c_Divides_Odivmod__int(V_k_2,V_l_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))))) # label(fact_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1502 (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]. 8.92/8.90 1503 (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_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))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,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)) = 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)))))) # label(fact_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1504 (all V_r_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) & class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_r_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_r_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))))) # label(fact_half__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1505 (all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1506 (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]. 8.92/8.90 1507 (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]. 8.92/8.90 1508 (all V_n V_n != hAPP(c_Nat_OSuc,V_n)) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1509 (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]. 8.92/8.90 1510 (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]. 8.92/8.90 1511 (all V_a_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_OImage(T_c,T_c,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_equiv__class__self) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1512 (all V_y_2 all V_x_2 all V_b_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Oone__class_Oone(T_c)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_b_2),V_x_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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]. 8.92/8.90 1513 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_f1_2 all T_c all T_d c_Hoare__Mirabelle_Otriple_Otriple__rec(T_d,T_c,V_f1_2,hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_d),V_fun1_2),V_com_2),V_fun2_2)) = hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2)) # label(fact_triple_Orecs) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1514 (all V_c_2 all T_c -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.92/8.90 1515 (all V_A_2 all V_f_2 all T_d all V_r_2 all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_r_2),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),hAPP(c_Groups_Otimes__class_Otimes(T_c),V_r_2)),V_f_2)),V_A_2))) # label(fact_setsum__right__distrib) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 1516 (all V_zs_2 all T_d all T_c V_zs_2 = hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_c),c_Product__Type_Ofst(T_c,T_d)),V_zs_2)),hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_d),c_Product__Type_Osnd(T_c,T_d)),V_zs_2))) # label(fact_zip__map__fst__snd) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 1517 (all V_n_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_c),hAPP(c_List_Olexn(T_c,V_r_2),V_n_2)))) # label(fact_wf__lexn) # label(axiom) # label(non_clause). [assumption]. 8.92/8.91 1518 (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]. 8.92/8.91 1519 (all V_M_2 all V_n_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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]. 8.92/8.91 1520 (all V_x_2 all T_c c_Set_Othe__elem(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_x_2) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1521 (all V_B_2 all V_A_2 all V_l_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_l_2),c_List_Olists(T_c,V_A_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_l_2),c_List_Olists(T_c,V_B_2))) -> hBOOL(hAPP(c_List_Olistsp(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_l_2))))) # label(fact_lists__IntI) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1522 (all V_A_2 all T_c -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_not__psubset__empty) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1523 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(V_b_2,V_a_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2) = c_Groups_Ozero__class_Ozero(T_d))))) # label(fact_setsum__delta) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1524 (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]. 8.93/8.91 1525 (all V_f2_2 all V_f1_2 all T_c c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_c,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]. 8.93/8.91 1526 (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]. 8.93/8.91 1527 (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_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)),V_c) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_a)))) # label(fact_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1528 (all V_z_2 all V_y_2 all V_x_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,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]. 8.93/8.91 1529 (all V_N_2 all V_M_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_M_2),V_N_2)) -> (V_M_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_M_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_N_2)))))))) # label(fact_Max__mono) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1530 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2),hAPP(c_List_Orev(T_d),V_ys_2)))) # label(fact_list__all2__rev) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1531 (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]. 8.93/8.91 1532 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),hAPP(c_COMBK(tc_HOL_Obool,T_c),c_fTrue))) # label(fact_UNIV__def) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1533 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,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(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__insert2) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1534 (all V_xs_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)))) # label(fact_length__sort) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1535 (all V_F_2 all V_E_2 all V_B_2 all V_A_2 all T_d all T_c all T_f all T_e hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_prod(T_e,T_f),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_e,T_f),V_A_2),hAPP(c_COMBK(tc_fun(T_f,tc_HOL_Obool),T_e),V_B_2))),hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_f,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),tc_fun(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_f,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),tc_fun(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),T_e),c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_f)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_e),c_Product__Type_OSigma(T_c,T_d)),V_E_2))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),T_f),c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c)),V_F_2)))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_E_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_F_2)))) # label(fact_UN__Times__distrib) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1536 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (((exists 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)))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)))))) # label(fact_card_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1537 (all V_k_2 all V_M_2 all V_n_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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_c,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)))) # label(fact_UN__le__add__shift) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1538 (all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1539 (all V_B_2 all V_a_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2))))) # label(fact_UN__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1540 (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_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),V_n))) # label(fact_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1541 (all V_r_2 all T_c ((all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))) <-> c_Wellfounded_Oacyclic(T_c,V_r_2))) # label(fact_acyclic__def) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1542 (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)),V_b),V_c))) # label(fact_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1543 (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) -> c_List_Olist_ONil(T_a) = 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))))) # label(fact_take__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1544 (all V_xs all T_a c_List_Oremdups(T_a,V_xs) = c_List_Oremdups(T_a,c_List_Oremdups(T_a,V_xs))) # label(fact_remdups__remdups) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1545 (all V_Qa_2 all V_Pa_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_Set_OCollect(T_c),V_Pa_2)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),c_Set_OCollect(T_d)),V_Qa_2)) = hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),V_Pa_2))),V_Qa_2)))) # label(fact_SetCompr__Sigma__eq) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1546 (all T_c all B_P ((exists B_x hBOOL(hAPP(B_P,B_x))) -> hBOOL(hAPP(B_P,c_Hilbert__Choice_OEps(T_c,B_P))))) # label(fact_tfl__some) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1547 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_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),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)))) # label(fact_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1548 (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]. 8.93/8.91 1549 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__bot__left) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1550 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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)),V_B_2),V_A_2)) -> V_B_2 = V_A_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1551 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),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]. 8.93/8.91 1552 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),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))))))) & (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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)))))) # label(fact_min__max_OInf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1553 (all V_w hAPP(c_Groups_Ouminus__class_Ouminus(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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1554 (all V_Qa_2 all V_Pa_2 (-hBOOL(V_Pa_2) | hBOOL(V_Qa_2) | -hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)))) # label(help_c__fimplies__3) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1555 (all V_g_2 all V_x_2 all V_f_2 all T_c all T_d all T_e (hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_c,T_e),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_c,T_e),V_x_2)) <-> hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),V_x_2) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_g_2),V_x_2))) # label(fact_apsnd__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1556 (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_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) & (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) != c_Groups_Ozero__class_Ozero(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)) = 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)))))) # label(fact_nth__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1557 (all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))) # label(fact_neg__0__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1558 (all V_a_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_a_2),V_A_2)))) # label(fact_Sup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1559 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) | V_c_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1560 (all V_B_2 all V_a_2 all T_c hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)))) # label(fact_insertI1) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1561 (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),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_inf__sup__aci_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1562 (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]. 8.93/8.91 1563 (all V_z_2 all V_w_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_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_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]. 8.93/8.91 1564 (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]. 8.93/8.91 1565 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2)))) # label(fact_insort__insert__insort) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1566 (all V_y all V_x hAPP(hAPP(c_Nitpick_Oint__gcd,V_x),V_y) = 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))))) # label(fact_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1567 (all V_r_2 all T_c (c_Wellfounded_Oacyclic(T_c,V_r_2) -> c_Relation_Oantisym(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) # label(fact_acyclic__impl__antisym__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1568 (all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1569 (all V_f_2 all V_A_2 all V_a_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> 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,tc_Nat_Onat),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2))))) # label(fact_setsum__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1570 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_d_2)) -> (V_d_2 = V_b_2 & V_a_2 = V_c_2 <-> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2)))))) # label(fact_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1571 (all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Set_Ovimage(T_c,T_d,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]. 8.93/8.91 1572 (all V_f_2 all T_d all T_c all V_S_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) != V_S_2 -> ((all B_A (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_A),V_S_2)) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,B_A))) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_S_2))))) # label(fact_inj__on__Inter) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1573 (all V_a_2 all T_c 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_c),hAPP(c_fequal,V_a_2))) # label(fact_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1574 (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]. 8.93/8.91 1575 (all V_B_2 all 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) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2) <-> V_B_2 = V_A_2)) # label(fact_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1576 (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(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),V_n))) # label(fact_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1577 (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]. 8.93/8.91 1578 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,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)),V_A_2),V_B_2)))) # label(fact_psubset__imp__subset) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1579 (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]. 8.93/8.91 1580 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Oid(T_c) = hAPP(c_Fun_Ocomp(T_d,T_c,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),V_f_2))) # label(fact_inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1581 (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_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.93/8.91 1582 (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,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))) = 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)) & (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]. 8.94/8.91 1583 (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]. 8.94/8.91 1584 (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]. 8.94/8.91 1585 (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]. 8.94/8.91 1586 (all V_r_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) <-> c_Relation_Orefl__on(T_c,V_A_2,V_r_2))) # label(fact_refl__on__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1587 (all V_xs_2 all V_n_2 all T_c c_Enum_On__lists(T_c,hAPP(c_Nat_OSuc,V_n_2),V_xs_2) = c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),tc_List_Olist(T_c)),c_List_Omap(T_c,tc_List_Olist(T_c))),hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)))),V_xs_2)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)))) # label(fact_n__lists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1588 (all V_B_2 all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> (hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2) <-> V_B_2 = V_A_2)))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1589 (all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_OtakeWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),hAPP(c_List_Orev(T_c),V_xs_2)) = hAPP(c_List_Orev(T_c),hAPP(c_List_Otl(T_c),c_List_OdropWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),V_xs_2)))))) # label(fact_takeWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1590 (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]. 8.94/8.91 1591 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_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_c)),V_ys_2))) # label(fact_listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1592 (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]. 8.94/8.91 1593 (all V_n_2 all V_m_2 ((exists B_k V_n_2 = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1594 (all V_xs_2 all V_n_2 all T_c (hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_rotate__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1595 (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]. 8.94/8.91 1596 (all V_n all V_m all V_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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))) -> (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_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m),V_n))))) # label(fact_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1597 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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__eq(T_c),B_x),V_x_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)),V_x_2))))))) # label(fact_fold1__below__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1598 (all V_R_2 all T_c (c_Predicate_Otransp(T_c,V_R_2) & c_Predicate_Osymp(T_c,V_R_2) & c_Predicate_Oreflp(T_c,V_R_2) <-> c_Equiv__Relations_Oequivp(T_c,V_R_2))) # label(fact_equivp__reflp__symp__transp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1599 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omax(T_c)),V_A_2) = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)))) # label(fact_Max_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1600 (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]. 8.94/8.91 1601 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,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))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2) = V_x_2))))) # label(fact_Min_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1602 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Orderings_Obot__class_Obot(T_c) = c_Complete__Lattice_OInf__class_OInf(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Inf__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1603 (all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_d,T_c)),V_r_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))))) # label(fact_finite__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1604 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),V_ys_2) = c_List_OtakeWhile(tc_prod(T_c,T_d),hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,tc_prod(T_c,T_d),V_Pa_2),c_Product__Type_Ofst(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2))) # label(fact_zip__takeWhile__fst) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1605 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_List_Olinorder__class_Osorted__list__of__set(T_c,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))))) = c_List_Oremove1(T_c,V_x_2,c_List_Olinorder__class_Osorted__list__of__set(T_c,V_A_2))))) # label(fact_sorted__list__of__set__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1606 (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]. 8.94/8.91 1607 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> c_Relation_Otrans(tc_List_Olist(T_c),c_List_Olistrel(T_c,V_r_2)))) # label(fact_listrel__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1608 (all V_g_2 all V_f_2 all T_d all T_c hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2) = c_Finite__Set_Ofold(T_d,T_c,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),V_f_2),V_g_2))) # label(fact_fold__image__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1609 (all V_nat c_Nat_Onat_Onat__size(hAPP(c_Nat_OSuc,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)))) # label(fact_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1610 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1611 (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_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_u_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_x_2))),hAPP(hAPP(c_List_Oappend(T_c),V_u_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_b_2),V_y_2)))),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__append__left__rightI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1612 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_Orotate1(T_c),V_xs_2)) = hAPP(c_List_Oset(T_c),V_xs_2)) # label(fact_set__rotate1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1613 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (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_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1614 (all V_b_2 all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_b_2),V_A_2))))))) # label(fact_min__max_Osup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1615 (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__eq(tc_Int_Oint),V_k_2),V_i_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)))))) # label(fact_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1616 (all V_k_2 (V_k_2 = c_Int_OPls <-> c_Int_OPls = c_Int_OBit0(V_k_2))) # label(fact_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1617 (all T_c hAPP(c_Set_OPow(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Pow__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1618 (all V_A_2 all T_c hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),c_HOL_OEx(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBS(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_fconj)),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool))),c_fequal)),c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),c_Set_OPow(T_c))))) = hAPP(c_Set_OPow(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Pow__Compl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1619 (all V_bs_2 all V_f_2 all T_e all V_as_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_as_2,hAPP(hAPP(c_List_Omap(T_e,T_d),V_f_2),V_bs_2)) <-> c_List_Olist__all2(T_c,T_e,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_e,T_d),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_e,T_d),tc_fun(T_e,tc_HOL_Obool)),T_c),c_COMBB(T_d,tc_HOL_Obool,T_e)),V_Pa_2)),V_f_2),V_as_2,V_bs_2))) # label(fact_list__all2__map2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1620 (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]. 8.94/8.91 1621 (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]. 8.94/8.91 1622 (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]. 8.94/8.91 1623 (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]. 8.94/8.91 1624 (all V_k_2 all V_R_2 all V_t_2 all T_d all V_s_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_s_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_t_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_t_2)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_OCollect(T_c),hAPP(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_s_2))),hAPP(hAPP(c_COMBC(T_c,T_d,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_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_Nat_Onat,T_c),c_Finite__Set_Ocard(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),c_Set_OCollect(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_t_2)))),V_R_2)))),V_s_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,tc_Nat_Onat),V_k_2),V_t_2))))) # label(fact_setsum__multicount__gen) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1625 (all V_n 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_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))) = 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))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1626 (all V_R_2 all V_Y_2 all V_X_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_X_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Y_2)) -> (V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_X_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_Y_2)) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)))))) -> c_Wellfounded_Omax__extp(T_c,V_R_2,V_X_2,V_Y_2)))))) # label(fact_max__extp_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1627 (all V_c_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))))) # label(fact_subsetD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1628 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Relation_OId__on(T_c,V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2))))) # label(fact_Id__on__subset__Times) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1629 (all V_R_2 all V_x_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_x_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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]. 8.94/8.91 1630 (all V_com_2 all V_fun_2 hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OWhile(V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_com_Osize_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1631 (all V_xs all V_n all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_take__tl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1632 (all V_xs_2 all T_c hAPP(hAPP(c_List_Omap(T_c,T_c),c_COMBI(T_c)),V_xs_2) = V_xs_2) # label(fact_map__ident) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1633 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = V_m -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n | V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1634 (all V_x_2 all V_a_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_x_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_a_2)),V_r_2))))) # label(fact_wf__not__sym) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1635 (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_r = V_r_H)))) # label(fact_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1636 (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]. 8.94/8.91 1637 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_d,T_c),V_A_2)),hAPP(c_Relation_ORange(T_d,T_c),V_B_2)) = hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1638 (all V_v_2 all V_d_2 all V_c_2 all T_f all V_b_2 all V_a_2 all T_c all T_d all T_e (hAPP(c_Fun_Ocomp(T_f,T_d,T_c,V_c_2),V_d_2) = hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_a_2),V_b_2) -> hAPP(V_a_2,hAPP(V_b_2,V_v_2)) = hAPP(V_c_2,hAPP(V_d_2,V_v_2)))) # label(fact_o__eq__dest) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1639 (all V_x_2 all T_c all V_B_2 all V_A_2 (V_B_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_x_2),V_B_2))))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1640 (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]. 8.94/8.91 1641 (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),c_Int_OPls),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_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]. 8.94/8.91 1642 (all V_com2 all V_com1 hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OSemi(V_com1,V_com2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com1)),hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),V_com2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_com_Osize_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1643 (all V_A_2 all V_r_2 all T_c (V_r_2 = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)),V_r_2) -> (hAPP(c_Relation_ODomain(T_c,T_c),V_r_2) = V_A_2 -> c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2)))) # label(fact_comp__equivI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1644 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),c_Groups_Oone__class_Oone(T_c)))))) # label(fact_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1645 (all V_xs_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_y_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2)))) # label(fact_insort__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1646 (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__eq(tc_Int_Oint),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2))))) # label(fact_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1647 (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]. 8.94/8.91 1648 (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]. 8.94/8.91 1649 (all V_Pa_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> hBOOL(hAPP(V_Pa_2,V_a_2)))) # label(fact_CollectD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1650 (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]. 8.94/8.91 1651 (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]. 8.94/8.91 1652 (all V_v_2 all V_m_2 (hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2) <-> hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)) = V_m_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]. 8.94/8.91 1653 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) -> ((all B_M (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_M),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_M)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)))))) # label(fact_finite__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1654 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2) <-> V_n_2 = V_m_2)) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1655 (all V_k_2 all V_N_2 all T_c (class_Groups_Olinordered__ab__semigroup__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_k_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_N_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(T_c),V_k_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_N_2))))))))) # label(fact_add__Max__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1656 (all V_Pa_2 all V_xs_2 all T_c 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_c)),V_xs_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_n_2)))))) # label(fact_list__ball__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1657 (all V_xs_2 all T_c 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_c))),c_List_Otranspose(T_c,V_xs_2)))) -> hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_Nat_Onat,T_c),c_List_Onth(T_c)),V_i_2)),hAPP(c_List_Ofilter(tc_List_Olist(T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_xs_2)) = hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),c_List_Otranspose(T_c,V_xs_2)),V_i_2))) # label(fact_nth__transpose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1658 (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]. 8.94/8.91 1659 (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]. 8.94/8.91 1660 (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]. 8.94/8.91 1661 (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]. 8.94/8.91 1662 (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_q = c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_self__quotient) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1663 (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 V_x = 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)))))) # label(fact_negD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1664 (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]. 8.94/8.91 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)) -> (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]. 8.94/8.91 1666 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_ys_2) -> (c_Fun_Oinj__on(T_d,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> V_ys_2 = V_xs_2))) # label(fact_map__injective) # label(axiom) # label(non_clause). [assumption]. 8.94/8.91 1667 (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]. 8.94/8.92 1668 (all V_B_2 all V_A_2 all T_c (class_Lattices_Odistrib__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBS(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_c),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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal))),c_Lattices_Osemilattice__sup__class_Osup(T_c)))))),hAPP(hAPP(c_COMBC(T_c,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_c),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_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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))))))) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2)))))))) # label(fact_sup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1669 (all V_g_2 all V_x_2 all V_f_2 all T_d all T_c all T_e (hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)) <-> hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_g_2),V_x_2) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),V_x_2))) # label(fact_apfst__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1670 (all V_k_2 all V_Pa_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_Pa_2,B_x)) -> hBOOL(hAPP(V_Pa_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_Pa_2,B_x)) -> hBOOL(hAPP(V_Pa_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]. 8.94/8.92 1671 (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]. 8.94/8.92 1672 (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 = V_a_H -> V_b != V_b_H))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1673 (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]. 8.94/8.92 1674 (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]. 8.94/8.92 1675 (all V_xs all T_b all T_a c_List_Olist_ONil(tc_prod(T_a,T_b)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),c_List_Olist_ONil(T_b))) # label(fact_zip_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1676 (all V_x all T_a (class_Groups_Osgn__if(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_x)) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) = c_Groups_Osgn__class_Osgn(T_a,V_x)) & (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))) & (V_x = c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_sgn__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1677 (all V_R_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Quotient_ORespects(T_c,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]. 8.94/8.92 1678 (all V_x_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Wellfounded_Oacc(T_c,V_r_2))))) # label(fact_acc__wfD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1679 (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]. 8.94/8.92 1680 (all V_ys_2 all V_xs_2 all V_x_2 all T_d all T_c c_Enum_Oproduct(T_c,T_d,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2),V_ys_2) = hAPP(hAPP(c_List_Oappend(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Omap(T_d,tc_prod(T_c,T_d)),hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2)),V_ys_2)),c_Enum_Oproduct(T_c,T_d,V_xs_2,V_ys_2))) # label(fact_product_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1681 (all V_n_2 all T_c (class_Rings_Osemiring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),c_Nat_Osemiring__1__class_ONats(T_c))))) # label(fact_of__nat__in__Nats) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1682 (all V_Qa_2 all V_Pa_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2),V_Qa_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),B_x)),c_Transitive__Closure_Ortrancl(T_c,V_Pa_2))) -> (all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_Qa_2)) -> B_y = B_x)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_Pa_2)))))) # label(fact_rtrancl__Un__separatorE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1683 (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]. 8.94/8.92 1684 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_x_2),V_A_2))))) # label(fact_Max_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1685 (all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_a_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_2)) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_a_2),V_B_2))) # label(fact_Inter__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1686 (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]. 8.94/8.92 1687 (all V_xs all V_n all T_a 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) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_On__lists(T_a,V_n,V_xs))) # label(fact_length__n__lists) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1688 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_c_2,V_d_2)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)))) # label(fact_Int__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1689 (all V_y_2 all V_k_2 all V_xs_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Olist__update(T_d,V_xs_2),V_k_2),V_y_2)) = hAPP(hAPP(c_List_Olist__update(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),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]. 8.94/8.92 1690 (all V_c_2 all V_b_2 all V_A_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (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)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_x),V_b_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_c_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_b_2),V_c_2))))))) # label(fact_fold__sup__le__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1691 (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_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1692 (all V_R_2 all V_n_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)) -> c_Wellfounded_Owf(T_c,V_R_2))) # label(fact_wf__exp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1693 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1694 (all V_a_2 all V_b_2 all T_c (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_c) -> (V_b_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_a_2) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1695 (all V_k all V_n all V_m 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)) = 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)) # label(fact_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1696 (all V_n 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(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1697 (all V_rb_2 all T_d all V_ra_2 all T_c (c_Wellfounded_Owf(T_c,V_ra_2) -> (c_Wellfounded_Owf(T_d,V_rb_2) -> c_Wellfounded_Owf(tc_prod(T_c,T_d),c_Wellfounded_Olex__prod(T_c,T_d,V_ra_2,V_rb_2))))) # label(fact_wf__lex__prod) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1698 (all V_a_2 all T_c (class_Rings_Oring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),c_Int_Oring__1__class_OInts(T_c)))))) # label(fact_Ints__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1699 (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]. 8.94/8.92 1700 (all V_com_2 all V_fun_2 all V_loc_2 all V_com2_H_2 all V_com1_H_2 c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2)) # label(fact_com_Osimps_I35_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1701 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(c_Relation_ORange(T_c,T_d),V_r_2))))) # label(fact_RangeI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1702 (all V_S_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_S_2 -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),V_S_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_S_2)))))) # label(fact_minus__Max__eq__Min) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1703 (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]. 8.94/8.92 1704 (all V_g_2 all V_f_2 all V_xs_2 all T_c all T_d (class_Groups_Omonoid__add(T_d) & class_Groups_Oordered__ab__semigroup__add(T_d) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_g_2),V_xs_2))))))) # label(fact_listsum__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1705 (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_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1706 (all V_x all T_a c_List_Olist_ONil(T_a) = hAPP(c_List_OremoveAll(T_a,V_x),c_List_Olist_ONil(T_a))) # label(fact_removeAll_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1707 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_list__all__iff__raw) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1708 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> ((V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x_2 != V_z_2) -> -(all B_y all B_m (V_n_2 = hAPP(c_Nat_OSuc,B_m) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),B_m),V_R_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_z_2)),V_R_2)))))))) # label(fact_rel__pow__E) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1709 (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]. 8.94/8.92 1710 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k))) # label(fact_number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1711 (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]. 8.94/8.92 1712 (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]. 8.94/8.92 1713 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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(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),V_C_2))) # label(fact_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1714 (all V_x all V_k all V_i all T_a c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_k),V_i),V_x) = hAPP(hAPP(c_List_Odrop(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x))) # label(fact_drop__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1715 (all V_f_2 all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),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_c,V_f_2,V_xs_2)))))) # label(fact_list__size__estimation_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1716 (all V_y_2 all V_x_2 all V_z_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_x_2))))) # label(fact_min__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1717 (all V_B_2 all V_k_2 all V_A_2 all T_c ((all B_n 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(tc_Nat_Onat,tc_fun(T_c,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_c,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_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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_c,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]. 8.94/8.92 1718 (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]. 8.94/8.92 1719 (all V_xs_2 all V_y_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> hAPP(c_List_Ofoldl(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_d),V_f_2),V_y_2),c_List_Oremdups(T_c,V_xs_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_y_2),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_fun__left__comm_Ofold__set__remdups) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1720 (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]. 8.94/8.92 1721 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oordered__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_b_2),V_a_2)),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2)))))) # label(fact_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1722 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(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),V_B_2)) = V_A_2) # label(fact_Un__Diff__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1723 (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),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a_H),V_b)))) # label(fact_mod__minus__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1724 (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]. 8.94/8.92 1725 (all V_h_2 all V_g_2 all T_f all V_f_2 all T_c all T_d all T_e hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_2),hAPP(c_Fun_Ocomp(T_f,T_e,T_c,V_g_2),V_h_2)) = hAPP(c_Fun_Ocomp(T_f,T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_d,T_f,V_f_2),V_g_2)),V_h_2)) # label(fact_o__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1726 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omin(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_Min__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1727 (all V_A_2 all T_c c_Fun_Obij__betw(T_c,T_c,c_Fun_Oid(T_c),V_A_2,V_A_2)) # label(fact_bij__betw__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1728 (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]. 8.94/8.92 1729 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) -> ((c_Groups_Ozero__class_Ozero(T_c) != V_c_2 -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)) & (c_Groups_Ozero__class_Ozero(T_c) = V_c_2 -> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)) <-> V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))) # label(fact_eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1730 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),V_A_2))))))) # label(fact_Min_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1731 (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]. 8.94/8.92 1732 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (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_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))))) # label(fact_converse__rtrancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1733 (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]. 8.94/8.92 1734 (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]. 8.94/8.92 1735 (all T_1 all T_2 (class_Enum_Oenum(T_2) & class_Enum_Oenum(T_1) -> class_Enum_Oenum(tc_prod(T_2,T_1)))) # label(arity_prod__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1736 (all V_B_2 all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),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))))),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_2)))))) # label(fact_diff__single__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1737 (all V_f_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (exists B_x1 hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,V_A_2),B_x1)))))) # label(fact_finite__nonempty__imp__fold1Set) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1738 (all V_x_2 all V_A_2 all T_c (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(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),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)))))) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__Suc__Diff1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1739 (all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> c_Fun_Oid(T_d) = hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_folding_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1740 (all V_A_2 all T_c (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_subset__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1741 (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_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_xs)) = 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))) # label(fact_replicate__app__Cons__same) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1742 (all V_y_2 all V_x_2 all V_r_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),T_c),hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_d),c_COMBC(T_e,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_d,T_e)))),c_Product__Type_OPair(T_d,T_e))))),V_r_2)),V_x_2),V_y_2)))) # label(fact_SUP__UN__eq2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1743 (all V_xs_2 all V_f_2 all T_c hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_f_2),V_xs_2)),hAPP(c_List_Ofilter(T_c,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,T_c,c_fNot),V_f_2)),V_xs_2)) = c_List_Opartition(T_c,V_f_2,V_xs_2)) # label(fact_partition__filter__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1744 (all V_f_2 all V_B_2 all V_A_2 all T_c (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_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),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(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_c,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_setsum__Un__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1745 (all V_x_2 all V_g_2 all T_f all V_f_2 all T_d all T_c all T_e hAPP(c_Product__Type_Oapfst(T_f,T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_c,T_f,V_f_2),V_g_2)),V_x_2) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),hAPP(c_Product__Type_Oapfst(T_f,T_e,T_d,V_g_2),V_x_2))) # label(fact_apfst__compose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1746 (all V_B_2 all V_A_2 all V_c_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1747 (all V_qr_2 all V_n_2 all V_m_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2) = c_Groups_Ozero__class_Ozero(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_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))) & (-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))))) & 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)) <-> 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]. 8.94/8.92 1748 (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]. 8.94/8.92 1749 (all V_q all T_a all T_b (class_Rings_Oring__1(T_b) & class_Rings_Oinverse(T_b) -> c_Nitpick_Oof__frac(T_a,T_b,V_q) = 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))))) # label(fact_Nitpick_Oof__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1750 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,V_r_2) <-> c_Relation_Ototal__on(T_c,V_A_2,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_total__on__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1751 (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]. 8.94/8.92 1752 (all V_F_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_x_2),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_x_2),V_F_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_xa),hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),V_x_2),V_F_2))) -> (B_x != B_xa -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),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)),V_x_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_F_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_insert__partition) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1753 (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))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_c -> 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]. 8.94/8.92 1754 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord__class_Omin(T_c) = c_Orderings_Oord_Omin(T_c,c_Orderings_Oord__class_Oless__eq(T_c)))) # label(fact_min__ord__min) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1755 (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]. 8.94/8.92 1756 (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 hAPP(c_Code__Numeral_Onat__of,B_x) != V_y_2))) # label(fact_nat__of__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1757 (all V_C_2 all V_B_2 all V_A_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,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_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_e),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_UN__UN__flatten) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1758 (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),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_y,V_m)),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m)) # label(fact_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1759 (all V_a_2 all V_B_2 all V_A_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_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))) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,V_a_2))) -> -hBOOL(hAPP(hAPP(c_member(T_d),V_a_2),V_A_2))))) # label(fact_INT__E) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1760 (all V_xs_2 all V_f_2 all T_d all T_c c_List_Omaps(T_d,T_c,V_f_2,V_xs_2) = c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(T_d,tc_List_Olist(T_c)),V_f_2),V_xs_2))) # label(fact_concat__map__maps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1761 (all V_xs_2 all V_g_2 all T_e all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_e,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_g_2),V_xs_2))) # label(fact_map__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1762 (all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> c_Equiv__Relations_Opart__equivp(T_c,V_R_2))) # label(fact_equivp__implies__part__equivp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1763 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_divide__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1764 (all V_A_2 all V_B_2 all V_c_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnCI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1765 (all V_x_2 all V_g_2 all T_f all V_f_2 all T_c all T_d all T_e hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),hAPP(c_Product__Type_Oapsnd(T_f,T_e,T_c,V_g_2),V_x_2)) = hAPP(c_Product__Type_Oapsnd(T_f,T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_d,T_f,V_f_2),V_g_2)),V_x_2)) # label(fact_apsnd__compose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1766 (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]. 8.94/8.92 1767 (all V_y_2 all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Oremove1(T_c,V_y_2,V_xs_2)))))) # label(fact_notin__set__remove1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1768 (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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1769 (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))) -> 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))) = c_Divides_OnegDivAlg(V_a,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))) -> 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)) = c_Divides_OnegDivAlg(V_a,V_b)))) # label(fact_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1770 (all V_xs_2 all V_Pa_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2) = c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))) # label(fact_takeWhile__eq__filter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1771 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_divide__zero__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1772 (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),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),V_u_2))) # label(fact_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1773 (all V_f_2 all T_e all V_g_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Oimage(T_d,T_c,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_e),hAPP(c_Set_Oimage(T_d,T_e,hAPP(hAPP(c_COMBB(T_c,T_e,T_d),V_f_2),V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))))) # label(fact_finite__range__imageI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1774 (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)) -> (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)) <-> V_q_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1775 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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))) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)))) # label(fact_bij__image__Compl__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1776 (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]. 8.94/8.92 1777 (all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (class_Groups_Ominus(T_c) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(V_A_2,V_x_2)),hAPP(V_B_2,V_x_2)) = hAPP(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,T_c)),V_A_2),V_B_2),V_x_2))) # label(fact_fun__diff__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1778 (all V_g_2 all V_f_2 all T_d all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,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_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_g_2)))))) # label(fact_INT__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1779 (all V_x all V_k all V_i all T_a hAPP(hAPP(c_List_Otake(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_i),V_k),V_x)) # label(fact_take__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1780 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Ouminus__class_Ouminus(T_c)),V_f_2)),V_A_2))) # label(fact_setsum__negf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1781 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_y_2) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_x_2) <-> V_y_2 = V_x_2))) # label(fact_greaterThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1782 (all T_e all V_f_2 all V_r2_2 all T_d all V_r1_2 all T_c ((all B_y1 all B_z1 (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y1),B_z1)),V_r1_2)) -> (all B_y2 all B_z2 (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),B_y2),B_z2)),V_r2_2)) -> hAPP(hAPP(V_f_2,B_y1),B_y2) = hAPP(hAPP(V_f_2,B_z1),B_z2))))) -> c_Equiv__Relations_Ocongruent2(T_c,T_d,T_e,V_r1_2,V_r2_2,V_f_2))) # label(fact_congruent2I_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1783 (all V_x_2 all V_z_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_x_2)) -> V_z_2 = V_x_2)) # label(fact_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1784 (all V_y_2 all V_x_2 all V_r_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),T_c),hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_d),c_COMBC(T_e,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_d,T_e)))),c_Product__Type_OPair(T_d,T_e))))),V_r_2)),V_x_2),V_y_2)))) # label(fact_INF__INT__eq2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1785 (all V_f_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__inf__class_Oinf(T_d)),V_f_2)),c_Orderings_Otop__class_Otop(T_d)),V_A_2)))) # label(fact_INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1786 (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_Int_Oint),V_w_2),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)))))) # label(fact_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1787 (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]. 8.94/8.92 1788 (all V_B_2 all V_A_2 all T_d all V_C_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_C_2)) -> (hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)) <-> V_A_2 = V_B_2))) # label(fact_Times__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1789 (all V_a_2 all V_f_2 all V_r2_2 all T_e all T_d all V_r1_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r1_2) -> (c_Equiv__Relations_Ocongruent2(T_c,T_d,T_e,V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> c_Equiv__Relations_Ocongruent(T_d,T_e,V_r2_2,hAPP(V_f_2,V_a_2)))))) # label(fact_congruent2__implies__congruent) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1790 (all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),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_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_c,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omin__ext(T_c,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)))))),c_Wellfounded_Omin__ext(T_c,V_R_2))))) # label(fact_min__ext__compat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1791 (all V_a_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c)) = V_a_2 | c_Groups_Oone__class_Oone(T_c) = V_a_2 <-> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) # label(fact_power2__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1792 (all V_B_2 all V_A_2 all V_a_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1793 (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]. 8.94/8.92 1794 (all T_1 class_Nat_Osize(tc_Option_Ooption(T_1))) # label(arity_Option__Ooption__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1795 (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_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)))) # label(fact_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1796 (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]. 8.94/8.92 1797 (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)) -> 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))) = c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j))))) # label(fact_upt__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1798 (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_int__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1799 (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]. 8.94/8.92 1800 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) <-> (exists 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_c)),V_xs_2))) & hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n))))))) # label(fact_list__ex__length) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1801 (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]. 8.94/8.92 1802 (all V_a_2 all V_r_2 all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,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_c,V_r_2),B_x)))))) # label(fact_accp_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1803 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_remove1__filter__not) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1804 (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]. 8.94/8.92 1805 (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]. 8.94/8.92 1806 (all V_y_2 all V_x_2 all V_r_2 all T_c ((all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_a),B_b)),V_r_2)) | B_a = B_b | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_b),B_a)),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olexord(T_c,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_y_2),V_x_2)),c_List_Olexord(T_c,V_r_2))) | V_y_2 = V_x_2)) # label(fact_lexord__linear) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1807 (all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Range__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1808 (all V_m_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.92 1809 (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]. 8.94/8.92 1810 (all V_f_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (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_c),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_c),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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]. 8.94/8.93 1811 (all V_n_2 all V_m_2 all T_c (class_Nat_Osemiring__char__0(T_c) -> (V_n_2 = V_m_2 <-> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_m_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)))) # label(fact_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1812 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Un__upper1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1813 (all V_f_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,T_c),c_Int_Onat),V_f_2)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,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 1814 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2) = c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)))) # label(fact_Min_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1815 (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),c_Int_OBit0(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))) # label(fact_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1816 (all V_i all V_d ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),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)))) = 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_Oempty(tc_Int_Oint) = c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i)))) # label(fact_small__lazy_H_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1817 (all V_B_2 all V_A_2 all T_d all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),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_OSUPR(T_d,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_d),c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_Compl__INT) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1818 (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]. 8.94/8.93 1819 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Min_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1820 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2))))) # label(fact_in__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1821 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_x_2),V_y_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1822 (all V_r_2 all T_c (c_Relation_Osym(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) <-> c_Relation_Osym(T_c,V_r_2))) # label(fact_sym__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1823 (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]. 8.94/8.93 1824 (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_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1825 (all V_y_2 all V_x_2 all V_s_2 all V_r_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(c_Predicate_Opred__comp(T_c,T_d,T_e,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_r_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_d),c_COMBC(T_e,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_d,T_e)))),c_Product__Type_OPair(T_d,T_e)))),V_s_2)),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_e),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_2),V_s_2))))) # label(fact_pred__comp__rel__comp__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1826 (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(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)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_atLeastLessThanSuc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1827 (all T_d all V_f_2 all V_r_2 all T_c ((all B_y all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_z)),V_r_2)) -> hAPP(V_f_2,B_z) = hAPP(V_f_2,B_y))) -> c_Equiv__Relations_Ocongruent(T_c,T_d,V_r_2,V_f_2))) # label(fact_congruentI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1828 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),V_A_2))))))) # label(fact_Max_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1829 (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]. 8.94/8.93 1830 (all V_com_H_2 all V_fun_H_2 all V_loc_H_2 c_Com_Ocom_OSKIP != c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1831 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1832 (all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))) # label(fact_trancl__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1833 (all V_xq_2 all V_s_2 all T_c c_Option_Ooption_Ooption__case(tc_Nat_Onat,tc_prod(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),hAPP(hAPP(c_COMBC(T_c,tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),T_c),c_COMBC(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),T_c),hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Lazy__Sequence_Olazy__sequence(T_c)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),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_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),T_c),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Lazy__Sequence_Olazy__sequence(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),T_c),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_s_2))),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_s_2))))),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_Lazy__Sequence_Oyield(T_c),V_xq_2)) = hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_s_2),V_xq_2)) # label(fact_lazy__sequence__size__code) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1834 (all V_n_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) -> (hBOOL(hAPP(V_Pa_2,V_n_2)) -> (exists B_k ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_k)) -> -hBOOL(hAPP(V_Pa_2,B_i)))) & hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_k),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_k),V_n_2))))))) # label(fact_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1835 (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(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]. 8.94/8.93 1836 (all V_xs_2 all V_Pa_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,V_x_2)))) # label(fact_set__takeWhileD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1837 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2) = hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_i_2,V_j_2))) # label(fact_set__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1838 (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_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1839 (all V_Pa_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_Pa_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_Pa_2,B_x)))))) # label(fact_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1840 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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]. 8.94/8.93 1841 (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]. 8.94/8.93 1842 (all V_r_2 all V_ys_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_c)),hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)),B_x)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,V_r_2))))) # label(fact_listrel__iff__zip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1843 (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_List_Onth(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),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),hAPP(hAPP(c_List_Onth(T_b),V_ys),V_i))))) # label(fact_nth__zip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1844 (all V_B_2 all V_A_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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_2),V_B_2) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_B_2))))))))) # label(fact_fold1__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1845 (all 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))) = c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b)) # label(fact_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1846 (all V_A_2 all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2)) = V_A_2)) # label(fact_surj__image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1847 (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]. 8.94/8.93 1848 (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]. 8.94/8.93 1849 (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]. 8.94/8.93 1850 (all T_c c_Relation_Oantisym(T_c,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))) # label(fact_antisym__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1851 (all V_p_2 all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_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_d)),V_ys_2))) -> hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_p_2)),hAPP(hAPP(c_List_Onth(T_d),V_ys_2),V_p_2)))))) # label(fact_list__all2__nthD2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1852 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)))))) # label(fact_less__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1853 (all V_A_2 all V_a_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2))))) # label(fact_inj__image__mem__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1854 (all V_n_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_c)),hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),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_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),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_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_n_2))))))) # label(fact_finite__lists__length__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1855 (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]. 8.94/8.93 1856 (all V_r_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Fun_Oinj__on(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),T_c),c_Equiv__Relations_Oquotient(T_c)),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))))),V_r_2),V_A_2) -> hAPP(c_Finite__Set_Ocard(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2)) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__quotient__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1857 (all V_b all V_a ((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(tc_Int_Oint),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) = c_Divides_OposDivAlg(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(tc_Int_Oint),V_a),V_b))) -> 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]. 8.94/8.93 1858 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2))))) # label(fact_setprod_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1859 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,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_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)))) # label(fact_INT__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1860 (all V_a3_2 all V_a2_2 all V_a1_2 all V_f_2 all T_c ((exists B_a exists B_A (-hBOOL(hAPP(hAPP(c_member(T_c),B_a),B_A)) & hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,V_f_2,B_a,B_A),V_a3_2)) & hAPP(hAPP(c_Set_Oinsert(T_c),V_a1_2),V_a2_2) = hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_A))) <-> hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a1_2),V_a2_2)),V_a3_2)))) # label(fact_fold1Set_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1861 (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]. 8.94/8.93 1862 (all T_c 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_c,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1CI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1863 (all V_i_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,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_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1864 (all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)) <-> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1865 (all V_a_2 all V_r_2 all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Wellfounded_Oacc(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Wellfounded_Oacc(T_c,V_r_2)))))) # label(fact_acc__downward) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1866 (all V_b_2 all V_A_2 all T_c ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) -> c_Partial__Function_Oflat__lub(T_c,V_b_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(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) -> c_HOL_OThe(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),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_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = c_Partial__Function_Oflat__lub(T_c,V_b_2,V_A_2)))) # label(fact_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1867 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> c_Groups_Oone__class_Oone(T_c) = c_Groups_Osgn__class_Osgn(T_c,V_a_2)))) # label(fact_sgn__1__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1868 (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]. 8.94/8.93 1869 (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)) -> c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_drop__all) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1870 (all V_p_2 all T_c all T_d hAPP(c_Product__Type_Osnd(T_d,T_c),V_p_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,T_c),hAPP(c_COMBK(tc_fun(T_c,T_c),T_d),c_COMBI(T_c))),V_p_2)) # label(fact_snd__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1871 (all V_zs all V_y all V_x all T_a c_List_Oremove1(T_a,V_x,c_List_Oremove1(T_a,V_y,V_zs)) = c_List_Oremove1(T_a,V_y,c_List_Oremove1(T_a,V_x,V_zs))) # label(fact_remove1__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1872 (all V_n all V_m all V_k all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),V_m)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),V_n)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),c_Divides_Odiv__class_Omod(T_a,V_m,V_n))))))) # label(fact_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1873 (all V_y_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(c_List_Orev(T_c),V_xs_2)) -> (c_List_Olist_ONil(T_c) != V_xs_2 -> c_List_Ofoldr(T_c,T_c,c_Orderings_Oord__class_Omax(T_c),V_xs_2,V_y_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_y_2)) & (V_xs_2 = c_List_Olist_ONil(T_c) -> V_y_2 = c_List_Ofoldr(T_c,T_c,c_Orderings_Oord__class_Omax(T_c),V_xs_2,V_y_2))))) # label(fact_foldr__max__sorted) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1874 (all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) <-> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_Compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1875 (all V_w_2 all V_b_2 all V_a_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Ofield__inverse__zero(T_c) -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)) <-> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) -> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)) & (c_Groups_Ozero__class_Ozero(T_c) != hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))) # label(fact_eq__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1876 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> V_x_2 = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,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)))))) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Min_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1877 (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]. 8.94/8.93 1878 (all T_a (class_Rings_Oring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_of__int__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1879 (all V_B_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),V_x_2)),V_B_2)))))) # label(fact_the__inv__into__into) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1880 (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]. 8.94/8.93 1881 (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]. 8.94/8.93 1882 (all V_x_2 all V_X_2 all V_a_2 all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_X_2)),V_x_2)) -> -(all B_a all B_A (hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_X_2) = hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_A) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,V_f_2,B_a,B_A),V_x_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),B_a),B_A))))))) # label(fact_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1883 (all V_q all V_p all T_b all T_a (hAPP(c_Product__Type_Ofst(T_a,T_b),V_p) = hAPP(c_Product__Type_Ofst(T_a,T_b),V_q) -> (hAPP(c_Product__Type_Osnd(T_a,T_b),V_q) = hAPP(c_Product__Type_Osnd(T_a,T_b),V_p) -> V_q = V_p))) # label(fact_prod__eqI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1884 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1885 (all V_x_2 all V_i_2 all V_xs_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2))),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__update__subset__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1886 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_N_2))))))) # label(fact_min__max_OInf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1887 (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]. 8.94/8.93 1888 (all V_B_2 all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_B_2),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2))))) # label(fact_inf__Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1889 (all V_M_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_M_2)) -> (exists B_h c_Fun_Obij__betw(tc_Nat_Onat,T_c,B_h,c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_M_2)),V_M_2)))) # label(fact_ex__bij__betw__nat__finite__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1890 (all V_xss_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xss_2))) -> B_x = c_List_Olist_ONil(T_c))) <-> c_List_Oconcat(T_c,V_xss_2) = c_List_Olist_ONil(T_c))) # label(fact_concat__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1891 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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]. 8.94/8.93 1892 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_B_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_2))))) # label(fact_Inter__anti__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1893 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),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_min__max_OSup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1894 (all V_x_2 all T_c (class_Groups_Ozero(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1895 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2)))))) # label(fact_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1896 (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),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),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_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1897 (all V_A_2 all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_A_2)) # label(fact_Compl__partition2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1898 (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]. 8.94/8.93 1899 (all T_d all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_c,T_d),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)))) # label(fact_Domain__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1900 (all V_no_2 all V_yes_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Opartition(T_c,V_Pa_2,V_xs_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_yes_2),V_no_2) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_yes_2)),hAPP(c_List_Oset(T_c),V_no_2)) = hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_partition__set) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1901 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,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))))))) & (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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)))))) # label(fact_Max_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1902 (all V_r_2 all V_x_2 all T_c all V_y_2 ((exists B_b exists B_z hAPP(hAPP(c_List_Olist_OCons(T_c),B_b),B_z) = V_y_2) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_y_2))),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__append__rightI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1903 (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]. 8.94/8.93 1904 (all V_S_2 all V_R_2 all V_Pa_2 all T_c (c_FunDef_Oreduction__pair(T_c,V_Pa_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_S_2),hAPP(c_Product__Type_Osnd(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2))) -> (c_Wellfounded_Owf(T_c,V_S_2) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2))))))) # label(fact_reduction__pair__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1905 (all V_xs_2 all V_b_2 all V_a_2 all T_c (V_a_2 = V_b_2 & hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),hAPP(c_List_Oset(tc_prod(T_c,T_c)),hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_xs_2)))))) # label(fact_zip__same) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1906 (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]. 8.94/8.93 1907 (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]. 8.94/8.93 1908 (all V_com_2 all V_fun_2 all V_loc_2 all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2)) # label(fact_com_Osimps_I39_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1909 (all V_f_2 all V_a_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2))))) # label(fact_setsum__diff1_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1910 (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]. 8.94/8.93 1911 (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]. 8.94/8.93 1912 (all V_xs_2 all T_c hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremdups(T_c,V_xs_2))) # label(fact_length__remdups__card__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1913 (all V_a_2 all V_f_2 all T_c all V_g_2 (c_Finite__Set_Ofold1(T_c,V_f_2) = V_g_2 -> hAPP(V_g_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_a_2)) # label(fact_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1914 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,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_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1915 (all V_A_2 all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)),V_A_2)) # label(fact_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1916 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_e,T_c,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Fun_Ocomp(T_c,T_e,T_d,c_Hilbert__Choice_Oinv__into(T_e,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)),V_g_2)),c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)) = c_Hilbert__Choice_Oinv__into(T_e,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)),hAPP(c_Fun_Ocomp(T_c,T_d,T_e,V_f_2),V_g_2))))) # label(fact_o__inv__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1917 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(tc_fun(T_c,tc_HOL_Obool),c_Wellfounded_Omax__ext(T_c,V_r_2)))) # label(fact_max__ext__wf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1918 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,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__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1919 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Otimes__class_Otimes(T_c)),V_f_2)),V_g_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_g_2),V_A_2)))) # label(fact_setprod__timesf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1920 (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]. 8.94/8.93 1921 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,V_r_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_listrel__eq__len) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1922 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_sup__compl__top) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1923 (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]. 8.94/8.93 1924 (all V_B_2 all V_A_2 all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_2))) # label(fact_Inter__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1925 (all V_b_2 all V_a_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1926 (all V_r_2 all T_c (c_Relation_Oantisym(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) <-> c_Relation_Oantisym(T_c,V_r_2))) # label(fact_antisym__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1927 (all T_c hAPP(c_List_Oset(T_c),c_List_Olist_ONil(T_c)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_List_Oset_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1928 (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]. 8.94/8.93 1929 (all V_z_2 all V_f_2 all T_d all T_c hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_z_2))) # label(fact_fold__graph_OemptyI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1930 (all V_A_2 all T_d all V_I_2 all V_k_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_UN__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1931 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) <-> c_List_Olist__all(T_c,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2)))) # label(fact_list__all__rev) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1932 (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]. 8.94/8.93 1933 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(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)) # label(fact_Int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1934 (all V_f_2 all T_d all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))))) # label(fact_imageI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1935 (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]. 8.94/8.93 1936 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2))))) # label(fact_ivl__disj__un_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1937 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2))))))) # label(fact_min__max_OInf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1938 (all V_k_2 all V_n_2 all V_Pa_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_k_2 -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_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_k_2),B_i)),B_j) -> hBOOL(hAPP(V_Pa_2,B_i)))))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) <-> hBOOL(hAPP(V_Pa_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]. 8.94/8.93 1939 (all V_k c_Int_Opred(V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_pred__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1940 (all V_A_2 all T_d all T_c 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),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_Sigma__empty2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1941 (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]. 8.94/8.93 1942 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_UN__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1943 (all V_z_2 all V_y_2 all V_x_2 all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,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]. 8.94/8.93 1944 (all V_M_2 all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_M_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_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_M_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_M_2))) # label(fact_UN__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1945 (all V_a all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield(T_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))) = V_a)) # label(fact_divide__numeral__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1946 (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]. 8.94/8.93 1947 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) <-> V_y_2 = V_x_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_x_2))))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.93 1948 (all V_R_2 all V_Qa_2 all V_Pa_2 all T_c all T_e all T_d hAPP(hAPP(V_Pa_2,V_R_2),V_Qa_2) = hAPP(hAPP(hAPP(c_COMBC(T_d,T_e,T_c),V_Pa_2),V_Qa_2),V_R_2)) # label(help_c__COMBC__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1949 (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_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1950 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(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(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))))) # label(fact_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1951 (all V_A_2 all T_c 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_A_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) # label(fact_Compl__eq__Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1952 (all V_A_2 all T_c c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),T_c),hAPP(c_COMBS(tc_fun(T_c,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_c,tc_HOL_Obool)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_c,tc_HOL_Obool))),V_A_2)))),c_member(T_c))))) # label(fact_Union__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1953 (all V_b_2 all 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)) <-> -(hBOOL(hAPP(hAPP(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))))) # label(fact_dvd_OatLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1954 (all V_A_2 all V_f_2 all T_e all T_d all T_c all V_I_2 (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_I_2 -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_I_2)) -> c_Fun_Oinj__on(T_d,T_e,V_f_2,hAPP(V_A_2,B_i)))) -> c_Fun_Oinj__on(T_d,T_e,V_f_2,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_inj__on__INTER) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1955 (all V_B_2 all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),hAPP(hAPP(c_COMBS(T_c,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_c),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_B_2))) # label(fact_Sigma__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1956 (all V_b_2 all V_a_2 all V_f_2 all T_c all T_e all T_d hAPP(hAPP(V_f_2,V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2))) # label(fact_split__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1957 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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))) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_e),c_Set_Oimage(T_c,T_d,V_f_2)),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_bij__image__INT) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1958 (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]. 8.94/8.94 1959 (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]. 8.94/8.94 1960 (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]. 8.94/8.94 1961 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1962 (all V_z_2 all V_w_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),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]. 8.94/8.94 1963 (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]. 8.94/8.94 1964 (all V_R_2 all T_c ((all B_A (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),B_A),hAPP(c_Relation_OImage(T_c,T_c,V_R_2),B_A))) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = B_A)) -> c_Wellfounded_Owf(T_c,V_R_2))) # label(fact_wfI__pf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1965 (all V_z_2 all V_g_2 all V_a_2 all V_A_2 all T_c all T_d (class_Groups_Oab__semigroup__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_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),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(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)))))) # label(fact_fold__image__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1966 (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]. 8.94/8.94 1967 (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]. 8.94/8.94 1968 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Oab__group__add(T_d) -> (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_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setsum__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1969 (all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Relation_OId(T_c))) -> -(all B_x V_p_2 != hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)))) # label(fact_IdE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1970 (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]. 8.94/8.94 1971 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),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)))))) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))) # label(fact_card__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1972 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_finite__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1973 (all V_xs_2 all T_c (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xs_2))) -> (c_List_Olist_ONil(tc_List_Olist(T_c)) = V_xs_2 -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2))) & (c_List_Olist_ONil(tc_List_Olist(T_c)) != V_xs_2 -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2))))) # label(fact_length__transpose__sorted) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1974 (all T_d all T_c 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_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2))))) # label(fact_inf2I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1975 (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_Omax(T_a),V_x),V_y)),V_z) = 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)))) # label(fact_max__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1976 (all V_k_2 all V_i_2 all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_i_2),V_k_2))))) # label(fact_atMost__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1977 (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_a -> 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]. 8.94/8.94 1978 (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]. 8.94/8.94 1979 (all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_c,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]. 8.94/8.94 1980 (all T_g all T_f all T_e all T_c all T_d 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_H_2,hAPP(V_g_H_2,V_x_H_2)) = hAPP(V_f_2,hAPP(V_g_2,V_x_2)) -> hAPP(hAPP(c_Fun_Ocomp(T_f,T_c,T_g,V_f_H_2),V_g_H_2),V_x_H_2) = hAPP(hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2),V_x_2))) # label(fact_comp__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1981 (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]. 8.94/8.94 1982 (all V_i_2 all V_xs_2 all T_c (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),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_c))),V_xs_2))) -> hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),V_i_2) = hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_Nat_Onat,T_c),c_List_Onth(T_c)),V_i_2)),hAPP(c_List_Ofilter(tc_List_Olist(T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),c_List_Otranspose(T_c,V_xs_2)))))) # label(fact_transpose__column) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1983 (all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Inter__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1984 (all V_A_2 all V_C_2 all V_X_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),V_X_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2)))))) # label(fact_UnionI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1985 (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]. 8.94/8.94 1986 (all V_x_2 all V_g_2 all V_f_2 all T_f all T_e all T_c all T_d hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_e,T_f),V_x_2)) = hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(c_Product__Type_Omap__pair(T_e,T_d,T_f,T_c,V_f_2,V_g_2),V_x_2))) # label(fact_snd__prod__fun) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1987 (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]. 8.94/8.94 1988 (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]. 8.94/8.94 1989 (all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),hAPP(c_COMBK(tc_HOL_Obool,T_c),c_fFalse))) # label(fact_empty__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1990 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (c_Fun_Oinj__on(T_e,T_c,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_e,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_e,V_f_2),V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)))))) # label(fact_inj__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1991 (all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_R_2),V_S_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2))))) # label(fact_pred__subset__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1992 (all V_I_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__sublistD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1993 (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]. 8.94/8.94 1994 (all V_B_2 all V_A_2 all T_c all T_d c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_A_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_UNION__eq__Union__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1995 (all V_b_2 all V_a_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_d,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]. 8.94/8.94 1996 (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]. 8.94/8.94 1997 (all V_code__numeral_H c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H)) # label(fact_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1998 (all V_ys_2 all V_Pa_2 all V_xs_2 all V_x_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_ys_2) <-> (exists B_us exists B_vs (hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_vs)) = V_ys_2 & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),B_us))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) & hAPP(c_List_Ofilter(T_c,V_Pa_2),B_vs) = V_xs_2 & hBOOL(hAPP(V_Pa_2,V_x_2)))))) # label(fact_Cons__eq__filter__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 1999 (all T_c hAPP(c_Relation_ORange(T_c,T_c),c_Relation_OId(T_c)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Range__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2000 (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]. 8.94/8.94 2001 (all V_r_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_xs_2)),c_List_Olistrel(T_c,V_r_2))) -> V_xs_2 = c_List_Olist_ONil(T_c))) # label(fact_listrel__Nil1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2002 (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]. 8.94/8.94 2003 (all V_r_2 all T_c (V_r_2 = hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2) <-> c_Relation_Osym(T_c,V_r_2))) # label(fact_sym__conv__converse__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2004 (all V_v_2 all V_f_2 all V_a_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,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)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),V_a_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_c,T_c),tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Int_Oint),c_Nat_Onat_Onat__rec(T_c,V_a_2,V_f_2)),c_Int_Onat)))) = hAPP(c_Nat_Onat_Onat__rec(T_c,V_a_2,V_f_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_nat__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2005 (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]. 8.94/8.94 2006 (all V_g_2 all T_e all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_c,T_e,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(T_d,T_e,hAPP(c_Fun_Ocomp(T_c,T_e,T_d,V_g_2),V_f_2)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool))))) # label(fact_comp__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2007 (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]. 8.94/8.94 2008 (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 2009 (all V_A_2 all T_c hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),V_A_2)) = hAPP(c_Set_OPow(T_c),V_A_2)) # label(fact_Pow__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2010 (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]. 8.94/8.94 2011 (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]. 8.94/8.94 2012 (all V_xs_2 all T_d all V_y_2 all V_x_2 all V_f_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(V_f_2,V_x_2) != hAPP(V_f_2,V_y_2) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_y_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_xs_2))))) # label(fact_insort__key__left__comm) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2013 (all V_xss_2 all T_c hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xss_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oconcat(T_c,V_xss_2))) # label(fact_length__concat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2014 (all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2) = hAPP(c_Set_OCollect(T_c),hAPP(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_fdisj),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),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_insert__compr) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2015 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_r_2) = c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))) # label(fact_rtrancl__r__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2016 (all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> (c_Fun_Oinj__on(T_c,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,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_c,tc_HOL_Obool))))) # label(fact_finite__UNIV__inj__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2017 (all V_xs_2 all V_p_2 all T_c c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Ofilter(T_c,V_p_2)),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_p_2),c_List_Oconcat(T_c,V_xs_2))) # label(fact_filter__concat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2018 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(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_fdisj),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_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2019 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__top(T_c) -> (V_x_2 = c_Orderings_Otop__class_Otop(T_c) & V_y_2 = c_Orderings_Otop__class_Otop(T_c) <-> c_Orderings_Otop__class_Otop(T_c) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_y_2)))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2020 (all V_ys_2 all T_d all V_xs_2 all T_c (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) -> hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_c),c_Product__Type_Ofst(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)) = V_xs_2)) # label(fact_map__fst__zip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2021 (all V_A_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)))) # label(fact_Inf__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2022 (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_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))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs))) # label(fact_take__hd__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2023 (all V_s_2 all V_r_2 all T_e all T_c all T_d c_Predicate_Opred__comp(T_c,T_e,T_d,c_Predicate_Oconversep(T_e,T_c,V_s_2),c_Predicate_Oconversep(T_d,T_e,V_r_2)) = c_Predicate_Oconversep(T_d,T_c,c_Predicate_Opred__comp(T_d,T_e,T_c,V_r_2,V_s_2))) # label(fact_converse__pred__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2024 (all V_r_2 all V_xs_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_xs_2)),c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2))))) # label(fact_listrel__rtrancl__refl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2025 (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_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2026 (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),V_b),V_c) = 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)) # label(fact_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2027 (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]. 8.94/8.94 2028 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)) = hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_x_2)))) # label(fact_image__uminus__atMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2029 (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]. 8.94/8.94 2030 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) <-> c_Relation_Otrans(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_trans__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2031 (all V_n_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),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)))) # label(fact_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2032 (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]. 8.94/8.94 2033 (all V_B_2 all V_A_2 all T_c (c_Fun_Obij__betw(T_c,T_c,c_Fun_Oid(T_c),V_A_2,V_B_2) <-> V_B_2 = V_A_2)) # label(fact_bij__betw__id__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2034 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_c_2,V_d_2)))) # label(fact_Int__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2035 (all V_n_2 all V_a_2 all T_c (class_Rings_Oring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_n_2)),c_Int_Oring__1__class_OInts(T_c)))))) # label(fact_Ints__power) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2036 (all V_zs all V_xs all V_ys all V_xs1 all V_x all T_a (V_ys = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs1) -> (hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_zs) = V_xs -> 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]. 8.94/8.94 2037 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> 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]. 8.94/8.94 2038 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls)))) # label(fact_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2039 (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]. 8.94/8.94 2040 (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]. 8.94/8.94 2041 (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_Ominus__class_Ominus(tc_Int_Oint),V_w),V_z)) = 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)))) # label(fact_of__int__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2042 (all V_f_2 all V_x_2 all T_c all T_d all V_i_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_i_2 -> hAPP(c_New__DSequence_Oneg__decr__bind(T_d,T_c,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Ohit__bound(T_c)) & (V_i_2 != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(c_New__DSequence_Oneg__decr__bind(T_d,T_c,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Ohb__bind(T_d,T_c,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_d,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c))),V_f_2),V_i_2))))) # label(fact_New__DSequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2043 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)))) # label(fact_Int__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2044 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2),B_x)))))) # label(fact_list__all2__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2045 (all V_m_2 all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> 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]. 8.94/8.94 2046 (all V_b_2 all V_B_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2))))) # label(fact_insertI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2047 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)))))) # label(fact_pos__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2048 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2049 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))) <-> c_Wellfounded_OwfP(T_c,V_r_2))) # label(fact_wfP__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2050 (all V_ys_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Obutlast(T_c,V_ys_2)))) | hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Obutlast(T_c,V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Obutlast(T_c,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))))) # label(fact_in__set__butlast__appendI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2051 (all V_A_2 all T_c (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_c),V_A_2))) <-> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_card__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2052 (all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Orderings_Otop__class_Otop(T_a)))) # label(fact_compl__top__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2053 (all V_a5_2 all V_a1_2 all V_a2_2 all V_a4_2 all V_a3_2 all T_d all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_a2_2 & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = V_a4_2 | (exists B_x exists B_y (c_Nitpick_Ofold__graph_H(T_c,T_d,V_a3_2,V_a4_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_a2_2),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),B_y) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a2_2)) & hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y))) <-> c_Nitpick_Ofold__graph_H(T_c,T_d,V_a3_2,V_a4_2,V_a2_2,hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2)))) # label(fact_fold__graph_H_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2054 (all V_n all T_a (class_Rings_Osemiring__0(T_a) & class_Power_Opower(T_a) -> (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n)))) # label(fact_power__0__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2055 (all V_f_2 all T_d all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_insert__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2056 (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)) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_c)))) # label(fact_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2057 (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]. 8.94/8.94 2058 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_A_2)))) # label(fact_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2059 (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]. 8.94/8.94 2060 (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]. 8.94/8.94 2061 (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]. 8.94/8.94 2062 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2)))) # label(fact_ivl__disj__un_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2063 (all V_r_H_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_H_2),V_r_2)))) # label(fact_wf__Int2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2064 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_diff__card__le__card__Diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2065 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2066 (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]. 8.94/8.94 2067 (all V_zs_2 all V_x_2 all T_c (class_Groups_Omonoid__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_zs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_x_2),V_zs_2))) # label(fact_foldl__absorb0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2068 (all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_prod(T_c,T_d)),c_Product__Type_OPair(T_c,T_d)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_split__Pair) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2069 (all V_x_2 all V_f_2 all T_e all T_c all T_d hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_d,T_e),V_x_2)) = hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(c_Product__Type_Oapsnd(T_e,T_c,T_d,V_f_2),V_x_2))) # label(fact_snd__apsnd) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2070 (all V_y_2 all V_x_2 (V_x_2 = V_y_2 <-> hAPP(c_Code__Numeral_Onat__of,V_y_2) = hAPP(c_Code__Numeral_Onat__of,V_x_2))) # label(fact_code__numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2071 (all V_Pa_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_Pa_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_Pa_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2072 (all V_n_2 all T_c all V_xs_2 ((c_List_Olist_ONil(tc_List_Olist(T_c)) = 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_c))),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),B_i)) = V_n_2)) -> c_List_Otranspose(T_c,V_xs_2) = hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_List_Olist(T_c)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(tc_Nat_Onat),tc_List_Olist(T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_List_Olist(tc_Nat_Onat),tc_List_Olist(T_c)),tc_Nat_Onat),c_List_Omap(tc_Nat_Onat,T_c)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_Nat_Onat,T_c),tc_Nat_Onat),c_List_Onth(T_c)),hAPP(c_List_Onth(tc_List_Olist(T_c)),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_c))),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]. 8.94/8.94 2073 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2))))) # label(fact_zero__less__divide__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2074 (all V_x_2 all T_e all T_d all T_c hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_e,tc_prod(T_d,T_e)),V_x_2),c_Product__Type_OPair(T_d,T_e)) = V_x_2) # label(fact_scomp__Pair) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2075 (all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> V_xs_2 = c_List_Oremove1(T_c,V_x_2,V_xs_2))) # label(fact_remove1__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2076 (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]. 8.94/8.94 2077 (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]. 8.94/8.94 2078 (all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Oring__1(T_c) & class_Groups_Ocomm__monoid__mult(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_c,T_d),c_Int_Oring__1__class_Oof__int(T_c)),V_f_2)),V_A_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,tc_Int_Oint),V_f_2),V_A_2)))) # label(fact_of__int__setprod) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2079 (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]. 8.94/8.94 2080 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Fields_Ofield(T_d) -> (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_f_2,B_x) != c_Groups_Ozero__class_Ozero(T_d))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),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(c_Rings_Oinverse__class_Odivide(T_d),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_setprod__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2081 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_trancl__converseD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2082 (all V_R2_2 all V_R1_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),V_R1_2),V_R2_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Wellfounded_Oaccp(T_c,V_R2_2)),c_Wellfounded_Oaccp(T_c,V_R1_2))))) # label(fact_accp__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2083 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,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_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))))) # label(fact_inj__image__Compl__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2084 (all V_x_2 all T_c all V_n_2 (V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_x_2)))) # label(fact_set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2085 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)))) # label(fact_sup__top__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2086 (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]. 8.94/8.94 2087 (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 2088 (all T_c 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_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2089 (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]. 8.94/8.94 2090 (all V_A_2 all T_c c_Fun_Oinj__on(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Orev(T_c),V_A_2)) # label(fact_inj__on__rev) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2091 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) <-> -(exists B_f all B_i hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(B_f,hAPP(c_Nat_OSuc,B_i))),hAPP(B_f,B_i))),V_r_2))))) # label(fact_wf__iff__no__infinite__down__chain) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2092 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Pa_2) | hBOOL(V_Qa_2) | -hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)))) # label(help_c__fdisj__3) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2093 (all V_b_2 all V_f_2 all T_d all T_c all V_a_2 all V_x_2 ((V_a_2 != V_x_2 -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ofun__upd(T_c,T_d,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_c,T_d,V_f_2,V_a_2,V_b_2),V_x_2)))) # label(fact_fun__upd__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2094 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2))) # label(fact_Compl__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2095 (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),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a))) # label(fact_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2096 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),c_Groups_Oone__class_Oone(T_c)),V_a_2)),V_a_2)),c_Groups_Ozero__class_Ozero(T_c))))))) # label(fact_Ints__odd__less__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2097 (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]. 8.94/8.94 2098 (all V_A_2 all V_r_2 all T_d all V_b_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_x),V_b_2)),V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_OImage(T_d,T_c,V_r_2),V_A_2))))) # label(fact_Image__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2099 (all V_C_2 all V_B_2 all T_e all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_d),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_INT__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2100 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2))))))) # label(fact_Inf__le__Sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2101 (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]. 8.94/8.94 2102 (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),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_minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2103 (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)) -> V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 8.94/8.94 2104 (all V_fun2_H_2 all V_com_H_2 all V_fun1_H_2 all V_fun2_2 all V_com_2 all V_fun1_2 all T_c (hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_fun1_H_2),V_com_H_2),V_fun2_H_2) = hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_fun1_2),V_com_2),V_fun2_2) <-> V_fun1_2 = V_fun1_H_2 & V_com_H_2 = V_com_2 & V_fun2_2 = V_fun2_H_2)) # label(fact_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2105 (all V_R_2 all T_c c_Relation_OId(T_c) = hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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]. 8.94/8.95 2106 (all T_c hAPP(c_Finite__Set_Ocard(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2107 (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]. 8.94/8.95 2108 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(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_I30_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2109 (all V_A_2 all T_c hAPP(c_Set_OPow(T_c),V_A_2) != c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))) # label(fact_Pow__not__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2110 (all V_n all V_x 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)) = 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)) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2111 (all V_S_2 all V_R_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_x_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_x_2),c_Transitive__Closure_Ortrancl(T_c,V_S_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_x_2),c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)))))) # label(fact_in__rtrancl__UnI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2112 (all V_ys_2 all V_y_2 all V_xs_2 all T_c (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),hAPP(c_List_Orev(T_c),V_ys_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),c_List_Olist_ONil(T_c))) <-> hAPP(c_List_Orev(T_c),V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))) # label(fact_rev__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2113 (all T_a c_Lazy__Sequence_Oflat(T_a,c_Lazy__Sequence_Olazy__sequence_OEmpty(tc_Lazy__Sequence_Olazy__sequence(T_a))) = c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a)) # label(fact__04) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2114 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) | hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_Un__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2115 (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__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]. 8.94/8.95 2116 (all T_c (class_Orderings_Owellorder(T_c) -> c_Wellfounded_Owf(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),c_Orderings_Oord__class_Oless(T_c)))))) # label(fact_wf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2117 (all V_f_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(c_Wellfounded_Omeasure(T_c),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))))) # label(fact_in__measure) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2118 (all V_xs_2 all V_x_2 all T_c ((exists B_i (hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i) = V_x_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2119 (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]. 8.94/8.95 2120 (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)) -> 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) = V_t_2)))) # label(fact_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2121 (all V_s_2 all V_c_2 all T_e all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),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),V_b_2),V_c_2)),V_s_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_e),V_a_2),V_c_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_2),V_s_2)))))) # label(fact_rel__compI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2122 (all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,V_r_2) <-> (all B_x1 hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),B_x1))))) # label(fact_wfP__accp__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2123 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2) = V_xs_2))) # label(fact_insort__insert__triv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2124 (all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__empty__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2125 (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]. 8.94/8.95 2126 (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)) -> V_y = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2127 (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]. 8.94/8.95 2128 (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]. 8.94/8.95 2129 (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]. 8.94/8.95 2130 (all V_xs_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Orderings_Otop__class_Otop(T_c)),V_xs_2))) # label(fact_Inf__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2131 (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]. 8.94/8.95 2132 (all V_Pa_2 all T_c all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> hBOOL(hAPP(V_Pa_2,B_x)))) # label(fact_ball__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2133 (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_Oplus__class_Oplus(T_a),V_b),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2134 (all V_Pa_2 all V_a_2 all V_n_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_a_2)))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(V_Pa_2,V_a_2)))) # label(fact_Ball__set__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2135 (all V_x_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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_setprod_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2136 (all V_B_2 all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),hAPP(hAPP(c_COMBS(T_c,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_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_B_2))) # label(fact_Sigma__Diff__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2137 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_l_2)))) # label(fact_ivl__disj__un_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2138 (all V_S_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_COMBI(tc_fun(T_c,tc_HOL_Obool))) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_S_2)) # label(fact_Inter__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2139 (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]. 8.94/8.95 2140 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),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_C_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))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))) # label(fact_Times__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2141 (all V_Pa_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (hBOOL(hAPP(V_Pa_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> ((all B_x all B_F (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_F)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_F)) -> (hBOOL(hAPP(V_Pa_2,B_F)) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Set_Oinsert(T_c),B_x),B_F))))))) -> hBOOL(hAPP(V_Pa_2,V_F_2)))))) # label(fact_finite__induct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2142 (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]. 8.94/8.95 2143 (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]. 8.94/8.95 2144 (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]. 8.94/8.95 2145 (all V_z_2 hAPP(c_Int_Onat,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))))))) # label(fact_nat__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2146 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_Opred(V_k_2)),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2147 (all V_a2_2 all V_a1_2 all V_R_2 all T_c (c_Wellfounded_Omax__extp(T_c,V_R_2,V_a1_2,V_a2_2) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_2)) & V_a2_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_a2_2)) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)))))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a2_2)))) # label(fact_max__extp_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2148 (all B_p (c_Hoare__Mirabelle_Ohoare__derivs(t_b,v_G,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_b)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(t_b),hAPP(v_P_H,B_p)),hAPP(v_c0,B_p)),hAPP(v_Q_H,B_p))),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,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_b)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(t_b),hAPP(v_P,B_p)),hAPP(v_c0,B_p)),hAPP(v_Q,B_p))),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_b),tc_HOL_Obool)))))) # label(conj_0) # label(hypothesis) # label(non_clause). [assumption]. 8.94/8.95 2149 (all V_r_2 all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))) # label(fact_r__into__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2150 (all V_Pa_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))))) # label(fact_distinct__takeWhile) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2151 (all V_k_2 all V_R_2 all V_T_2 all T_d all V_S_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_T_2)) -> V_k_2 = hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_OCollect(T_c),hAPP(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_S_2))),hAPP(hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),V_R_2),B_x)))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_Nat_Onat,T_c),c_Finite__Set_Ocard(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),c_Set_OCollect(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),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_d),V_T_2)))))) # label(fact_setsum__multicount) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2152 (all V_a_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_2)),c_Relation_OId(T_c)))) # label(fact_IdI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2153 (all V_g_2 all V_f_2 all T_d all V_A_2 all V_a_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_g_2,V_a_2) = hAPP(c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,V_A_2),V_a_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,V_A_2),V_a_2) = hAPP(V_f_2,V_a_2)))) # label(fact_override__on__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2154 (all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2) = hAPP(c_Set_OCollect(T_c),hAPP(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_fdisj),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),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_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2155 (all V_f_2 all V_n_2 all V_m_2 all T_c hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)),V_f_2) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_m_2),V_f_2)),hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2))) # label(fact_funpow__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2156 (all V_y_2 all V_x_2 all V_ys_2 all V_Pa_2 all T_d all V_xs_2 all T_c 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_c)),V_xs_2))) -> (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) -> (hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2)) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2),hAPP(hAPP(c_List_Olist__update(T_d,V_ys_2),V_i_2),V_y_2)))))) # label(fact_list__all2__update__cong) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2157 (all V_g_2 all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) & class_Groups_Oordered__cancel__ab__semigroup__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_d),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,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_A_2)))))))) # label(fact_setsum__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2158 (all V_z_2 hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),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)))))))) # label(fact_minus__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2159 (all V_x_2 all T_c hBOOL(hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_x_2))) # label(fact_top1I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2160 (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]. 8.94/8.95 2161 (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 2162 (all V_y_2 all V_x_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (V_y_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2163 (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) = c_List_Olist_ONil(tc_Int_Oint)) & (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))))) # label(fact_upto_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2164 (all V_a1_2 all V_a2_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_a1_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_a1_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a2_2),B_x)))) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_a2_2),V_a1_2))))) # label(fact_sorted_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2165 (all V_A_2 all V_f_2 all T_c hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2) = c_HOL_OThe(T_c,c_Finite__Set_Ofold1Set(T_c,V_f_2,V_A_2))) # label(fact_fold1__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2166 (all V_n_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_n_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_SetInterval_Oord__class_OatMost(T_c),V_n_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_n_2)))) # label(fact_atMost__Int__atLeast) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2167 (all V_y_2 all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Int_Oring__char__0(T_c) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) <-> V_x_2 = V_y_2))) # label(fact_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2168 (all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,c_List_Olist_ONil(T_c)) = c_List_Olist_ONil(T_c))) # label(fact_sort__key__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2169 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Oremdups(T_c,V_xs_2)) = c_List_Oremdups(T_c,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_remdups__filter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2170 (all V_i_2 all V_x_2 all V_A_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2))),V_A_2))))) # label(fact_set__update__subsetI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2171 (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]. 8.94/8.95 2172 (all V_X_2 all T_c c_HOL_OThe(T_c,hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),hAPP(c_fequal,V_X_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))))) = c_Set_Othe__elem(T_c,V_X_2)) # label(fact_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2173 (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]. 8.94/8.95 2174 (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_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2175 (all V_n all V_k all 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_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) # label(fact_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2176 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_c (class_Groups_Oab__group__add(T_c) -> (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_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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_c),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_c),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]. 8.94/8.95 2177 (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]. 8.94/8.95 2178 (all V_x all V_xs all T_a 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)))) = V_xs) # label(fact_butlast__snoc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2179 (all V_a_2 all V_xs_2 all T_c (class_Groups_Oab__semigroup__add(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_xs_2) = c_List_Ofoldr(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_xs_2,V_a_2))) # label(fact_foldr__conv__foldl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2180 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> 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))) # label(fact_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2181 (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]. 8.94/8.95 2182 (all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> V_a_2 = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Inf__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2183 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_2) = c_Transitive__Closure_Ortrancl(T_c,c_Transitive__Closure_Otrancl(T_c,V_R_2))) # label(fact_trancl__rtrancl__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2184 (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(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2185 (all V_f_2 all V_S_2 all V_T_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_d))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_T_2)))))) # label(fact_setsum__mono__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2186 (all T_d all T_c 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_c,T_d,V_r_2),V_a_2)))) # label(fact_DomainP_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2187 (all V_f_2 all V_xs_2 all V_x_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_f_2,V_x_2)),hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),c_List_Oremove1(T_c,V_x_2,V_xs_2))))))) # label(fact_listsum__map__remove1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2188 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c c_List_Opartition(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),tc_fun(tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),tc_List_Olist(T_c)),c_COMBS(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(tc_List_Olist(T_c),tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_List_Olist(T_c)),hAPP(c_If(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(V_Pa_2,V_x_2)))),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_List_Olist(T_c)),c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(c_List_Olist_OCons(T_c),V_x_2))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)))),tc_List_Olist(T_c)),c_COMBB(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_List_Olist(T_c))),c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)))),hAPP(c_List_Olist_OCons(T_c),V_x_2)))),c_List_Opartition(T_c,V_Pa_2,V_xs_2))) # label(fact_partition_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2189 (all V_l all V_k ((-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)) & (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)))) # label(fact_beyond__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2190 (all V_k_2 all V_Pa_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_Pa_2,B_x)) -> hBOOL(hAPP(V_Pa_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_Pa_2,B_x)) -> hBOOL(hAPP(V_Pa_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]. 8.94/8.95 2191 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I22_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2192 (all T_c (class_Lattices_Osemilattice__sup(T_c) -> c_Finite__Set_Ofun__left__comm__idem(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)))) # label(fact_fun__left__comm__idem__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2193 (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_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2194 (all V_x_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_x_2)),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]. 8.94/8.95 2195 (all V_f_2 all V_B_2 all T_e all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_e,V_f_2),V_A_2)),V_B_2)) # label(fact_UN__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2196 (all T_a all V_xs (c_List_Olist_ONil(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))) = hAPP(c_List_Orotate1(T_a),V_xs))) # label(fact_rotate1__hd__tl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2197 (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]. 8.94/8.95 2198 (all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oord(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2)) <-> (all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_x))))))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2199 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (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))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_B_2))))))) # label(fact_setsum_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2200 (all V_n_2 all V_v_2 all V_f_2 all V_a_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,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)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),c_Nat_Onat_Onat__case(T_c,V_a_2,V_f_2,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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_c,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]. 8.94/8.95 2201 (all V_k_2 all V_i_2 all T_c (class_Orderings_Oord(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_i_2),V_k_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2)))))) # label(fact_lessThan__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2202 (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]. 8.94/8.95 2203 (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]. 8.94/8.95 2204 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_d hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),hAPP(c_Relation_ORange(T_d,T_c),V_r_2)) = hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_a_2),V_b_2)),V_r_2))) # label(fact_Range__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2205 (all V_r_2 all T_c c_List_Olenlex(T_c,V_r_2) = hAPP(c_Set_OCollect(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBS(tc_List_Olist(T_c),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fdisj)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBS(tc_List_Olist(T_c),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fconj)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBC(tc_List_Olist(T_c),tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(tc_List_Olist(T_c),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_HOL_Obool)),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))))),c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c))))),c_List_Olex(T_c,V_r_2))))))) # label(fact_lenlex__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2206 (all V_xss_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xss_2))) -> B_x = c_List_Olist_ONil(T_c))) <-> c_List_Oconcat(T_c,V_xss_2) = c_List_Olist_ONil(T_c))) # label(fact_Nil__eq__concat__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2207 (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]. 8.94/8.95 2208 (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_b)) | 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))) -> 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]. 8.94/8.95 2209 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2210 (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]. 8.94/8.95 2211 (all V_lazy__sequence_H_2 all V_a_H_2 all V_lazy__sequence_2 all V_a_2 all T_c (V_a_H_2 = V_a_2 & V_lazy__sequence_H_2 = V_lazy__sequence_2 <-> c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_a_2,V_lazy__sequence_2) = c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_a_H_2,V_lazy__sequence_H_2))) # label(fact_lazy__sequence_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2212 (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]. 8.94/8.95 2213 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_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__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2214 (all V_ys all V_xs all T_a (class_Groups_Omonoid__add(T_a) -> 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)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_listsum__append) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2215 (all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2216 (all V_xs_2 all V_Pa_2 all T_c 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_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_j_2))) # label(fact_takeWhile__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2217 (all V_z_2 all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olexord(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_y_2),V_z_2)),c_List_Olexord(T_c,V_r_2))) -> (c_Relation_Otrans(T_c,V_r_2) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_z_2)),c_List_Olexord(T_c,V_r_2))))))) # label(fact_lexord__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2218 (all T_c (class_Orderings_Olinorder(T_c) -> c_Finite__Set_Ofun__left__comm__idem(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)))) # label(fact_min__max_Ofun__left__comm__idem__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2219 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2220 (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))) & (V_xs = c_List_Olist_ONil(T_a) -> 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]. 8.94/8.95 2221 (all V_B_2 all V_a_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_B_2)) -> hAPP(c_Finite__Set_Ocard(T_c),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_a_2),V_B_2))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Groups_Oone__class_Oone(tc_Nat_Onat)))))) # label(fact_card__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2222 (all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_trancl_Or__into__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2223 (all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,V_r_2) -> c_Wellfounded_Oacyclic(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))))) # label(fact_wfP__acyclicP) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2224 (all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,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]. 8.94/8.95 2225 (all V_y all V_x 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)) = 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))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2226 (all V_Pa_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_Pa_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_Pa_2,B_d),B_i))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2227 (all V_k_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,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))) # label(fact_lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2228 (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)) -> V_a = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2229 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),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]. 8.94/8.95 2230 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),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_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2231 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_distinct__card) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2232 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> (c_Relation_Oantisym(T_c,V_r_2) -> c_Relation_Otrans(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))))) # label(fact_trans__diff__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2233 (all V_r all V_q all T_a 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))) = c_Nitpick_Otimes__frac(T_a,V_q,V_r)) # label(fact_Nitpick_Otimes__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2234 (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]. 8.94/8.95 2235 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> c_Groups_Osgn__class_Osgn(T_c,V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c))))) # label(fact_sgn__1__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2236 (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]. 8.94/8.95 2237 (all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,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]. 8.94/8.95 2238 (all T_c (class_Rings_Osemiring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Groups_Ozero__class_Ozero(T_c)),c_Nat_Osemiring__1__class_ONats(T_c))))) # label(fact_Nats__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2239 (all V_s_2 all V_r_2 all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Otrancl(T_c,V_s_2)))))) # label(fact_trancl__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2240 (all V_xs_2 all V_x_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),c_List_Oremove1(T_c,V_x_2,V_xs_2))),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_set__remove1__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2241 (all V_P all T_a hAPP(c_COMBI(T_a),V_P) = V_P) # label(help_c__COMBI__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2242 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_y_2),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_min__max_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2243 (all V_xs_2 all V_n_2 all T_c hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_List_Oset(T_c),hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2))) # label(fact_set__rotate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2244 (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]. 8.94/8.95 2245 (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]. 8.94/8.95 2246 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (c_List_Olist__ex1(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) <-> (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2)) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist__all(T_c,hAPP(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_fdisj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2))),hAPP(c_fequal,V_x_2)),V_xs_2)))) # label(fact_list__ex1__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.95 2247 (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__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]. 8.94/8.95 2248 (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]. 8.94/8.96 2249 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Groups_Oordered__ab__semigroup__add__imp__le(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_b_2)))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 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_H,V_c) = c_Divides_Odiv__class_Omod(T_a,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) = 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]. 8.94/8.96 2251 (all V_S_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_A_2)) -> V_S_2 = hAPP(c_Set_Oimage(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_S_2))))) # label(fact_inv__into__image__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2252 (all V_x_2 all V_n_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Oreplicate(T_d,V_n_2,V_x_2)) = c_List_Oreplicate(T_c,V_n_2,hAPP(V_f_2,V_x_2))) # label(fact_map__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2253 (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]. 8.94/8.96 2254 (all V_a_2 all V_b_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2))))) # label(fact_neg__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2255 (all V_s_2 all T_c all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Product__Type_Ofst(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Random_Oselect(T_c,V_xs_2),V_s_2))),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_select) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2256 (all V_B_2 all V_A_2 all V_C_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_Int__greatest) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2257 (all V_Pa_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_Pa_2,B_y),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B_x,B_y)))) -> hBOOL(hAPP(hAPP(V_Pa_2,B_x),B_y))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2258 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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_c),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]. 8.94/8.96 2259 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_finite__Diff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2260 (all V_A_2 (c_Nat__Transfer_Onat__set(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)) = V_A_2)) # label(fact_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2261 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2262 (all V_ts_2 all V_zs_2 all V_ys_2 all V_xs_2 all T_c ((exists B_us (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ts_2) & V_zs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),B_us) | V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_zs_2),B_us) & V_ts_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ys_2))) <-> hAPP(hAPP(c_List_Oappend(T_c),V_zs_2),V_ts_2) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_append__eq__append__conv2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2263 (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_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)))) = c_Int_Oint__ge__less__than(V_d_2)) # label(fact_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2264 (all V_A_2 all V_f_2 all T_c hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),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,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_c),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]. 8.94/8.96 2265 (all V_Pa_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (-hBOOL(hAPP(V_Pa_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_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))))) # label(fact_length__filter__less) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2266 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (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_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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_c),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_c),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]. 8.94/8.96 2267 (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]. 8.94/8.96 2268 (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]. 8.94/8.96 2269 (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),V_y) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x))) # label(fact_min__max_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2270 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(T_c))) # label(fact_Sup__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2271 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Predicate_ODomainP(T_c,T_d,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_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_DomainP__Domain__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2272 (all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) = c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_UNIV__Times__UNIV) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2273 (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]. 8.94/8.96 2274 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Int__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2275 (all V_f_2 all V_xs_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(c_List_Ofoldl(T_c,T_d,hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_f_2)),c_Orderings_Otop__class_Otop(T_c)),V_xs_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,T_c),hAPP(c_List_Oset(T_d),V_xs_2)),V_f_2))) # label(fact_INFI__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2276 (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]. 8.94/8.96 2277 (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]. 8.94/8.96 2278 (all V_i_2 all V_xq_2 all V_f_2 all T_c all T_d c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,hAPP(V_xq_2,V_i_2)) = c_New__DSequence_Opos__map(T_d,T_c,V_f_2,V_xq_2,V_i_2)) # label(fact_New__DSequence_Opos__map__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2279 (all V_xs_2 all V_x_2 all V_ys_2 all V_Pa_2 all T_c ((exists B_us exists B_vs ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),B_us))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) & hAPP(c_List_Ofilter(T_c,V_Pa_2),B_vs) = V_xs_2 & hBOOL(hAPP(V_Pa_2,V_x_2)) & hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_vs)) = V_ys_2)) <-> hAPP(c_List_Ofilter(T_c,V_Pa_2),V_ys_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_filter__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2280 (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]. 8.94/8.96 2281 (all V_a_2 all V_b_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),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]. 8.94/8.96 2282 (all V_a_2 all V_b_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(c_Relation_OImage(T_c,T_c,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)))))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)))))) # label(fact_subset__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2283 (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(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_nonzero__minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2284 (all V_R_2 all V_g_2 all V_f_2 all T_f all T_e all V_c_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),V_c_2),hAPP(c_Set_Oimage(tc_prod(T_e,T_f),tc_prod(T_c,T_d),c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2)),V_R_2))) -> -(all B_x all B_y (V_c_2 = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_y)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_e,T_f)),hAPP(hAPP(c_Product__Type_OPair(T_e,T_f),B_x),B_y)),V_R_2)))))) # label(fact_prod__fun__imageE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2285 (all V_g_2 all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),hAPP(c_List_Oset(T_d),V_xs_2))) -> hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x))))) # label(fact_map__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2286 (all V_f_2 all V_A_2 all V_B_2 all T_c all T_d (class_Groups_Oordered__comm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_b (hBOOL(hAPP(hAPP(c_member(T_c),B_b),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_b))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2)))))))) # label(fact_setsum__mono2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2287 (all V_Q all V_P all T_b all T_a hAPP(hAPP(c_COMBK(T_a,T_b),V_P),V_Q) = V_P) # label(help_c__COMBK__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2288 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Olast(tc_Nat_Onat,c_List_Oupt(V_i,V_j)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_last__upt) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2289 (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]. 8.94/8.96 2290 (all V_A_2 all V_f_2 all T_d all V_y_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) -> hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,V_f_2),V_y_2)) = V_y_2)) # label(fact_f__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2291 (all V_B_2 all V_a_2 all T_d all T_c all V_C_2 ((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_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_UN__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2292 (all V_xs_2 all T_c 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_c),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_c)),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Random_Opick(T_c,V_xs_2),V_i_2)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Omap(tc_prod(tc_Code__Numeral_Ocode__numeral,T_c),T_c),c_Product__Type_Osnd(tc_Code__Numeral_Ocode__numeral,T_c)),V_xs_2)))))) # label(fact_pick__member) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2293 (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_x = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)))) # label(fact_min__max_Oinf__absorb1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2294 (all V_u_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_ivl__disj__un_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2295 (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]. 8.94/8.96 2296 (all T_c c_Hilbert__Choice_Oinv__into(T_c,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Fun_Oid(T_c)) = c_Fun_Oid(T_c)) # label(fact_inv__id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2297 (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(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)))) & (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(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)))) # label(fact_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2298 (all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oord(T_d) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_g_2),V_f_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,T_d)),V_f_2),V_g_2))))) # label(fact_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2299 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_f all T_e all T_d hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_d,T_e,T_f,T_c),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,T_c),V_g_2),hAPP(V_f_2,V_x_2))) # label(fact_scomp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2300 (all V_xs_2 all T_c 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_c)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_n_2)),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_nth__mem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2301 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2302 (all V_Ga_2 all V_ts_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)),V_ts_2),V_Ga_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2))) # label(fact_asm) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2303 (all V_y_2 all V_R_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Relation_ODomain(T_c,T_c),V_R_2))) -> (V_x_2 = V_y_2 <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2)))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2304 (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]. 8.94/8.96 2305 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d hAPP(hAPP(hAPP(c_New__Random__Sequence_Oneg__bind(T_d,T_c,V_R_2,V_f_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Oneg__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,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_c)))),hAPP(hAPP(c_COMBC(T_d,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_c))))),hAPP(hAPP(c_COMBC(T_d,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_c)))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_neg__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2306 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,T_c,c_fNot),V_Pa_2)),V_xs_2) = hAPP(c_Product__Type_Osnd(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Opartition(T_c,V_Pa_2,V_xs_2))) # label(fact_partition__filter2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2307 (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]. 8.94/8.96 2308 (all V_a2_2 all V_a1_2 all T_d all T_c c_Nitpick_Ofold__graph_H(T_c,T_d,V_a1_2,V_a2_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),V_a2_2)) # label(fact_fold__graph_H_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2309 (all T_d all V_f_2 all T_c (c_FunDef_Ois__measure(T_c,V_f_2) -> c_FunDef_Ois__measure(tc_prod(T_c,T_d),hAPP(hAPP(c_COMBB(T_c,tc_Nat_Onat,tc_prod(T_c,T_d)),V_f_2),c_Product__Type_Ofst(T_c,T_d))))) # label(fact_measure__fst) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2310 (all V_xs_2 all V_f_2 all T_d all V_n_2 all T_c hAPP(hAPP(c_List_Odrop(T_c),V_n_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Odrop(T_d),V_n_2),V_xs_2))) # label(fact_drop__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2311 (all V_b_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),V_b_2))) # label(fact_Sup__binary) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2312 (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_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]. 8.94/8.96 2313 (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]. 8.94/8.96 2314 (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))) -> c_List_Obutlast(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),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs))) # label(fact_butlast__take) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2315 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2316 (all V_a1_2 all V_a2_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(V_r_2,V_a1_2),V_a2_2)) <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_d,V_r_2),V_a2_2),V_a1_2)))) # label(fact_conversep_Oequations) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2317 (all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osorted(T_c,c_List_Olinorder__class_Osort__key(T_c,T_c,c_COMBI(T_c),V_xs_2)))) # label(fact_sorted__sort) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2318 (all V_R_2 all V_f_2 all T_c c_Wellfounded_Omlex__prod(T_c,V_f_2,V_R_2) = hAPP(hAPP(c_Relation_Oinv__image(tc_prod(tc_Nat_Onat,T_c),T_c),c_Wellfounded_Olex__prod(tc_Nat_Onat,T_c,c_Wellfounded_Oless__than,V_R_2)),hAPP(hAPP(c_COMBS(T_c,T_c,tc_prod(tc_Nat_Onat,T_c)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_c,tc_prod(tc_Nat_Onat,T_c)),T_c),c_Product__Type_OPair(tc_Nat_Onat,T_c)),V_f_2)),c_COMBI(T_c)))) # label(fact_mlex__prod__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2319 (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]. 8.94/8.96 2320 (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]. 8.94/8.96 2321 (all V_n 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))) # label(fact_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2322 (all V_x_2 all V_R_2 all T_c (hBOOL(hAPP(c_Quotient_ORespects(T_c,V_R_2),V_x_2)) <-> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)))) # label(fact_Respects__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2323 (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]. 8.94/8.96 2324 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2325 (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]. 8.94/8.96 2326 (all V_u_2 all V_l_2 c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2)),V_u_2)) # label(fact_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2327 (all V_Pa_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Quotient_ORespects(T_c,V_R_2)))))))) # label(fact_bex__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2328 (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]. 8.94/8.96 2329 (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(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2)))))) # label(fact_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2330 (all V_yq all V_xq all V_x all T_a c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,V_xq),V_yq) = c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,c_Lazy__Sequence_Oappend(T_a,V_xq,V_yq))) # label(fact__01) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2331 (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]. 8.94/8.96 2332 (all V_Pa_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_Pa_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_Pa_2),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,V_m_2,V_n_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_Pa_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2))))) # label(fact_setsum__head) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2333 (all V_R_2 all V_Y_2 all V_X_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_X_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Y_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_Y_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_X_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_xa)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_Y_2)))))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_Y_2)),c_Wellfounded_Omax__ext(T_c,V_R_2)))))))) # label(fact_max__ext_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2334 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_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_c,T_d,V_f_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)))))) # label(fact_bijI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2335 (all V_xs all V_x all T_a V_x = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_hd_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2336 (all V_ys_2 all V_xs_2 all V_g_2 all T_f all V_f_2 all T_c all T_e all T_d hAPP(hAPP(c_List_Omap(tc_prod(T_f,T_e),T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_f,T_e,T_c),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(T_e,T_c),T_f),hAPP(c_COMBB(tc_prod(T_d,T_e),T_c,T_e),V_f_2)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_e,tc_prod(T_d,T_e)),T_f),c_Product__Type_OPair(T_d,T_e)),V_g_2)))),hAPP(hAPP(c_List_Ozip(T_f,T_e),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_e),T_c),V_f_2),hAPP(hAPP(c_List_Ozip(T_d,T_e),hAPP(hAPP(c_List_Omap(T_f,T_d),V_g_2),V_xs_2)),V_ys_2))) # label(fact_map__zip__map) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2337 (all V_f_2 all V_B_2 all T_e all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),T_c,T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,T_c),V_B_2)),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,T_c),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_c),T_c,T_e),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2)),hAPP(c_COMBC(T_d,T_e,T_c),V_f_2))))) # label(fact_SUP__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2338 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(V_f_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__one__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2339 (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(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n)))) # label(fact_hd__drop__conv__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2340 (all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_Abs_2),V_A_2))) # label(fact_type__definition_OAbs__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2341 (all V_Pa_2 ((exists B_x hBOOL(hAPP(V_Pa_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))) <-> (exists B_x (hBOOL(hAPP(V_Pa_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)))))) # label(fact_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2342 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_DiffD1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2343 (all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hAPP(c_List_Oset(T_c),c_List_Oremove1(T_c,V_x_2,V_xs_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_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_set__remove1__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2344 (all V_b_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),V_b_2) = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_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_Inf__binary) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2345 (all V_r_2 all V_s_2 all T_c (c_Wellfounded_Oacyclic(T_c,V_s_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> c_Wellfounded_Oacyclic(T_c,V_r_2)))) # label(fact_acyclic__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2346 (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]. 8.94/8.96 2347 (all V_n_2 all V_m_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (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_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)))))) # label(fact_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2348 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2349 (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]. 8.94/8.96 2350 (all V_b_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2))))) # label(fact_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2351 (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]. 8.94/8.96 2352 (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]. 8.94/8.96 2353 (all V_B_2 all V_A_2 all V_C_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_C_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,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_inj__on__image__set__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2354 (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]. 8.94/8.96 2355 (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]. 8.94/8.96 2356 (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) -> (V_b = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) -> 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]. 8.94/8.96 2357 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),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_c),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_card_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2358 (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]. 8.94/8.96 2359 (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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = 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)))) # label(fact_diff__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2360 (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_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]. 8.94/8.96 2361 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_c,V_r_2) -> c_Wellfounded_Owf(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2))))) # label(fact_finite__acyclic__wf__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2362 (all T_c c_Set_OBex(T_c) = c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_HOL_Obool)) # label(fact_SUPR__bool__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2363 (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]. 8.94/8.96 2364 (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]. 8.94/8.96 2365 (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]. 8.94/8.96 2366 (all V_f_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hAPP(c_Finite__Set_Ocard(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2)))) # label(fact_eq__card__imp__inj__on) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2367 (all V_w all V_z2 all V_z1 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) = 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))) # label(fact_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2368 (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]. 8.94/8.96 2369 (all V_Qa_2 all T_c all V_x_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_Pa_2),V_Qa_2)) -> hBOOL(hAPP(V_Qa_2,V_x_2))))) # label(fact_rev__predicate1D) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2370 (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]. 8.94/8.96 2371 (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]. 8.94/8.96 2372 (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]. 8.94/8.96 2373 (all V_y_2 all V_x_2 all V_z_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_z_2),V_x_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_z_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2)))))) # label(fact_le__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2374 (all V_xs_2 all V_f_2 all T_c all T_d (c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) <-> V_xs_2 = c_List_Olist_ONil(T_d))) # label(fact_map__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2375 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omax(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_Max__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2376 (all V_h_2 all V_g_2 all V_f_2 all T_h all T_g all T_d all T_f all T_e all T_c hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_e,T_f,T_d),hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_g,T_h,tc_prod(T_e,T_f)),V_f_2),V_g_2)),V_h_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_g,T_h,T_d),V_f_2),hAPP(hAPP(c_COMBC(T_g,tc_fun(T_e,tc_fun(T_f,T_d)),tc_fun(T_h,T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_h,tc_prod(T_e,T_f)),tc_fun(tc_fun(T_e,tc_fun(T_f,T_d)),tc_fun(T_h,T_d)),T_g),c_Product__Type_Oscomp(T_h,T_e,T_f,T_d)),V_g_2)),V_h_2))) # label(fact_scomp__scomp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2377 (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]. 8.94/8.96 2378 (all V_x all V_k all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> (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))) & (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_x = 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]. 8.94/8.96 2379 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Osgn__class_Osgn(T_c,V_a_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sgn__0__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2380 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_c) <-> V_a_2 = V_b_2))) # label(fact_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2381 (all V_Pa_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_c),V_Pa_2))) # label(fact_Collect__neg__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2382 (all V_X_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r_2),V_X_2)),V_X_2)) -> hAPP(c_Relation_OImage(T_c,T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_X_2) = V_X_2)) # label(fact_Image__closed__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2383 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2)))))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2384 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2385 (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]. 8.94/8.96 2386 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2))))) # label(fact_Sup__fin__Sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2387 (all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2388 (all V_f_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> -(all B_k 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,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]. 8.94/8.96 2389 (all T_c hAPP(c_Relation_Oconverse(T_c,T_c),c_Relation_OId(T_c)) = c_Relation_OId(T_c)) # label(fact_converse__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2390 (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_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),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_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2391 (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]. 8.94/8.96 2392 (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),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),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)))) & (V_n = c_Groups_Ozero__class_Ozero(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_x))) # label(fact_nth__Cons_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2393 (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]. 8.94/8.96 2394 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_c,V_r_2) <-> c_Wellfounded_Owf(T_c,V_r_2)))) # label(fact_wf__iff__acyclic__if__finite) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2395 (all V_Qa_2 all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Qa_2))) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(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),V_Pa_2)),V_Qa_2)))))) # label(fact_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2396 (all V_n all V_m ((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) = 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 -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))))) # label(fact_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2397 (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]. 8.94/8.96 2398 (all V_a all T_a (class_Rings_Ocomm__semiring__1(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_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2399 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__compl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2400 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_y_2),V_y_2)) = c_Groups_Ozero__class_Ozero(T_c) <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_c) & c_Groups_Ozero__class_Ozero(T_c) = V_y_2))) # label(fact_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2401 (all V_Pa_2 all V_A_2 all T_d all T_c 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_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Set_OCollect(T_c)),hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),V_Pa_2))) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OAll(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fimplies),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_A_2)))),V_Pa_2)))) # label(fact_Collect__ball__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2402 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_H_2),V_b_H_2) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2) <-> V_b_H_2 = V_b_2 & V_a_H_2 = V_a_2)) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2403 (all V_no_2 all V_yes_2 all V_xs_2 all V_Pa_2 all T_c (hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_yes_2),V_no_2) = c_List_Opartition(T_c,V_Pa_2,V_xs_2) -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_no_2))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_yes_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_partition__P) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2404 (all V_b_2 all V_f_2 all T_d all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))) <-> hAPP(V_f_2,V_a_2) = V_b_2)) # label(fact_vimage__singleton__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.96 2405 (all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2),V_z_2),V_A_2)))) # label(fact_folding__image__simple_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2406 (all V_xs_2 all V_Pa_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_length__filter__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2407 (all V_n_2 all V_m_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2408 (all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_insort__key_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2409 (all V_c 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_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_c)) = 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))) # label(fact_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2410 (all V_Pa_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2)) = c_HOL_OThe(tc_prod(T_c,T_d),hAPP(hAPP(c_COMBS(tc_prod(T_c,T_d),T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_HOL_Obool),tc_prod(T_c,T_d)),V_Pa_2),c_Product__Type_Ofst(T_c,T_d))),c_Product__Type_Osnd(T_c,T_d)))) # label(fact_The__split) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2411 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,hAPP(c_List_Orev(T_d),V_ys_2)) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2),V_ys_2))) # label(fact_list__all2__rev1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2412 (all V_k_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2)))) # label(fact_single__Diff__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2413 (all V_A_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2)))) -> c_Relation_Orefl__on(T_c,V_A_2,V_r_2)))) # label(fact_refl__onI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2414 (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]. 8.94/8.97 2415 (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]. 8.94/8.97 2416 (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]. 8.94/8.97 2417 (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]. 8.94/8.97 2418 (all V_xs_2 all V_x_2 all T_c c_List_Oreplicate(T_c,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,hAPP(c_fequal,V_x_2)),V_xs_2)),V_x_2) = hAPP(c_List_Ofilter(T_c,hAPP(c_fequal,V_x_2)),V_xs_2)) # label(fact_replicate__length__filter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2419 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls)))) # label(fact_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2420 (all V_x all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> 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]. 8.94/8.97 2421 (all V_f_2 all V_m_2 all V_n_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (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_c),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_c)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)))))) # label(fact_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2422 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c ((exists B_x (hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_fun(T_e,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]. 8.94/8.97 2423 (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]. 8.94/8.97 2424 (all V_A_2 all V_B_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnI2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2425 (all V_p_2 all V_c_2 all T_c all T_e all T_d hAPP(hAPP(V_c_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_p_2)),hAPP(c_Product__Type_Osnd(T_d,T_e),V_p_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_c_2),V_p_2)) # label(fact_split__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2426 (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]. 8.94/8.97 2427 (all V_x_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oremdups(T_c,V_x_2) <-> V_x_2 = c_List_Olist_ONil(T_c))) # label(fact_remdups__eq__nil__right__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2428 (all V_l_2 hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),c_Int_Onat),V_l_2) = c_List_Oreturn__list(V_l_2)) # label(fact_return__list__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2429 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),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_c)),V_xs_2))))) # label(fact_length__pos__if__in__set) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2430 (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),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(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))) & (-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)),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))))) # label(fact_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2431 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)),hAPP(c_Set_OPow(T_c),V_B_2))))) # label(fact_Pow__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2432 (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]. 8.94/8.97 2433 (all V_xs_2 all T_c 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_xs_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2))) # label(fact_length__transpose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2434 (all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_c,V_R_2))) -> (exists B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_z),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_z)),V_R_2)))))) # label(fact_tranclD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2435 (all V_x all T_a all V_xs ((V_xs = c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = V_x) & (c_List_Olist_ONil(T_a) != V_xs -> c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olast(T_a,V_xs)))) # label(fact_last_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2436 (all V_f_2 all T_d all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_image__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2437 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (V_B_2 = V_A_2 <-> hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)))) # label(fact_inj__on__Un__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2438 (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_Otake(T_a),hAPP(c_Nat_OSuc,V_i)),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))))) # label(fact_take__Suc__conv__app__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2439 (all V_A_H_H_2 all V_f_H_2 all T_e all V_A_H_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_d,T_e,V_f_H_2,V_A_H_2,V_A_H_H_2) <-> c_Fun_Obij__betw(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,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]. 8.94/8.97 2440 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2))))) # label(fact_Min_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2441 (all V_r_2 all V_a_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),V_a_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Wellfounded_Oacc(T_c,V_r_2))))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Wellfounded_Oacc(T_c,V_r_2))))) # label(fact_acc_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2442 (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(tc_Code__Numeral_Ocode__numeral),V_i),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)))) -> 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]. 8.94/8.97 2443 (all V_x_2 all T_d all T_c (class_Orderings_Otop(T_c) -> hAPP(c_Orderings_Otop__class_Otop(tc_fun(T_d,T_c)),V_x_2) = c_Orderings_Otop__class_Otop(T_c))) # label(fact_top__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2444 (all V_n_2 all V_m_2 (V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_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)) = 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]. 8.94/8.97 2445 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Odivision__ring(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_c_2 -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2) <-> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))) # label(fact_nonzero__eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2446 (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]. 8.94/8.97 2447 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)) = c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) # label(fact_sort__key__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2448 (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]. 8.94/8.97 2449 (all V_i_2 all V_xs_2 all V_A_2 all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_c,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_c,tc_Nat_Onat)),c_member(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_prod(T_c,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_c,tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),V_i_2))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_c,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_c)),V_xs_2))))) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_c,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_c,tc_Nat_Onat)),c_member(tc_Nat_Onat)),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_c,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_c)),V_xs_2))))))) # label(fact_sublist__shift__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2450 (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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z)) # label(fact_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2451 (all V_n 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)),V_n) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2452 (all V_B_2 all V_A_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),T_d),c_Set_OPow(T_c)),V_B_2))),hAPP(c_Set_OPow(T_c),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_UN__Pow__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2453 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> -(all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),V_R_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))))))) # label(fact_rel__pow__Suc__E2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2454 (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]. 8.94/8.97 2455 (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]. 8.94/8.97 2456 (all V_r_2 all T_c (c_Relation_Oantisym(T_c,V_r_2) <-> c_Relation_Oantisym(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c))))) # label(fact_antisym__reflcl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2457 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2458 (all V_l_2 all V_k_2 all T_c (class_Orderings_Oorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_k_2),V_l_2)) <-> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2459 (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))))) -> (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]. 8.94/8.97 2460 (all V_B_2 all T_d all V_A_2 all T_c (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)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_finite__UN) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2461 (all V_xs_2 all V_x_2 all V_j_2 all V_i_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) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_j_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))) # label(fact_upt__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2462 (all V_h_2 all T_d all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (c_Fun_Oinj__on(T_d,T_c,V_h_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_Ovimage(T_d,T_c,V_h_2),V_F_2)))))) # label(fact_finite__vimageI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2463 (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(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(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))) & (c_Divides_Odiv__class_Omod(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),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]. 8.94/8.97 2464 (all 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))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2465 (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]. 8.94/8.97 2466 (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]. 8.94/8.97 2467 (all V_n_2 all V_l_2 all T_c c_List_Osublist(T_c,V_l_2,hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_l_2)) # label(fact_sublist__upt__eq__take) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2468 (all V_d_2 all V_Qa_2 all V_c_2 all V_b_2 all V_Pa_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,V_b_2)),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Com_Ostate,c_fNot),V_b_2))),V_d_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),c_Com_Ocom_OCond(V_b_2,V_c_2,V_d_2)),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))))) # label(fact_hoare__derivs_OIf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2469 (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]. 8.94/8.97 2470 (all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> V_x = 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)))) # label(fact_mult__sgn__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2471 (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]. 8.94/8.97 2472 (all V_Pa_2 ((all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (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_Pa_2,hAPP(c_Int_Onat,B_x))))))) # label(fact_all__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2473 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2474 (all V_y_2 all V_x_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),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)))) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)))) # label(fact_eq__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2475 (all T_c c_Relation_Oantisym(T_c,c_Relation_OId(T_c))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2476 (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]. 8.94/8.97 2477 (all V_Ga_2 all V_ts_2 all V_G_H_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_G_H_2,V_ts_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)),V_G_H_2),V_Ga_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2)))) # label(fact_thin) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2478 (all V_x V_x = c_Int_Osucc(c_Int_Opred(V_x))) # label(fact_succ__pred) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2479 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olex(T_c,V_r_2))) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olex(T_c,V_r_2))) & V_x_2 = V_y_2)) # label(fact_Cons__in__lex) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2480 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2))) <-> (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_c),V_x_2),B_x))))))))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2481 (all V_m_2 all V_n_2 all V_k_2 all T_c (class_Divides_Osemiring__div(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_k_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_k_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_k_2),c_Divides_Odiv__class_Omod(T_c,V_m_2,V_n_2))))))) # label(fact_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2482 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c all T_e hAPP(c_Set_Oimage(T_e,tc_prod(T_c,T_d),hAPP(hAPP(c_COMBS(T_e,T_d,tc_prod(T_c,T_d)),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_f_2)),V_g_2)),V_A_2) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_Set_Oimage(T_e,T_c,V_f_2),V_A_2)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),c_Set_Oimage(T_e,T_d,V_g_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_e,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool),T_c),c_Set_Ovimage(T_e,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)))))),V_A_2)))) # label(fact_image__split__eq__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2483 (all V_s_2 all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> (c_Relation_Osym(T_c,V_s_2) -> c_Relation_Osym(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_sym__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2484 (all V_p_2 all T_d all T_c hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_c),c_COMBK(T_c,T_d)),V_p_2) = hAPP(c_Product__Type_Ofst(T_c,T_d),V_p_2)) # label(fact_fst__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2485 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2))))))) # label(fact_Max__ge) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2486 (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]. 8.94/8.97 2487 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_H_2,V_b_H_2) = c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2) <-> V_a_2 = V_a_H_2 & V_b_H_2 = V_b_2)) # label(fact_pair__box_Oinject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2488 (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]. 8.94/8.97 2489 (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]. 8.94/8.97 2490 (all V_A_2 all V_B_2 all V_f_2 all T_e all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_prod(T_d,T_e),T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2)),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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),tc_fun(tc_fun(T_e,tc_HOL_Obool),T_c),T_d),c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_e,T_c)),V_f_2)),V_B_2)),V_A_2))) # label(fact_setprod__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2491 (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]. 8.94/8.97 2492 (all V_r_2 all V_S_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_S_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_e),c_Relation_Oconverse(T_d,T_c)),V_r_2)) = hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_S_2),V_r_2))) # label(fact_converse__INTER) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2493 (all V_w_2 all V_c_2 all V_b_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) & class_Int_Onumber(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))) # label(fact_divide__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2494 (all V_xs_2 all T_c hAPP(c_List_Orev(T_c),V_xs_2) = hAPP(c_List_Ofoldl(tc_List_Olist(T_c),T_c,hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c)),V_xs_2)) # label(fact_rev__foldl__cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2495 (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]. 8.94/8.97 2496 (all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osorted__list__of__set(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_List_Olist_ONil(T_c))) # label(fact_sorted__list__of__set__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2497 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = 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)) # label(fact_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2498 (all V_z_2 all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_z_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_rtrancl__trancl__trancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2499 (all T_a hAPP(c_List_Orev(T_a),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_rev_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2500 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))))) # label(fact_Inf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2501 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_x_2),V_A_2))))) # label(fact_min__max_OSup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2502 (all V_xa_2 all V_x_2 all V_r_2 all T_c (c_List_Olistrelp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2),V_x_2,V_xa_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_xa_2)),c_List_Olistrel(T_c,V_r_2))))) # label(fact_listrelp__listrel__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2503 (all V_r_2 all T_c hAPP(c_Relation_ODomain(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)) = hAPP(c_Relation_ODomain(T_c,T_c),V_r_2)) # label(fact_trancl__domain) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2504 (all V_S_2 all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)) = c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_R_2)),c_Transitive__Closure_Ortrancl(T_c,V_S_2)))) # label(fact_rtrancl__Un__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2505 (all V_k_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OatLeast(T_c,V_less__eq_2,V_k_2))) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_k_2),V_i_2)))) # label(fact_ord_OatLeast__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2506 (all V_z_2 all V_f_2 all T_c all T_d V_z_2 = hAPP(hAPP(c_Finite__Set_Ofold(T_d,T_c,V_f_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_fold__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2507 (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_c = V_b -> 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]. 8.94/8.97 2508 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),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_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel(T_c,V_r_2)),c_List_Olistrel(T_c,V_s_2))))) # label(fact_listrel__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2509 (all V_m all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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))) # label(fact_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2510 (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__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]. 8.94/8.97 2511 (all V_y_2 all V_f_2 all T_d all T_c all V_x_2 all V_z_2 (V_z_2 != V_x_2 -> hAPP(c_Fun_Ofun__upd(T_c,T_d,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]. 8.94/8.97 2512 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2) = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2513 (all V_xs all V_x all T_b all T_a (c_List_Olist_ONil(T_a) = hAPP(c_List_Orotate1(T_a),c_List_Olist_ONil(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))))) # label(fact_rotate__simps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2514 (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]. 8.94/8.97 2515 (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]. 8.94/8.97 2516 (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]. 8.94/8.97 2517 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2)))) # label(fact_Compl__atMost) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2518 (all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2519 (all V_r_2 all V_x_2 all T_c ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_x_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),B_y),c_Wellfounded_Oacc(T_c,V_r_2))))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Wellfounded_Oacc(T_c,V_r_2))))) # label(fact_acc_OaccI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2520 (all V_M_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_M_2)) -> (exists B_h c_Fun_Obij__betw(T_c,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_c),V_M_2)))))) # label(fact_ex__bij__betw__finite__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2521 (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))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2522 (all V_b_2 all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((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_c),B_x),V_b_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)),V_b_2))))) # label(fact_Sup__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2523 (all V_B_2 all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),hAPP(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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2524 (all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 all V_com2_2 all V_com1_2 all V_fun_2 (V_com1_H_2 = V_com1_2 & V_com2_H_2 = V_com2_2 & V_fun_2 = V_fun_H_2 <-> c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2) = c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2))) # label(fact_com_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2525 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2) = hAPP(c_Groups_Oabs__class_Oabs(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2)))) # label(fact_abs__setsum__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2526 (all V_B_2 all V_A_2 all V_a_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2527 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_rtrancl__idemp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2528 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_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(hAPP(c_Finite__Set_Ofold(T_c,T_c,V_f_2),V_x_2),V_A_2) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2529 (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]. 8.94/8.97 2530 (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]. 8.94/8.97 2531 (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]. 8.94/8.97 2532 (all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,tc_Nat_Onat),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,T_d),c_Nat_Osemiring__1__class_Oof__nat(T_c)),V_f_2)),V_A_2))) # label(fact_of__nat__setsum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2533 (all V_x_2 all V_A_2 all V_a_2 all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,V_f_2,V_a_2,V_A_2),V_x_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),V_x_2))))) # label(fact_fold1Set_Ointros) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2534 (all V_b_2 all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),V_b_2))))) # label(fact_abs__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2535 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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),c_Groups_Oone__class_Oone(T_d)),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2) = c_Groups_Oone__class_Oone(T_d)))) # label(fact_setprod_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2536 (all V_x_2 (c_Nat__Transfer_Ois__nat(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)))) # label(fact_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2537 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),V_y_2) = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_min__max_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2538 (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]. 8.94/8.97 2539 (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]. 8.94/8.97 2540 (all V_A_2 all T_c 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_c,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2)))) # label(fact_sup1I2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2541 (all V_f_2 all V_T_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_d))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_f_2),c_Groups_Oone__class_Oone(T_d)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_f_2),c_Groups_Oone__class_Oone(T_d)),V_S_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_f_2),c_Groups_Oone__class_Oone(T_d)),V_T_2))))))) # label(fact_fold__image__Un__one) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2542 (all V_a4_2 all V_a2_2 all V_a3_2 all V_a1_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(V_r_2,V_a1_2),V_a2_2)) & c_List_Olistrelp(T_c,V_r_2,V_a3_2,V_a4_2) <-> c_List_Olistrelp(T_c,V_r_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_a1_2),V_a3_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_a2_2),V_a4_2)))) # label(fact_listrelp_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2543 (all V_r_2 all T_c hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)) = c_Transitive__Closure_Otrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2))) # label(fact_trancl__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2544 (all V_a_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_2)),V_r_2)))) # label(fact_wf__irrefl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2545 (all V_B_2 all V_A_2 all T_c (V_A_2 = V_B_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Compl__eq__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2546 (all V_h_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Groups_Oplus__class_Oplus(T_d)),V_g_2)),V_h_2)),V_A_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,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_A_2))))) # label(fact_setsum_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.97 2547 (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]. 8.94/8.97 2548 (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]. 8.94/8.97 2549 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_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)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),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]. 8.94/8.98 2550 (all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_c (V_ys_2 = V_xs_2 & V_y_2 = V_x_2 <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c))) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),c_List_Olist_ONil(T_c))))) # label(fact_append1__eq__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2551 (all V_fs_2 all T_c 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_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,V_fs_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_c,tc_Nat_Onat)),V_f_2),V_fs_2))))))) # label(fact_measures__lesseq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2552 (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]. 8.94/8.98 2553 (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 2554 (all V_g_2 all V_f_2 all T_d all T_c all T_e all T_f hAPP(c_Fun_Ocomp(T_d,T_e,tc_prod(T_c,T_d),V_g_2),c_Product__Type_Osnd(T_c,T_d)) = hAPP(c_Fun_Ocomp(tc_prod(T_f,T_e),T_e,tc_prod(T_c,T_d),c_Product__Type_Osnd(T_f,T_e)),c_Product__Type_Omap__pair(T_c,T_f,T_d,T_e,V_f_2,V_g_2))) # label(fact_snd__comp__map__pair) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2555 (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 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2556 (all V_y all V_i all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_y)) # label(fact_list__update__code_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2557 (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]. 8.94/8.98 2558 (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]. 8.94/8.98 2559 (all V_A_2 all V_f_2 all T_d all V_B_2 all T_c ((exists B_AA (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),B_AA),V_A_2)) & hAPP(c_Set_Oimage(T_d,T_c,V_f_2),B_AA) = V_B_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_d,T_c,V_f_2),V_A_2))))) # label(fact_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2560 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBS(T_d,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_d),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_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_B_2))) # label(fact_INT__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2561 (all V_A_2 all T_c ((all B_x -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2562 (all V_r_2 all T_c c_List_Olistrelp(T_c,V_r_2,c_List_Olist_ONil(T_c),c_List_Olist_ONil(T_c))) # label(fact_listrelp_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2563 (all V_S_2 all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_Relation_ODomain(T_c,T_d))) # label(fact_Domain__Union) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2564 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_a_2),V_A_2)))) # label(fact_Min_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2565 (all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__Collect__not) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2566 (all V_D_2 all V_B_2 all T_d all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,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_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_C_2),V_D_2)))))) # label(fact_Sigma__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2567 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),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))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> V_x_2 = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2568 (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]. 8.94/8.98 2569 (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]. 8.94/8.98 2570 (all V_x_2 all V_r_2 all T_c (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_x_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_irrefl__tranclI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2571 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Olist_OCons(T_c),hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2))) # label(fact_map_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2572 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2573 (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]. 8.94/8.98 2574 (all V_r_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2)))))) # label(fact_finite__quotient) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2575 (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]. 8.94/8.98 2576 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d hAPP(hAPP(hAPP(c_New__Random__Sequence_Opos__bind(T_d,T_c,V_R_2,V_f_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Opos__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,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_c))),hAPP(hAPP(c_COMBC(T_d,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_c)))),hAPP(hAPP(c_COMBC(T_d,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_c))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_pos__bind__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2577 (all V_B_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2))))) # label(fact_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2578 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Oidom(T_c) -> (V_c_2 = c_Groups_Ozero__class_Ozero(T_c) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2)))))) # label(fact_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2579 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> c_Relation_Osym(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) # label(fact_sym__rtrancl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2580 (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]. 8.94/8.98 2581 (all V_k all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_k),c_Groups_Osgn__class_Osgn(T_a,V_k)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_k))) # label(fact_abs__sgn) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2582 (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]. 8.94/8.98 2583 (all V_r_2 all T_c (c_Predicate_Oreflp(T_c,V_r_2) <-> c_Relation_Orefl__on(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))))) # label(fact_reflp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2584 (all V_l all V_k c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_Osucc(V_l))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l))) # label(fact_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2585 (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]. 8.94/8.98 2586 (all V_m_2 all V_n_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_n_2),V_m_2)))) # label(fact_less__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2587 (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]. 8.94/8.98 2588 (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]. 8.94/8.98 2589 (all V_l all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OBit1(V_k)),V_l) = 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)) # label(fact_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2590 (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]. 8.94/8.98 2591 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_x_2 all T_c hAPP(hAPP(hAPP(hAPP(c_New__Random__Sequence_Opos__single(T_c),V_x_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Opos__single(T_c,V_x_2)) # label(fact_pos__single__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2592 (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)) -> 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) = c_Random_Ominus__shift(V_r,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),V_k),V_l)))) # label(fact_minus__shift__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2593 (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]. 8.94/8.98 2594 (all V_r_2 all T_c (c_Wellfounded_Oacyclic(T_c,V_r_2) <-> c_Wellfounded_Oacyclic(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_acyclic__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2595 (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]. 8.94/8.98 2596 (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]. 8.94/8.98 2597 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> -(all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_z_2)),V_R_2)))))) # label(fact_rel__pow__Suc__E) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2598 (all V_A_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> c_Fun_Oinj__on(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c),V_A_2))) # label(fact_inj__uminus) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2599 (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]. 8.94/8.98 2600 (all V_B_2 all T_d all V_A_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))) & (-(hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))))) # label(fact_card__Plus__conv__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2601 (all V_Pa_2 all V_a_2 all T_c (hBOOL(hAPP(V_Pa_2,V_a_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_OCollect(T_c),V_Pa_2))))) # label(fact_mem__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2602 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_R_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) # label(fact_rtrancl__idemp__self__comp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2603 (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]. 8.94/8.98 2604 (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_x),V_y)) = 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]. 8.94/8.98 2605 (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]. 8.94/8.98 2606 (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]. 8.94/8.98 2607 (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)),V_x_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)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2))))) # label(fact_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2608 (all T_c c_List_Oset(T_c) = c_List_Omember(T_c)) # label(fact_member__set) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2609 (all V_j_2 all V_i_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(tc_Int_Oint,T_c),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_c),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]. 8.94/8.98 2610 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Max_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2611 (all V_A_2 all V_f_2 all V_g_2 all T_e all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_g_2),V_f_2),V_A_2) -> c_Fun_Oinj__on(T_e,T_d,V_g_2,hAPP(c_Set_Oimage(T_c,T_e,V_f_2),V_A_2)))) # label(fact_inj__on__imageI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2612 (all V_S_2 all V_R_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_R_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_S_2),V_R_2))),V_S_2)) <-> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)))) # label(fact_wf__union__merge) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2613 (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]. 8.94/8.98 2614 (all V_f_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_card__image__le) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2615 (all V_v_2 all V_w_2 all T_c (class_Int_Onumber__ring(T_c) -> (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_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2))) = hAPP(hAPP(c_HOL_OLet(T_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_c),c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)))),c_COMBI(T_c)))) & (-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_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2))) = c_Groups_Oone__class_Oone(T_c)))) # label(fact_power__number__of__odd__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2616 (all V_y_2 all V_f_2 all T_d all T_c hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),V_y_2) = c_Hilbert__Choice_OEps(T_c,hAPP(hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_y_2))) # label(fact_inv__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2617 (all V_f_2 all T_d all T_c c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_inj__on__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2618 (all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Nitpick_Osetsum_H(T_c,T_d,V_f_2,V_A_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),c_Hilbert__Choice_OEps(tc_List_Olist(T_c),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),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_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_List_Oset(T_c))),V_A_2))),c_List_Odistinct(T_c)))))) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Ozero__class_Ozero(T_d) = c_Nitpick_Osetsum_H(T_c,T_d,V_f_2,V_A_2)))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2619 (all T_d all T_c 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_c,T_d,tc_HOL_Obool),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))))) # label(fact_splitI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2620 (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]. 8.94/8.98 2621 (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]. 8.94/8.98 2622 (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]. 8.94/8.98 2623 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_IntD1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2624 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> c_Fun_Oinj__on(T_d,T_c,c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_inj__on__the__inv__into) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2625 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (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(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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2) = V_x_2) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,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_Inf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2626 (all V_A_2 all V_x_2 all T_d all T_c hAPP(c_Finite__Set_Ocard(T_d),V_A_2) = hAPP(c_Finite__Set_Ocard(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_A_2)))) # label(fact_card__cartesian__product__singleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2627 (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]. 8.94/8.98 2628 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2629 (all V_B_2 all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_2))) <-> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),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))))),V_B_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_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_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2630 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),V_A_2) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_bij__betw__empty1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2631 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_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_f_2,V_y_2) = hAPP(V_f_2,V_x_2)))))) # label(fact_inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2632 (all V_y_2 all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),V_A_2)) -> (hAPP(V_Abs_2,V_y_2) = hAPP(V_Abs_2,V_x_2) <-> V_y_2 = V_x_2))))) # label(fact_type__definition_OAbs__inject) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2633 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_max__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2634 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_B_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Max__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2635 (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]. 8.94/8.98 2636 (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]. 8.94/8.98 2637 (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_Groups_Ouminus__class_Ouminus(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(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))),V_z))) # label(fact_minus__number__of__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2638 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_u_2))))) # label(fact_ivl__disj__un_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2639 (all V_x_2 all V_g_2 all V_f_2 all T_f all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_f),V_x_2))) # label(fact_fst__map__pair) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2640 (all T_c -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))))) # label(fact_infinite__UNIV__listI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2641 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> V_x_2 = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_min__max_OInf__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2642 (all V_f_2 all V_n_2 all T_d all T_c hAPP(hAPP(c_Nat_Ocompow(tc_fun(T_c,T_d),tc_fun(T_c,T_d)),V_n_2),c_Fun_Ocomp(T_d,T_d,T_c,V_f_2)) = c_Fun_Ocomp(T_d,T_d,T_c,hAPP(hAPP(c_Nat_Ocompow(T_d,T_d),V_n_2),V_f_2))) # label(fact_comp__funpow) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2643 (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)) -> V_m = 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))) # label(fact_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2644 (all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_finite__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2645 (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]. 8.94/8.98 2646 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(tc_List_Olist(T_c)),V_xs_2)) -> ((all B_ys (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_ys),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2))) -> hBOOL(hAPP(c_List_Odistinct(T_c),B_ys)))) -> ((all B_ys (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_ys),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2))) -> (all B_zs (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_zs),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2))) -> (B_zs != B_ys -> 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_List_Oset(T_c),B_ys)),hAPP(c_List_Oset(T_c),B_zs))))))) -> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Oconcat(T_c,V_xs_2))))))) # label(fact_distinct__concat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2647 (all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),V_r_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Relation_ORange(T_c,T_d),V_r_2))))) # label(fact_finite__Range) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2648 (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]. 8.94/8.98 2649 (all T_c 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_c,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2650 (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]. 8.94/8.98 2651 (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),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2652 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_2))) # label(fact_Un__Inter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2653 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2))))) # label(fact_set__mp) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2654 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_b_2) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) = V_a_2))) # label(fact_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2655 (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]. 8.94/8.98 2656 (all V_x_2 all V_f_2 all T_c -hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_x_2))) # label(fact_empty__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2657 (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]. 8.94/8.98 2658 (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_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)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> 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)))) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k)))) # label(fact_of__int__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2659 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice__class_OInf__fin(T_c,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(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2))))))) # label(fact_Inf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2660 (all V_I_2 all V_xs_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2))),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_set__sublist__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2661 (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]. 8.94/8.98 2662 (all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,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_c)),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_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i)),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_j)))))))))) # label(fact_sorted__equals__nth__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2663 (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]. 8.94/8.98 2664 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_ivl__disj__un_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2665 (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]. 8.94/8.98 2666 (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]. 8.94/8.98 2667 (all T_c all V_a_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_a_2)) -> hAPP(c_Set_OCollect(T_c),hAPP(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(c_fequal,V_a_2))),V_Pa_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) & (-hBOOL(hAPP(V_Pa_2,V_a_2)) -> hAPP(c_Set_OCollect(T_c),hAPP(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(c_fequal,V_a_2))),V_Pa_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2668 (all V_x_2 all V_xs_2 all T_c 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_c)),V_xs_2))) -> (hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_i_2) = V_x_2 <-> V_xs_2 = hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2)))) # label(fact_list__update__same__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2669 (all V_ys all V_y all V_xs all V_n all T_a (hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys) = hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs) -> V_y = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n))) # label(fact_nth__via__drop) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2670 (all V_n_2 all V_xs_2 all V_x_2 all T_c c_Nat_Onat_Onat__case(T_c,V_x_2,hAPP(c_List_Onth(T_c),V_xs_2),V_n_2) = hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_n_2)) # label(fact_nth_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2671 (all V_n_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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]. 8.94/8.98 2672 (all V_f_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_F_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_F_2)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2673 (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]. 8.94/8.98 2674 (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]. 8.94/8.98 2675 (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]. 8.94/8.98 2676 (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)) -> (c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) -> 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]. 8.94/8.98 2677 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)))) # label(fact_finite__PlusD_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2678 (all V_ys_2 all T_d all V_y_2 all V_x_2 all V_f_2 all T_c (class_Orderings_Olinorder(T_c) -> (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))) -> hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_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))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) = hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_ys_2))))) # label(fact_insort__key_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2679 (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]. 8.94/8.98 2680 (all V_ys_2 all V_xs_2 all T_c (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> V_xs_2 = c_List_Olist_ONil(T_c))) # label(fact_append__self__conv2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2681 (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]. 8.94/8.98 2682 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_R_2)),c_Relation_OId(T_c))) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2683 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(c_Finite__Set_Ocard(tc_Int_Oint),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)))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2684 (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__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_1),V_b_1))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> V_y = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_1),V_b_1))))) # label(fact_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2685 (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]. 8.94/8.98 2686 (all V_B_2 all V_A_2 all V_f_2 all T_e all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,hAPP(c_Fun_Ocomp(tc_prod(T_d,T_e),T_d,T_c,c_Product__Type_Ofst(T_d,T_e)),V_f_2)),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_e,hAPP(c_Fun_Ocomp(tc_prod(T_d,T_e),T_e,T_c,c_Product__Type_Osnd(T_d,T_e)),V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_c,tc_prod(T_d,T_e),V_f_2),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)))) # label(fact_vimage__Times) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2687 (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]. 8.94/8.98 2688 (all V_w_2 all V_z_2 all T_c (class_Int_Onumber__ring(T_c) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),c_Groups_Otimes__class_Otimes(T_c)),c_COMBI(T_c))))) # label(fact_power__number__of__even) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2689 (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]. 8.94/8.98 2690 (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]. 8.94/8.98 2691 (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]. 8.94/8.98 2692 (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]. 8.94/8.98 2693 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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]. 8.94/8.98 2694 (all V_A_2 all T_c ((exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))) <-> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2695 (all V_r_2 all V_A_2 all T_d all V_S_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) -> c_Relation_Orefl__on(T_d,hAPP(V_A_2,B_x),hAPP(V_r_2,B_x)))) -> c_Relation_Orefl__on(T_d,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_S_2),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),V_S_2),V_r_2)))) # label(fact_refl__on__INTER) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2696 (all V_y_2 all V_f_2 all T_d all V_A_2 all V_x_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,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))))))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = hAPP(c_Set_Oimage(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2)),V_A_2)))) # label(fact_fun__upd__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2697 (all V_f_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),V_S_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_x)))) -> ((all B_T1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_T1),V_S_2)) -> (all B_T2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_T2),V_S_2)) -> (B_T1 != B_T2 -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_T1)) -> (hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_T2)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_d))))))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2)),V_S_2)))))) # label(fact_setsum__UNION__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2698 (all V_r_2 all V_y_2 all V_b_2 all V_x_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olexord(T_c,V_r_2))) & V_a_2 = V_b_2 <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_x_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_b_2),V_y_2))),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__cons__cons) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2699 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_a))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_finite__UN__I) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2700 (all V_xs all T_a hAPP(hAPP(c_List_Otake(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_xs) = c_List_Olist_ONil(T_a)) # label(fact_take__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.98 2701 (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]. 8.94/8.98 2702 (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]. 8.94/8.99 2703 (all V_xs_2 all V_Pa_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_list__all__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2704 (all V_r_2 all V_v_2 all V_u_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_u_2)),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_v_2))),c_List_Olexord(T_c,V_r_2))) -> ((all B_a -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_a),B_a)),V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_u_2),V_v_2)),c_List_Olexord(T_c,V_r_2)))))) # label(fact_lexord__append__leftD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2705 (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]. 8.94/8.99 2706 (all V_n_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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_c),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]. 8.94/8.99 2707 (all V_i_2 all V_a_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_i_2)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) -> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_a_2)))))) # label(fact_distinct__list__update) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2708 (all V_xs_2 all V_n_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2))),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_set__drop__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2709 (all V_Pa_2 all T_c -c_List_Olist__ex(T_c,V_Pa_2,c_List_Olist_ONil(T_c))) # label(fact_list__ex__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2710 (all V_B_2 all V_A_2 all T_c (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)) <-> 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)),V_B_2))))) # label(fact_disjoint__eq__subset__Compl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2711 (all V_x_2 all V_A_2 all T_c (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(c_Finite__Set_Ocard(T_c),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_c),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_card_Oremove) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2712 (all V_m all V_n c_Code__Numeral_Odiv__mod__code__numeral(V_n,V_m) = 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))) # label(fact_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2713 (all V_nat_2 all V_f2_2 all V_f1_2 all T_c hAPP(c_Nat_Onat_Onat__rec(T_c,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_c,V_f1_2,V_f2_2),V_nat_2))) # label(fact_nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2714 (all V_m all V_n hAPP(hAPP(c_Orderings_Oord__class_Omax(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_n),V_m)),V_m)) # label(fact_nat__minus__add__max) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2715 (all T_c c_Finite__Set_Ofun__left__comm__idem(T_c,tc_fun(T_c,tc_HOL_Obool),c_Set_Oinsert(T_c))) # label(fact_fun__left__comm__idem__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2716 (all V_n_2 all V_m_2 all T_c hAPP(c_Fun_Ocomp(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Orotate(T_c,V_m_2)),c_List_Orotate(T_c,V_n_2)) = c_List_Orotate(T_c,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_rotate__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2717 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olistrel1(T_c,V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olistrel1(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))))) # label(fact_in__listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2718 (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_c ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_092_060_094isub_0621_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2))) -> V_xs_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),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_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_092_060_094isub_0621_2)),V_ys_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_c)),V_xs_092_060_094isub_0621_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2))) -> V_ys_092_060_094isub_0621_2 = hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2) & V_ys_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2)),V_xs_092_060_094isub_0622_2)) <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_092_060_094isub_0621_2),V_xs_092_060_094isub_0622_2) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_092_060_094isub_0621_2),V_ys_092_060_094isub_0622_2))) # label(fact_append__eq__append__conv__if) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2719 (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]. 8.94/8.99 2720 (all V_R_2 all V_p_2 all T_c ((exists B_n hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),B_n),V_R_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))))) # label(fact_rtrancl__power) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2721 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_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_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_inj__imp__surj__inv) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2722 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),V_A_2))))))) # label(fact_min__max_OInf__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2723 (all V_z_2 all V_w_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (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(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_w_2)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)))))) # label(fact_of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2724 (all V_Pa_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_Pa_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_Pa_2),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2725 (all V_Pa_2 all T_c -(exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_bex__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2726 (all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Olinorder__class_Osort__key(T_d,T_c,V_f_2,V_xs_2))))) # label(fact_sorted__sort__key) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2727 (all V_A_2 all T_d all T_c 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_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2I2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2728 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_B_2)))))))) # label(fact_min__max_OSup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2729 (all V_r_2 all V_q_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),c_Nitpick_Oint__lcm(hAPP(c_Nitpick_Odenom(T_c),V_q_2),hAPP(c_Nitpick_Odenom(T_c),V_r_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,T_c),tc_Int_Oint),c_Nitpick_Ofrac(T_c)),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_c),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_c),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_c),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_c),V_r_2)))))),c_COMBI(tc_Int_Oint))) = c_Nitpick_Oplus__frac(T_c,V_q_2,V_r_2)) # label(fact_Nitpick_Oplus__frac__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2730 (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)) -> hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x) = V_xs)) # label(fact_list__update__beyond) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2731 (all V_C_2 all V_A_2 all T_c ((all B_n 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(tc_Nat_Onat,tc_fun(T_c,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_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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]. 8.94/8.99 2732 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2733 (all V_F_2 all V_h_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Ovimage(T_c,T_d,V_h_2),V_F_2))) -> (c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_d,V_h_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_F_2))))) # label(fact_finite__vimageD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2734 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_y_2) <-> V_y_2 = V_x_2))) # label(fact_atLeast__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2735 (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]. 8.94/8.99 2736 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_card__Diff__subset__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2737 (all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (class_Groups_Ominus(T_c) -> hAPP(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,T_c)),V_A_2),V_B_2),V_x_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(V_A_2,V_x_2)),hAPP(V_B_2,V_x_2)))) # label(fact_minus__apply) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2738 (all V_f_2 all T_d all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> c_Relation_Osym(T_d,hAPP(hAPP(c_Relation_Oinv__image(T_c,T_d),V_r_2),V_f_2)))) # label(fact_sym__inv__image) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2739 (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))) -> V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i))) # label(fact_nth__list__update__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2740 (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]. 8.94/8.99 2741 (all V_l_2 (V_l_2 = c_Int_OMin <-> c_Int_OBit1(V_l_2) = c_Int_OMin)) # label(fact_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2742 (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]. 8.94/8.99 2743 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2744 (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]. 8.94/8.99 2745 (all V_j_2 all V_B_2 all V_A_2 all T_e all V_C_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_C_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_e),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_member(T_e),V_j_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_e),c_Set_Oimage(T_c,T_d,V_f_2)),V_B_2)))))) # label(fact_image__INT) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2746 (all V_z V_z = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)) # label(fact_zmult__1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2747 (all V_B_2 all V_J_2 all T_e all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),T_d),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e)),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_d),c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool))),V_A_2))),V_B_2)))) # label(fact_Un__INT__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2748 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2749 (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),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)),V_b),V_c))) # label(fact_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2750 (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)) -> 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),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]. 8.94/8.99 2751 (all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) <-> c_Relation_Orefl__on(T_c,V_A_2,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_refl__on__converse) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2752 (all V_n all T_a (class_Rings_Osemiring__0(T_a) & class_Power_Opower(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_OSuc,V_n)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2753 (all T_a (class_Groups_Osgn__if(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Osgn__class_Osgn(T_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_sgn0) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2754 (all V_a_2 all V_f_2 all T_d all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_c,tc_fun(T_d,tc_HOL_Obool),V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(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_r_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_UN__equiv__class) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2755 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2756 (all V_com2_H all V_com1_H c_Com_Ocom_OSemi(V_com1_H,V_com2_H) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2757 (all V_C_2 all V_J_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_J_2),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2)) # label(fact_Sigma__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2758 (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]. 8.94/8.99 2759 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))))) # label(fact_equiv__type) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2760 (all V_xs_2 all T_c all V_ys_2 (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = V_ys_2 <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_self__append__conv2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2761 (all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__UNIV__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2762 (all V_a_2 all V_y_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (exists B_y_H (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),B_y_H)) & hAPP(hAPP(V_f_2,V_a_2),B_y_H) = V_y_2)))))) # label(fact_fun__left__comm_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2763 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_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)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) <-> hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_r_2) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_r_2)))))) # label(fact_eq__equiv__class__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2764 (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]. 8.94/8.99 2765 (all T_1 all T_2 (class_Enum_Oenum(T_1) & class_Enum_Oenum(T_2) -> class_Enum_Oenum(tc_sum(T_2,T_1)))) # label(arity_sum__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2766 (all V_S_2 all V_R_2 all T_d all T_c (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_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_R_2 = V_S_2)) # label(fact_pred__equals__eq2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2767 (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]. 8.94/8.99 2768 (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]. 8.94/8.99 2769 (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]. 8.94/8.99 2770 (all V_x_2 all V_A_2 all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,V_A_2),V_x_2)) -> V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_fold1Set__nonempty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2771 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Ofinite__psubset(T_c))))) # label(fact_in__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2772 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_B_2))),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_image__diff__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2773 (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]. 8.94/8.99 2774 (all V_Pa_2 all T_c (class_Enum_Oenum(T_c) -> ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),V_Pa_2))))) # label(fact_exists__code) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2775 (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),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2776 (all V_xs_2 all V_x_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oremdups(T_c,V_xs_2) = c_List_Oremdups(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Oremdups(T_c,V_xs_2)) = c_List_Oremdups(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_remdups_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2777 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (V_y_2 = hAPP(V_f_2,V_x_2) -> V_x_2 = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),V_y_2)))) # label(fact_inv__f__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2778 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),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)))) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_finite__cartesian__productD1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2779 (all V_r_2 all V_f_2 all T_c ((all B_i 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,hAPP(c_Nat_OSuc,B_i))),hAPP(V_f_2,B_i))),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) -> (c_Wellfounded_Owf(T_c,c_Transitive__Closure_Otrancl(T_c,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]. 8.94/8.99 2780 (all V_D_2 all V_B_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_D_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Int__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2781 (all V_r_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)) = hAPP(c_List_Olexn(T_c,V_r_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_lexn_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2782 (all V_B_2 all V_C_2 all T_d all V_a_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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_d),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_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__extend__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2783 (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]. 8.94/8.99 2784 (all V_R_2 all V_n_2 all V_m_2 all T_c hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),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_c,T_c,T_c),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_m_2),V_R_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) # label(fact_rel__pow__add) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2785 (all V_XS_2 all V_A_2 all T_c hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_List_Olist(T_c)),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_List_Olist(T_c)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),c_COMBS(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),T_c),c_COMBS(tc_List_Olist(T_c),tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fconj))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),c_COMBB(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_List_Olist(T_c),tc_HOL_Obool,tc_List_Olist(T_c))),c_fequal))),c_List_Olist_OCons(T_c)))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),T_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_c))),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_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_c))),V_XS_2)))))) = c_List_Oset__Cons(T_c,V_A_2,V_XS_2)) # label(fact_set__Cons__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2786 (all V_B_2 all V_A_2 all T_c (V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,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)),V_A_2),V_B_2)))) # label(fact_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2787 (all V_k c_Int_OPls != c_Int_OBit1(V_k)) # label(fact_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2788 (all V_x_2 all V_f_2 all V_g_2 all T_e all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_g_2),V_f_2)),V_x_2) = hAPP(c_Set_Ovimage(T_c,T_e,V_f_2),hAPP(c_Set_Ovimage(T_e,T_d,V_g_2),V_x_2))) # label(fact_vimage__compose) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2789 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) <-> V_A_2 = V_B_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2790 (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__linear) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2791 (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]. 8.94/8.99 2792 (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]. 8.94/8.99 2793 (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]. 8.94/8.99 2794 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2))))) # label(fact_refl__onD2) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2795 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,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_x_2),V_y_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_x_2),V_z_2)),V_r_2)) -> V_y_2 = V_z_2)))) # label(fact_single__valuedD) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2796 (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]. 8.94/8.99 2797 (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]. 8.94/8.99 2798 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> c_Fun_Obij__betw(T_d,T_c,c_Fun_Othe__inv__into(T_c,T_d,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]. 8.94/8.99 2799 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> 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_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),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)),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),c_Groups_Oone__class_Oone(T_d)),V_B_2))))))) # label(fact_fold__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2800 (all V_g_2 all V_B_2 all T_d all V_A_2 all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (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)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,tc_prod(T_c,T_d),c_Groups_Otimes__class_Otimes(T_e)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_g_2)),c_Groups_Oone__class_Oone(T_e)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_c,c_Groups_Otimes__class_Otimes(T_e)),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBC(T_c,T_e,tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_e),tc_fun(T_e,tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e)),T_c),c_Finite__Set_Ofold__image(T_e,T_d,c_Groups_Otimes__class_Otimes(T_e))),V_g_2)),c_Groups_Oone__class_Oone(T_e))),V_B_2)),c_Groups_Oone__class_Oone(T_e)),V_A_2))))) # label(fact_fold__image__Sigma) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2801 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_c_2),V_d_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2802 (all V_b_2 all V_a_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hAPP(c_Relation_OImage(T_c,T_c,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_Relation_OImage(T_c,T_c,V_r_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_equiv__class__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2803 (all V_ys_2 all V_Pa_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(hAPP(c_List_Oappend(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)),V_ys_2) = c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))) # label(fact_dropWhile__append1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2804 (all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = V_A_2) # label(fact_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2805 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),V_k_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2))))) # label(fact_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2806 (all T_c (class_Orderings_Olinorder(T_c) -> c_Finite__Set_Ofun__left__comm__idem(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)))) # label(fact_min__max_Ofun__left__comm__idem__inf) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2807 (all V_s_2 all V_B_2 all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (c_Relation_Orefl__on(T_c,V_B_2,V_s_2) -> c_Relation_Orefl__on(T_c,hAPP(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(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_refl__on__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2808 (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]. 8.94/8.99 2809 (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]. 8.94/8.99 2810 (all 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))) = c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Code__Numeral_OSuc__code__numeral(V_code__numeral))) # label(fact_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2811 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2812 (all V_b_2 all V_a_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(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_Pa_2,B_d)))) | -hBOOL(hAPP(V_Pa_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))))) # label(fact_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2813 (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]. 8.94/8.99 2814 (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]. 8.94/8.99 2815 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__int_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2816 (all T_e all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Oinj__on(T_c,tc_fun(T_e,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_e,T_d),T_c),c_COMBK(T_d,T_e)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_inj__fun) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2817 (all V_x all V_n all T_a c_List_Oreplicate(T_a,V_n,V_x) = hAPP(c_List_Orev(T_a),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_rev__replicate) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2818 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)))) # label(fact_Inf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2819 (all V_a_2 all V_f_2 all T_d all T_c V_a_2 = hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),c_List_Olist_ONil(T_d))) # label(fact_foldl__Nil) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2820 (all V_k 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_k)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Osucc(V_k)))) # label(fact_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2821 (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]. 8.94/8.99 2822 (all V_z_2 all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_x_2),V_z_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]. 8.94/8.99 2823 (all V_x_2 all V_y_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (-hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> c_Fun_Oinj__on(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_A_2)))) # label(fact_inj__on__fun__updI) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2824 (all V_A_2 all V_B_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_insert__Diff1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2825 (all V_n_2 all V_m_2 all V_Pa_2 ((exists 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)) & 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_Pa_2,B_q)))) | hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_split__div_H) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2826 (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_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))))) <-> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Nat_OSuc,V_n_2))) # label(fact_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2827 (all V_F_2 all V_h_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_h_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))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_F_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_Ovimage(T_c,T_d,V_h_2),V_F_2)))))) # label(fact_finite__vimage__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2828 (all V_s_2 all V_B_2 all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (c_Relation_Orefl__on(T_c,V_B_2,V_s_2) -> c_Relation_Orefl__on(T_c,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(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_refl__on__Int) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2829 (all V_xs_2 all V_f_2 all V_c_2 all T_d all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2)),V_f_2)),V_xs_2)))) # label(fact_listsum__const__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2830 (all V_k all V_v ((V_v = V_k -> c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = c_Random_Oinc__shift(V_v,V_k)) & (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]. 8.94/8.99 2831 (all V_l (c_List_Onat__list(V_l) -> V_l = c_List_Oembed__list(c_List_Oreturn__list(V_l)))) # label(fact_transfer__nat__int__list__return__embed) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2832 (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)) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> V_z = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z))))) # label(fact_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2833 (all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_S_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) -> c_Transitive__Closure_Ortrancl(T_c,V_S_2) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2834 (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))))) -> 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)) # label(fact_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2835 (all V_B_2 all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_B_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_B_2),V_A_2)))) # label(fact_sup__Sup__fold__sup) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2836 (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_Oappend(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_ys) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_i),V_x))) # label(fact_list__update__append1) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2837 (all V_Pa_2 all V_Ga_2 all T_c c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),c_Com_Ocom_OSKIP),V_Pa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))) # label(fact_hoare__derivs_OSkip) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2838 (all V_y_2 all V_z_2 all V_f_2 all T_d all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Nitpick_Ofold__graph_H(T_c,T_d,V_f_2,V_z_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)))),V_y_2) -> c_Nitpick_Ofold__graph_H(T_c,T_d,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]. 8.94/8.99 2839 (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]. 8.94/8.99 2840 (all V_w hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),V_w)) # label(fact_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2841 (all V_y_2 all V_x_2 all V_Qa_2 all V_Pa_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_Pa_2),V_Qa_2)) -> (hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2)) -> hBOOL(hAPP(hAPP(V_Qa_2,V_x_2),V_y_2))))) # label(fact_predicate2D) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2842 (all V_y all V_x 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)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y))) # label(fact_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2843 (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]. 8.94/8.99 2844 (all V_x_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(c_Product__Type_Oapfst(T_e,T_d,T_c,V_f_2),V_x_2)) = hAPP(c_Product__Type_Osnd(T_e,T_c),V_x_2)) # label(fact_snd__apfst) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2845 (all V_A_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_A_2))) <-> V_b_2 = V_a_2 | hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2846 (all V_b_2 all V_a_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_c,T_d,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)) -> hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))))) # label(fact_curryE) # label(axiom) # label(non_clause). [assumption]. 8.94/8.99 2847 (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]. 8.94/9.00 2848 (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]. 8.94/9.00 2849 (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_OBit0(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2850 (all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2851 (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]. 8.94/9.00 2852 (all V_A_2 all V_C_2 all T_c (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),V_C_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)),B_x),V_A_2))))) # label(fact_Union__disjoint) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2853 (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]. 8.94/9.00 2854 (all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) # label(fact_insert__Diff__single) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2855 (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)),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_j),c_List_Olist_ONil(tc_Nat_Onat))) = c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)))) # label(fact_upt__Suc__append) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2856 (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]. 8.94/9.00 2857 (all V_f_2 all V_m_2 all V_n_2 all T_c hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),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_c,T_c),V_n_2),hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_m_2),V_f_2))) # label(fact_funpow__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2858 (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]. 8.94/9.00 2859 (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]. 8.94/9.00 2860 (all V_r_2 all T_c (c_Nitpick_Orefl_H(T_c,V_r_2) <-> (all B_x hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2))))) # label(fact_Nitpick_Orefl_H__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2861 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Groups_Oordered__ab__semigroup__add__imp__le(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_b_2)))))) # label(fact_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2862 (all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_d)) & (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Oplus__class_Oplus(T_d)),V_g_2),c_Groups_Ozero__class_Ozero(T_d)),V_A_2)))) # label(fact_setsum_OF__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2863 (all V_A_2 all T_c -(exists B_f hAPP(c_Set_OPow(T_c),V_A_2) = hAPP(c_Set_Oimage(T_c,tc_fun(T_c,tc_HOL_Obool),B_f),V_A_2))) # label(fact_Cantors__paradox) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2864 (all V_B_2 all V_A_2 all T_c c_List_Olistsp(T_c,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(tc_List_Olist(T_c),tc_HOL_Obool)),c_List_Olistsp(T_c,V_A_2)),c_List_Olistsp(T_c,V_B_2))) # label(fact_listsp__inf__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2865 (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]. 8.94/9.00 2866 (all V_a_2 all V_f_2 all T_c all T_d c_List_Ofoldr(T_d,T_c,V_f_2,c_List_Olist_ONil(T_d),V_a_2) = V_a_2) # label(fact_foldr_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2867 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> (c_Relation_Otrans(T_c,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)),V_r_2)),V_r_2))))) # label(fact_sym__trans__comp__subset) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2868 (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]. 8.94/9.00 2869 (all V_b all V_a 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_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2870 (all V_A_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),c_List_Olists(T_c,V_A_2))))) # label(fact_Cons__in__lists__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2871 (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]. 8.94/9.00 2872 (all V_A_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) | V_a_2 = V_b_2))) # label(fact_trancl__subset__Sigma__aux) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2873 (all V_y all V_q_1 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_q_1)),V_y) = 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__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]. 8.94/9.00 2874 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> V_r_2 = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)),V_r_2))) # label(fact_equiv__comp__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2875 (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]. 8.94/9.00 2876 (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]. 8.94/9.00 2877 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)),V_A_2))))))) # label(fact_min__max_OSup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2878 (all V_A_2 all T_c c_Relation_Osym(T_c,c_Relation_OId__on(T_c,V_A_2))) # label(fact_sym__Id__on) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2879 (all V_x all T_a (class_Rings_Olinordered__idom(T_a) & class_Int_Onumber__ring(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_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_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x))))) # label(fact_abs__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2880 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_x_2)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_y_2)))))) # label(fact_atMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2881 (all V_B_2 all V_A_2 all V_C_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2882 (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]. 8.94/9.00 2883 (all V_xs_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(c_List_Onth(T_c),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_c)),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),V_xs_2))) # label(fact_listsum__setsum__nth) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2884 (all V_ys_2 all T_c all V_xs_2 (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = V_xs_2 <-> c_List_Olist_ONil(T_c) = V_ys_2)) # label(fact_self__append__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2885 (all V_A_2 all V_l_H_2 all V_l_2 all T_c hAPP(hAPP(c_List_Oappend(T_c),c_List_Osublist(T_c,V_l_2,V_A_2)),c_List_Osublist(T_c,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_c)),V_l_2)))),V_A_2)))) = c_List_Osublist(T_c,hAPP(hAPP(c_List_Oappend(T_c),V_l_2),V_l_H_2),V_A_2)) # label(fact_sublist__append) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2886 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThanAtMost(T_c,V_less__eq_2,V_less_2,V_l_2,V_u_2))))) # label(fact_ord_OgreaterThanAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2887 (all V_z3 all V_z2 all V_z1 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) = 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))) # label(fact_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2888 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_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) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2889 (all V_x all T_a all V_xs (V_xs = c_List_Olist_ONil(T_a) -> 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]. 8.94/9.00 2890 (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]. 8.94/9.00 2891 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_b = 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)))) # label(fact_add__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2892 (all V_B_2 all V_A_2 all V_c_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),V_c_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_d),B_y),hAPP(V_B_2,B_x))) -> hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_y) != V_c_2)))))) # label(fact_SigmaE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2893 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_ys_2))) # label(fact_filter__append) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2894 (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]. 8.94/9.00 2895 (all V_n_2 all V_R_2 all T_c (c_Relation_Osingle__valued(T_c,T_c,V_R_2) -> c_Relation_Osingle__valued(T_c,T_c,hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)))) # label(fact_single__valued__rel__pow) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2896 (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]. 8.94/9.00 2897 (all V_x_2 all V_y_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> V_x_2 = V_y_2)))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2898 (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]. 8.94/9.00 2899 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2900 (all V_fun2_2 all V_com_2 all V_fun1_2 all T_c c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_fun1_2),V_com_2),V_fun2_2))) # label(fact_triple_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2901 (all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_not__in__set__insert) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2902 (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]. 8.94/9.00 2903 (all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_List_Olist(tc_List_Olist(T_c)),c_List_Otranspose__rel(T_c)),c_List_Olist_ONil(tc_List_Olist(T_c)))) -> c_List_Olist_ONil(tc_List_Olist(T_c)) = c_List_Otranspose(T_c,c_List_Olist_ONil(tc_List_Olist(T_c))))) # label(fact_transpose_Opsimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2904 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Un) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2905 (all V_a_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_a_2),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_Inf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2906 (all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) <-> (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)))))) # label(fact_equivp__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2907 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_f1_2 all T_c all T_d c_Hoare__Mirabelle_Otriple_Otriple__case(T_d,T_c,V_f1_2,hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_d),V_fun1_2),V_com_2),V_fun2_2)) = hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2)) # label(fact_triple_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2908 (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(T_a),V_x),V_y)) | V_x = V_y))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2909 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),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)))))),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_card__Diff1__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2910 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oordered__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),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]. 8.94/9.00 2911 (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),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(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)))) # label(fact_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2912 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2913 (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_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),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_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]. 8.94/9.00 2914 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(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_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))) # label(fact_drop__append) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2915 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (((exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2916 (all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_bij__is__surj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2917 (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]. 8.94/9.00 2918 (all T_d all V_A_2 all V_c_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_d,T_c,hAPP(c_COMBK(T_c,T_d),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(c_COMBK(T_c,T_d),V_c_2)),V_A_2)))) # label(fact_vimage__const) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2919 (all V_n all V_m 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)) = 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)) # label(fact_zpower__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2920 (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]. 8.94/9.00 2921 (all V_k_2 all V_j_2 all V_i_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)) <-> 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))))) # label(fact_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 8.94/9.00 2922 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) -> c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = c_List_OdropWhile(T_c,V_Pa_2,V_ys_2))) # label(fact_dropWhile__append2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2923 (all V_x_2 all V_f_2 all T_d all T_c c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,hAPP(V_f_2,V_x_2)) = V_f_2) # label(fact_fun__upd__triv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2924 (all V_y_2 all V_x_2 all V_R_2 all T_d all T_c (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)),V_R_2)) <-> hBOOL(hAPP(hAPP(c_FunDef_Oin__rel(T_c,T_d,V_R_2),V_x_2),V_y_2)))) # label(fact_in__rel__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2925 (all V_a2_2 all V_a1_2 all V_R_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_a2_2)) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)))))) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_a2_2 & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a2_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_2)) <-> c_Wellfounded_Omax__extp(T_c,V_R_2,V_a1_2,V_a2_2))) # label(fact_max__extp_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2926 (all V_A_2 all V_B_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_A_2))) # label(fact_Union__image__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2927 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)),V_B_2)) # label(fact_INT__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2928 (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]. 9.04/9.00 2929 (all V_f_2 all V_a_2 all V_F_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_F_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_F_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_F_2)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2930 (all V_r_2 all V_zb_2 all V_za_2 all V_xb_2 all V_xa_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_xa_2),V_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2))) -> (hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_za_2),V_zb_2) != hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_xa_2),V_xb_2) -> -(all B_a all B_b (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_xa_2),V_xb_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b))),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_za_2),V_zb_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2)))))))) # label(fact_converse__rtranclE2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2931 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OPow(T_c),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(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)),hAPP(c_Set_OPow(T_c),V_B_2))) # label(fact_Pow__Int__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2932 (all 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) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n))) # label(fact_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2933 (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]. 9.04/9.00 2934 (all V_w_2 all T_c (class_Int_Onumber__ring(T_c) & class_Int_Oring__char__0(T_c) -> (c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),c_Int_OBit0(V_w_2)))))) # label(fact_iszero__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2935 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2))))) # label(fact_Compl__subset__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2936 (all V_b_2 all V_a_2 (c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,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)) <-> -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2)))) # label(fact_dvd_OatLeastatMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2937 (all V_k_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = 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,T_c),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_c),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]. 9.04/9.00 2938 (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]. 9.04/9.00 2939 (all V_n all V_b all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> 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)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n))) # label(fact_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2940 (all V_r_2 all V_g_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Set_Oimage(T_e,T_c,V_f_2),hAPP(c_Set_Oimage(T_d,T_e,V_g_2),V_r_2)) = hAPP(c_Set_Oimage(T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_c,T_d,V_f_2),V_g_2)),V_r_2)) # label(fact_image__compose) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2941 (all V_xss_2 all T_c c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_c)),T_c,c_List_Olist_ONil(tc_List_Olist(T_c)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_OCons(tc_List_Olist(T_c))),c_List_Olist_ONil(tc_List_Olist(T_c)))))),V_xss_2)) = hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Otl(T_c)),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))) # label(fact_transpose__aux__filter__tail) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2942 (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(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(c_Int_Onumber__class_Onumber__of(T_b),V_u) = 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))))) # label(fact_max__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2943 (all V_B_2 all V_A_2 all V_f_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_c,tc_HOL_Obool),T_e),V_B_2),V_f_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_e,T_d,V_f_2),V_A_2)),V_B_2)) # label(fact_INT__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2944 (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_x),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]. 9.04/9.00 2945 (all V_f_2 all V_a_2 all V_A_2 all T_c all T_d (class_Rings_Oring(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_setsum__diff1__ring) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2946 (all V_x_2 all V_Pa_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_d),B_y),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) -> hBOOL(hAPP(V_Pa_2,hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),B_y))))) -> hBOOL(hAPP(V_Pa_2,V_x_2))))) # label(fact_inj__transfer) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2947 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_f_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_B_2),V_f_2)))))) # label(fact_SUP__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2948 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2))))))) # label(fact_card__inj__on__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2949 (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]. 9.04/9.00 2950 (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]. 9.04/9.00 2951 (all V_x_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> hAPP(hAPP(V_f_2,V_x_2),V_x_2) = V_x_2)) # label(fact_folding__image__simple__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2952 (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]. 9.04/9.00 2953 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Olist__all(tc_Nat_Onat,V_Pa_2,c_List_Oupt(V_i_2,V_j_2)) <-> c_List_Oall__interval__nat(V_Pa_2,V_i_2,V_j_2))) # label(fact_list__all__iff__all__interval__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2954 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2955 (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]. 9.04/9.00 2956 (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_r = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2957 (all V_l_2 all V_xs_2 all T_c all V_x_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_l_2))))) # label(fact_takeWhile__tail) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2958 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_div__by__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2959 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Opred(V_w)) = 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)))) # label(fact_number__of__pred) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2960 (all V_b_2 all V_a_2 all V_f_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Ocurry(T_d,T_e,T_c,V_f_2),V_a_2),V_b_2) = hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2))) # label(fact_curry__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2961 (all V_a_2 all V_B_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)))) # label(fact_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2962 (all V_h_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Groups_Otimes__class_Otimes(T_d)),V_g_2)),V_h_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,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_A_2))))) # label(fact_setprod_Odistrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2963 (all V_com_H_2 all V_fun_H_2 all V_com_2 all V_fun_2 all V_loc_2 c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I38_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2964 (all V_Pa_2 all V_r_2 all V_by_2 all V_bx_2 all V_ay_2 all V_ax_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_ax_2),V_ay_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2))) -> (hBOOL(hAPP(hAPP(V_Pa_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_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_aa),B_ba))),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_aa),B_ba)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_bx_2),V_by_2))),c_Transitive__Closure_Ortrancl(tc_prod(T_c,T_d),V_r_2))) -> (hBOOL(hAPP(hAPP(V_Pa_2,B_aa),B_ba)) -> hBOOL(hAPP(hAPP(V_Pa_2,B_a),B_b)))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_ax_2),V_ay_2)))))) # label(fact_converse__rtrancl__induct2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2965 (all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2966 (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]. 9.04/9.00 2967 (all V_xs_2 all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_Inf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2968 (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]. 9.04/9.00 2969 (all V_Pa_2 hAPP(c_Set_OCollect(tc_Nat_Onat),V_Pa_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_Pa_2),c_Int_Onat))))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2970 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (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_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_rtrancl__into__trancl1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2971 (all V_r_2 all V_ys_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_ys_2)),c_List_Olex(T_c,V_r_2)))) # label(fact_Nil__notin__lex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2972 (all V_M_2 all T_c ((exists B_x (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_x)) & hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),V_M_2)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_M_2))))) # label(fact_finite__Inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2973 (all V_R_2 all T_c 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_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Wellfounded_Omlex__prod(T_c,V_f_2,V_R_2))))) # label(fact_mlex__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2974 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)) = hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_x_2,V_y_2)))) # label(fact_image__uminus__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2975 (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]. 9.04/9.00 2976 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) <-> (all B_Q ((exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_Q))) -> (exists B_x ((all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_x)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(T_c),B_y),B_Q)))) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_Q)))))))) # label(fact_wf__eq__minimal) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2977 (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]. 9.04/9.00 2978 (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]. 9.04/9.00 2979 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_B_2)))))) # label(fact_setprod_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2980 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2)) # label(fact_length__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2981 (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]. 9.04/9.00 2982 (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__symmetric) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2983 (all V_xs_2 all T_c c_List_Olist_ONil(T_c) = c_List_Osublist(T_c,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]. 9.04/9.00 2984 (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]. 9.04/9.00 2985 (all V_n 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))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2986 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls)))) # label(fact_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2987 (all V_v_2 all V_c_2 all V_b_2 all V_a_2 all T_c all T_d all T_e (hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),V_c_2) = hAPP(c_Fun_Ocomp(T_e,T_d,T_c,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__id__dest) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2988 (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]. 9.04/9.00 2989 (all V_x all V_xs all T_a 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)))) = V_x) # label(fact_last__snoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2990 (all V_I_2 all V_xs_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Osublist(T_c,V_xs_2,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_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),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_length__sublist) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2991 (all V_x_2 all V_f_2 all V_A_2 all T_d all T_c c_HOL_OThe(T_c,hAPP(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(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_x_2))) = hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),V_x_2)) # label(fact_the__inv__into__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2992 (all T_d all T_c 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_c,T_d,V_r_2),V_b_2)))) # label(fact_RangeP_Ointros) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2993 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),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))) & (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),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = 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]. 9.04/9.00 2994 (all T_c c_Equiv__Relations_Oequivp(T_c,c_fequal)) # label(fact_identity__equivp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2995 (all V_A_2 all V_B_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_A_2))) # label(fact_Inter__image__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2996 (all V_f2_2 all V_f1_2 all T_c V_f1_2 = c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_c,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]. 9.04/9.00 2997 (all V_b_2 all 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)) <-> -(-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__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.00 2998 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2))) # label(fact_Diff__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 2999 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_x_2)))) # label(fact_in__set__member) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3000 (all V_c_2 all V_b_2 all V_Pa_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,V_b_2)),V_c_2),V_Pa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),c_Com_Ocom_OWhile(V_b_2,V_c_2)),c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Com_Ostate,c_fNot),V_b_2)))),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))))) # label(fact_Loop) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3001 (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_a = V_y)) # label(fact_snd__eqD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3002 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),c_List_Olists(T_c,V_A_2)),c_List_Olists(T_c,V_B_2))))) # label(fact_lists__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3003 (all V_xs_2 all V_y_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> hAPP(c_List_Ofoldl(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_d),V_f_2),V_y_2),V_xs_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_y_2),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_fun__left__comm__idem_Ofold__set) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3004 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)))) # label(fact_Diff__subset__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3005 (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]. 9.04/9.01 3006 (all V_u_2 V_u_2 = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_u_2))) # label(fact_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3007 (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]. 9.04/9.01 3008 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_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,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__one_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3009 (all V_n all V_m (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_m -> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3010 (all V_x all T_a all V_xs ((V_xs != c_List_Olist_ONil(T_a) -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Obutlast(T_a,V_xs))) & (c_List_Olist_ONil(T_a) = V_xs -> c_List_Olist_ONil(T_a) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_butlast_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3011 (all V_s_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> c_List_Osublist(T_c,V_xs_2,V_s_2) = hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_s_2)))),V_xs_2))) # label(fact_filter__in__sublist) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3012 (all 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)))) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_atLeast__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3013 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_inj__on__imp__bij__betw) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3014 (all V_xs_2 all V_n_2 all T_c (c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),V_n_2)))) # label(fact_drop__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3015 (all V_ys_2 all V_g_2 all T_f all V_xs_2 all V_f_2 all T_c all T_e all T_d hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_e),T_c),V_f_2),hAPP(hAPP(c_List_Ozip(T_d,T_e),V_xs_2),hAPP(hAPP(c_List_Omap(T_f,T_e),V_g_2),V_ys_2))) = hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_f),T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_f,T_c),hAPP(hAPP(c_COMBB(tc_fun(T_f,tc_prod(T_d,T_e)),tc_fun(T_f,T_c),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),T_c,T_f),V_f_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_f,T_e),tc_fun(T_f,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_f,T_e),tc_fun(T_f,tc_prod(T_d,T_e))),T_d),c_COMBB(T_e,tc_prod(T_d,T_e),T_f)),c_Product__Type_OPair(T_d,T_e))),V_g_2)))),hAPP(hAPP(c_List_Ozip(T_d,T_f),V_xs_2),V_ys_2))) # label(fact_map__zip__map2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3016 (all V_A_2 all T_c V_A_2 = hAPP(c_Relation_OImage(T_c,T_c,c_Relation_OId(T_c)),V_A_2)) # label(fact_Image__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3017 (all V_A_2 all V_f_2 all T_e all V_A_H_H_2 all V_A_H_2 all V_f_H_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,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_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_e,T_c,V_f_2),V_A_2)),V_A_H_2)) -> (c_Fun_Obij__betw(T_e,T_d,hAPP(c_Fun_Ocomp(T_c,T_d,T_e,V_f_H_2),V_f_2),V_A_2,V_A_H_H_2) <-> c_Fun_Obij__betw(T_e,T_c,V_f_2,V_A_2,V_A_H_2))))) # label(fact_bij__betw__comp__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3018 (all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_x_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_fold__graph__imp__finite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3019 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),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_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_inf__apply) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3020 (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]. 9.04/9.01 3021 (all V_a_2 all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(hAPP(V_f_2,V_x_2),c_List_Ofoldr(T_d,T_c,V_f_2,V_xs_2,V_a_2)) = c_List_Ofoldr(T_d,T_c,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2),V_a_2)) # label(fact_foldr_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3022 (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]. 9.04/9.01 3023 (all V_B_2 all V_A_2 all V_I_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_I_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(V_A_2,B_x)),V_B_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_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),V_B_2)))) # label(fact_UN__subset__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3024 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oordered__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),V_e_2)),V_c_2)),V_d_2))))) # label(fact_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3025 (all V_A_2 all V_a_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) # label(fact_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3026 (all V_Pa_2 all V_ys_2 all T_d all V_xs_2 all T_c ((hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),hAPP(hAPP(c_List_Otake(T_d),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),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))),V_ys_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_c)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2))),V_xs_2)) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Ozip(T_c,T_d),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_c)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2))),V_xs_2)),hAPP(hAPP(c_List_Otake(T_d),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),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))),V_ys_2))))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2))))) # label(fact_zip__obtain__same__length) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3027 (all V_ys_2 all V_xs_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,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_xs_2 = V_ys_2 <-> hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)))) # label(fact_inj__on__map__eq__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3028 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,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_Inf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3029 (all V_f_2 all T_d all V_xs_2 all T_c 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_c)),V_xs_2))) -> hAPP(hAPP(c_List_Onth(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)),V_n_2) = hAPP(V_f_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_n_2)))) # label(fact_nth__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3030 (all V_x_2 all V_A_2 all T_d all T_c (class_Groups_Ouminus(T_c) -> hAPP(hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,T_c)),V_A_2),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(V_A_2,V_x_2)))) # label(fact_uminus__apply) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3031 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),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(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_B_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)))))))) # label(fact_fold1__antimono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3032 (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]. 9.04/9.01 3033 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),c_Groups_Osgn__class_Osgn(T_c,V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_sgn__greater) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3034 (all V_r_2 all T_c c_List_Olistrelp(T_c,V_r_2,c_List_Olist_ONil(T_c),c_List_Olist_ONil(T_c))) # label(fact_listrelp_ONil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3035 (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_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]. 9.04/9.01 3036 (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_Int_Onumber__class_Onumber__of(T_a),V_v) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),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))) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_of__nat__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3037 (all V_C_2 all V_s_2 all T_e all V_B_2 all V_A_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_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)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool)),V_s_2),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_e),V_B_2),hAPP(c_COMBK(tc_fun(T_e,tc_HOL_Obool),T_d),V_C_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_e),V_r_2),V_s_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_C_2))))))) # label(fact_rel__comp__subset__Sigma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3038 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_m_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I18_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3039 (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]. 9.04/9.01 3040 (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]. 9.04/9.01 3041 (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]. 9.04/9.01 3042 (all V_s_2 all V_r_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),c_Predicate_Oconversep(T_d,T_c,V_r_2)),c_Predicate_Oconversep(T_d,T_c,V_s_2)) = c_Predicate_Oconversep(T_d,T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool))),V_r_2),V_s_2))) # label(fact_converse__join) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3043 (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]. 9.04/9.01 3044 (all V_f_2 all V_xs_2 all V_a_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) -> (V_a_2 = hAPP(c_List_Ohd(T_c),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(T_d,tc_HOL_Obool,T_c),hAPP(c_fequal,hAPP(V_f_2,V_a_2))),V_f_2)),V_xs_2)) -> V_xs_2 = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_a_2),c_List_Oremove1(T_c,V_a_2,V_xs_2))))))) # label(fact_insort__key__remove1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3045 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),B_y)))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> V_x_2 = c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)))))) # label(fact_Min__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3046 (all V_ys_2 all V_xs_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (V_xs_2 = V_ys_2 <-> hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)))) # label(fact_inj__map__eq__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3047 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_psubset__trans) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3048 (all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_Rep_2,V_x_2)),V_A_2)))) # label(fact_type__definition_ORep) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3049 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2))))) # label(fact_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3050 (all V_xs all T_a all V_y all V_x ((V_x != V_y -> 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),V_xs) = hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs))))) # label(fact_removeAll_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3051 (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]. 9.04/9.01 3052 (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]. 9.04/9.01 3053 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_f all T_e all T_d hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,T_c),V_g_2),hAPP(V_f_2,V_x_2)) = hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_d,T_e,T_f,T_c),V_f_2),V_g_2),V_x_2)) # label(fact_scomp__apply) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3054 (all V_Pa_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_Pa_2)))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3055 (all V_y all V_x all V_i all V_xs all T_a 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) = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_y)) # label(fact_list__update__overwrite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3056 (all V_x all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = V_x)) # label(fact_mult__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3057 (all V_n_2 all V_R_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) # label(fact_rel__pow__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3058 (all V_n_2 all V_v_2 all V_f_2 all V_a_2 all T_c hAPP(c_Nat_Onat_Onat__rec(T_c,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_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,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)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_Onat_Onat__rec(T_c,V_a_2,V_f_2),V_n_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(T_c,T_c),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_c,tc_Int_Oint),c_Nat_Onat_Onat__rec(T_c,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]. 9.04/9.01 3059 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_d all T_c (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(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_List_Oset(T_d),V_ys_2))))) # label(fact_set__zip__rightD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3060 (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]. 9.04/9.01 3061 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_subset__trans) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3062 (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]. 9.04/9.01 3063 (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]. 9.04/9.01 3064 (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(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))) # label(fact_minus__mult__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3065 (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]. 9.04/9.01 3066 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,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)))))),hAPP(V_f_2,V_x_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding_Oremove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3067 (all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,V_R_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)) <-> hBOOL(hAPP(hAPP(V_R_2,B_x),B_x)) & hAPP(V_R_2,B_y) = hAPP(V_R_2,B_x) & hBOOL(hAPP(hAPP(V_R_2,B_y),B_y)))) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))))) # label(fact_part__equivp__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3068 (all V_m_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m_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)))) # label(fact_dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3069 (all T_c (class_Rings_Oring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Groups_Ozero__class_Ozero(T_c)),c_Int_Oring__1__class_OInts(T_c))))) # label(fact_Ints__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3070 (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]. 9.04/9.01 3071 (all V_Pa_2 all T_d all T_c c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),hAPP(hAPP(c_COMBS(tc_prod(T_c,T_d),T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_HOL_Obool),tc_prod(T_c,T_d)),V_Pa_2),c_Product__Type_Ofst(T_c,T_d))),c_Product__Type_Osnd(T_c,T_d))) = c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2))) # label(fact_Eps__split) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3072 (all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osort__key(T_c,T_c,c_COMBI(T_c),c_List_Oremdups(T_c,V_xs_2)) = c_List_Olinorder__class_Osorted__list__of__set(T_c,hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_sorted__list__of__set__sort__remdups) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3073 (all V_A_2 all V_y_2 all V_x_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3074 (all V_A_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3075 (all V_R_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),V_R_2) = hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_c,c_Product__Type_Osnd(T_d,T_c)),V_R_2)) # label(fact_snd__eq__Range) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3076 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card__psubset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3077 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_A_2))) # label(fact_Int__lower1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3078 (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_List_Onth(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k))) # label(fact_nth__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3079 (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]. 9.04/9.01 3080 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))) # label(fact_rtrancl__converseD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3081 (all V_i_2 all V_inc_2 all T_c (class_Rings_Osemiring__1(T_c) -> c_Nat_Osemiring__1__class_Oof__nat__aux(T_c,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]. 9.04/9.01 3082 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))) & (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist_ONil(T_c) = c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_takeWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3083 (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),V_xs),V_n) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),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)))))) # label(fact_nth__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3084 (all V_xs_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> V_xs_2 = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_a_2),c_List_Oremove1(T_c,V_a_2,V_xs_2)))))) # label(fact_insort__remove1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3085 (all V_ys_2 all V_xs_2 all T_c (hAPP(c_List_Orev(T_c),V_ys_2) = V_xs_2 <-> V_ys_2 = hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__swap) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3086 (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]. 9.04/9.01 3087 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),B_x),V_x_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)),V_x_2))))))) # label(fact_fold1__strict__below__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3088 (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),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3089 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_d_2)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_d_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2)) <-> V_c_2 != V_d_2 & V_b_2 != V_a_2))) # label(fact_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3090 (all V_xs_2 all T_c hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),hAPP(c_List_Onth(T_c),V_xs_2)),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) = V_xs_2) # label(fact_map__nth) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3091 (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]. 9.04/9.01 3092 (all T_d all V_B_2 all T_c 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_c),V_b_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_d),V_a_2),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_B_2)))))) # label(fact_vimageI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3093 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_d,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]. 9.04/9.01 3094 (all V_f_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_f_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__sup__class_Osup(T_d)),V_f_2)),c_Orderings_Obot__class_Obot(T_d)),V_A_2)))) # label(fact_SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3095 (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]. 9.04/9.01 3096 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Transitive__Closure_Ortrancl(T_c,V_s_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),c_Transitive__Closure_Ortrancl(T_c,V_s_2))))) # label(fact_rtrancl__subset__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3097 (all T_1 all T_2 (class_Enum_Oenum(T_1) & class_Enum_Oenum(T_2) -> class_Enum_Oenum(tc_fun(T_2,T_1)))) # label(arity_fun__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3098 (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)) -> (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__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3099 (all V_h_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Groups_Otimes__class_Otimes(T_d)),V_g_2)),V_h_2)),c_Groups_Oone__class_Oone(T_d)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),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)),V_A_2)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_h_2),c_Groups_Oone__class_Oone(T_d)),V_A_2))))) # label(fact_fold__image__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3100 (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 -> V_t_2 != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2))))))) # label(fact_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3101 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2) = c_Groups_Oone__class_Oone(T_d)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2) = hAPP(V_b_2,V_a_2))))) # label(fact_setprod__delta) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3102 (all V_n_2 all V_d_2 all V_a_2 all T_c (class_Rings_Ocomm__semiring__1(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),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_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),c_Groups_Oone__class_Oone(T_c)),c_Groups_Oone__class_Oone(T_c))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,tc_Nat_Onat),hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),tc_Nat_Onat),c_Groups_Otimes__class_Otimes(T_c)),c_Nat_Osemiring__1__class_Oof__nat(T_c))),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]. 9.04/9.01 3103 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3104 (all T_d all V_h_2 all V_f_2 all V_S_2 all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),V_S_2)) -> (B_y != B_x -> (hAPP(V_f_2,B_x) = hAPP(V_f_2,B_y) -> hAPP(V_h_2,hAPP(V_f_2,B_x)) = c_Groups_Ozero__class_Ozero(T_e))))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_h_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_S_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2)),V_S_2))))) # label(fact_setsum__reindex__nonzero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3105 (all V_y_2 all V_x_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_x_2),V_y_2)) <-> 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)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)))))) # label(fact_bot__empty__eq2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3106 (all V_B_2 all V_a_2 all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2))) # label(fact_vimage__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3107 (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]. 9.04/9.01 3108 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))))) # label(fact_UnE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3109 (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_Ozero__class_Ozero(tc_Int_Oint) = V_r))) # label(fact_self__remainder) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3110 (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]. 9.04/9.01 3111 (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]. 9.04/9.01 3112 (all V_r_2 all V_ys_2 all V_xs_2 all T_c ((exists B_y exists 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_c)),V_xs_2))) & hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),B_n),B_y) = V_ys_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n)),B_y)),V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel1__iff__update) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3113 (all V_R_2 all T_d all T_e all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool))),V_R_2)) # label(fact_rel__comp__empty1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3114 (all V_x_2 all V_f_2 all V_c_2 all T_e all T_c all T_d hAPP(hAPP(c_Fun_Ocomp(T_d,T_c,T_e,hAPP(c_COMBK(T_c,T_d),V_c_2)),V_f_2),V_x_2) = V_c_2) # label(fact_K__record__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3115 (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_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3116 (all T_c hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_prod(T_c,T_c)),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(tc_prod(T_c,T_c),tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_HOL_Obool)),tc_prod(T_c,T_c)),c_COMBB(tc_prod(T_c,T_c),tc_HOL_Obool,T_c)),c_fequal)),hAPP(hAPP(c_COMBS(T_c,T_c,tc_prod(T_c,T_c)),c_Product__Type_OPair(T_c,T_c)),c_COMBI(T_c))))) = c_Relation_OId(T_c)) # label(fact_Id__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3117 (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]. 9.04/9.01 3118 (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]. 9.04/9.01 3119 (all V_r_H_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_r_H_2)) -> (c_Relation_Orefl__on(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_r_H_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel1(T_c,V_r_2)),c_List_Olistrel(T_c,V_r_H_2)))))) # label(fact_listrel1__subset__listrel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3120 (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),V_y) = 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)))) # label(fact_inf__sup__aci_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3121 (all T_c all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> hBOOL(hAPP(V_Pa_2,c_Hilbert__Choice_OEps(T_c,V_Pa_2))))) # label(fact_some__eq__ex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3122 (all V_A_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Nitpick_Ocard_H(T_c,V_A_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_Hilbert__Choice_OEps(tc_List_Olist(T_c),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),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_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_List_Oset(T_c))),V_A_2))),c_List_Odistinct(T_c))))) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Nitpick_Ocard_H(T_c,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_Nitpick_Ocard_H__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3123 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hBOOL(hAPP(c_List_Odistinct(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool)),c_List_Oset(T_c)),c_Enum_Osublists(T_c,V_xs_2)))))) # label(fact_distinct__set__sublists) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3124 (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_e all T_d all T_c (c_Equiv__Relations_Ocongruent2(T_c,T_d,T_e,V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y1_2),V_z1_2)),V_r1_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),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]. 9.04/9.01 3125 (all T_d all T_c c_Product__Type_Omap__pair(T_c,T_c,T_d,T_d,c_Fun_Oid(T_c),c_Fun_Oid(T_d)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_map__pair_Oid) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3126 (all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_d)))) # label(fact_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3127 (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_b = V_c -> 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]. 9.04/9.01 3128 (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),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))) <-> 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__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3129 (all V_xs_2 all T_d all V_x_2 all V_f_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_List_Oset(T_d),V_xs_2)))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_xs_2) = c_List_Olinorder__class_Oinsort__insert__key(T_d,T_c,V_f_2,V_x_2,V_xs_2)))) # label(fact_insort__insert__insort__key) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3130 (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]. 9.04/9.01 3131 (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(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)) -> 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 (hAPP(V_f_2,B_i) = V_k_2 & 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]. 9.04/9.01 3132 (all V_com2_H_2 all V_com1_H_2 all V_com_2 all V_fun_2 all V_loc_2 c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2)) # label(fact_com_Osimps_I34_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3133 (all V_xs_2 all T_d all V_x_2 all V_f_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_f_2,V_x_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_List_Oset(T_d),V_xs_2)))) -> c_List_Olinorder__class_Oinsort__insert__key(T_d,T_c,V_f_2,V_x_2,V_xs_2) = V_xs_2))) # label(fact_insort__insert__key__triv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3134 (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]. 9.04/9.01 3135 (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]. 9.04/9.01 3136 (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]. 9.04/9.01 3137 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) & -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2)))))) # label(fact_distinct__insort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3138 (all V_B_2 all V_C_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3139 (all V_g_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Set_Oimage(T_e,T_c,V_f_2),hAPP(c_Set_Oimage(T_d,T_e,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_d,T_c,hAPP(hAPP(c_COMBB(T_e,T_c,T_d),V_f_2),V_g_2)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_range__composition) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3140 (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]. 9.04/9.01 3141 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Oone__class_Oone(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2))))) # label(fact_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3142 (all V_n_2 all V_m_2 (c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3143 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_x_2),c_List_Olists(T_c,V_A_2))) <-> hBOOL(hAPP(c_List_Olistsp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)),V_x_2)))) # label(fact_listsp__lists__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3144 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2)))) # label(fact_ivl__disj__int_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3145 (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),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3146 (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]. 9.04/9.01 3147 (all V_A_2 all V_g_2 all T_e all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Oimage(T_e,T_d,V_g_2),V_A_2)) = hAPP(c_Set_Oimage(T_e,T_c,hAPP(hAPP(c_COMBB(T_d,T_c,T_e),V_f_2),V_g_2)),V_A_2)) # label(fact_image__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3148 (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]. 9.04/9.01 3149 (all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> V_xs = c_List_Oremdups(T_a,V_xs))) # label(fact_distinct__remdups__id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.01 3150 (all V_R_2 all T_c (c_Predicate_Osymp(T_c,V_R_2) & c_Predicate_Otransp(T_c,V_R_2) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) <-> c_Equiv__Relations_Opart__equivp(T_c,V_R_2))) # label(fact_part__equivp__refl__symp__transp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3151 (all V_xs_2 all V_x_2 all V_a_2 all V_f_2 all T_d all T_c hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_d,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]. 9.04/9.02 3152 (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_OBit0(V_k_2))))) # label(fact_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3153 (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]. 9.04/9.02 3154 (all V_m_2 all V_x_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_x_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)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2))) # label(fact_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3155 (all V_z_2 (V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> 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]. 9.04/9.02 3156 (all V_A_2 all V_f_2 all V_f_H_2 all T_e all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_H_2),V_f_2),V_A_2) -> c_Fun_Oinj__on(T_c,T_e,V_f_2,V_A_2))) # label(fact_inj__on__imageI2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3157 (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]. 9.04/9.02 3158 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2))))) # label(fact_atLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3159 (all V_f_2 all V_A_2 all V_B_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) & class_Rings_Odvd(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_B_2))))))) # label(fact_setprod__dvd__setprod__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3160 (all V_s_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_r_2),V_s_2)) -> (c_Relation_Osingle__valued(T_c,T_d,V_s_2) -> c_Relation_Osingle__valued(T_c,T_d,V_r_2)))) # label(fact_single__valued__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3161 (all V_B_2 all T_c all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_less__by__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3162 (all T_d all T_c all V_q_2 all V_Qa_2 all V_p_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,V_p_2)) -> (hBOOL(hAPP(V_Qa_2,V_q_2)) -> hBOOL(hAPP(V_Qa_2,hAPP(c_Product__Type_Osnd(T_c,T_d),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_p_2),V_q_2)))) & hBOOL(hAPP(V_Pa_2,hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_p_2),V_q_2))))))) # label(fact_conjI__realizer) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3163 (all V_zs_2 all V_y_2 all V_x_2 all T_c (class_Groups_Osemigroup__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_y_2),V_zs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),V_y_2)),V_zs_2))) # label(fact_foldl__assoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3164 (all V_R_2 all V_Qa_2 all V_Pa_2 all T_e all T_c all T_d hAPP(V_Pa_2,hAPP(V_Qa_2,V_R_2)) = hAPP(hAPP(hAPP(c_COMBB(T_d,T_c,T_e),V_Pa_2),V_Qa_2),V_R_2)) # label(help_c__COMBB__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3165 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_a_2),V_A_2) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_Max_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3166 (all V_y_2 all V_x_2 all V_f_2 all V_A_2 all T_d all T_c (hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_y_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> V_x_2 = V_y_2)))) # label(fact_inv__into__injective) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3167 (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]. 9.04/9.02 3168 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d c_New__Random__Sequence_Opos__decr__bind(T_d,T_c,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__decr__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,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_c))),hAPP(hAPP(c_COMBC(T_d,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_c)))),hAPP(hAPP(c_COMBC(T_d,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_c))))),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]. 9.04/9.02 3169 (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]. 9.04/9.02 3170 (all V_R2_2 all T_d all V_R1_2 all T_c (c_Relation_Otrans(T_c,V_R1_2) -> (c_Relation_Otrans(T_d,V_R2_2) -> c_Relation_Otrans(tc_prod(T_c,T_d),c_Wellfounded_Olex__prod(T_c,T_d,V_R1_2,V_R2_2))))) # label(fact_trans__lex__prod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3171 (all V_n_2 all V_d_2 all V_a_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)))) = 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)))) # label(fact_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3172 (all V_B_2 all V_A_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(V_B_2,B_x))) <-> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_UNION__empty__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3173 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> 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]. 9.04/9.02 3174 (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]. 9.04/9.02 3175 (all T_c all T_d all V_x_2 all V_y_2 all V_Pa_2 (hBOOL(hAPP(hAPP(V_Pa_2,V_y_2),V_x_2)) -> hBOOL(hAPP(hAPP(V_Pa_2,hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_x_2),V_y_2))),hAPP(c_Product__Type_Ofst(T_d,T_c),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_x_2),V_y_2)))))) # label(fact_exI__realizer) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3176 (all V_A_H_2 all T_d all T_c all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((exists B_g V_A_2 = hAPP(c_Set_Oimage(T_d,T_c,B_g),V_A_H_2)) <-> (exists B_f (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,B_f),V_A_2)),V_A_H_2)) & c_Fun_Oinj__on(T_c,T_d,B_f,V_A_2)))))) # label(fact_inj__on__iff__surj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3177 (all V_a_2 all V_b_2 all V_r_2 all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),V_b_2)) -> (hBOOL(hAPP(hAPP(V_r_2,V_a_2),V_b_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),V_a_2))))) # label(fact_accp__downward) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3178 (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_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)))) # label(fact_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3179 (all V_d_2 all V_b_2 all V_m_2 all T_d all T_c all V_c_2 all V_a_2 (V_a_2 != V_c_2 -> c_Fun_Ofun__upd(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_m_2,V_c_2,V_d_2),V_a_2,V_b_2) = c_Fun_Ofun__upd(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,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]. 9.04/9.02 3180 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_y),V_x_2)))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)))))) # label(fact_Max__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3181 (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]. 9.04/9.02 3182 (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]. 9.04/9.02 3183 (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]. 9.04/9.02 3184 (all T_c all V_R_2 all V_y_2 all V_y_H_2 all T_d all V_x_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,V_x_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),V_y_H_2),V_y_2)),hAPP(V_R_2,V_x_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_H_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2))),c_Recdef_Osame__fst(T_c,T_d,V_Pa_2,V_R_2)))))) # label(fact_same__fstI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3185 (all V_ys_2 all V_xs_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_ys_2)) & hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_xs_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))) # label(fact_append__in__listsp__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3186 (all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) <-> hBOOL(hAPP(hAPP(V_Pa_2,V_x_2),V_y_2)) & c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2))) # label(fact_list__all2__Cons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3187 (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]. 9.04/9.02 3188 (all V_x_2 all T_c all V_n_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_x_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 -> hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_x_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_set__replicate__conv__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3189 (all V_x_2 (-hBOOL(hAPP(c_Nat__Numeral_Oneg,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)))) # label(fact_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3190 (all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> V_x_2 = hAPP(c_Fun_Othe__inv__into(T_c,T_d,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]. 9.04/9.02 3191 (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]. 9.04/9.02 3192 (all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) | V_j_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> c_List_Oupt(V_i_2,V_j_2) = c_List_Olist_ONil(tc_Nat_Onat))) # label(fact_upt__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3193 (all V_y_2 all V_x_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(c_fequal,V_x_2)))),hAPP(c_fequal,V_y_2)))) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2)) # label(fact_The__split__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3194 (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_H),V_x_H)),V_i),V_x) = 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))) # label(fact_list__update__swap) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3195 (all V_com_H_2 all V_fun_H_2 all V_com2_2 all V_com1_2 c_Com_Ocom_OSemi(V_com1_2,V_com2_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I46_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3196 (all V_S_2 all V_R_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),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_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_R_2),V_S_2)))) # label(fact_pred__subset__eq2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3197 (all V_f_2 all V_S_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (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_c,tc_Nat_Onat),V_f_2),V_S_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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)))))))) # label(fact_setprod__pos__nat__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3198 (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]. 9.04/9.02 3199 (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 3200 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> 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]. 9.04/9.02 3201 (all V_Pa_2 all T_c ((all B_x -hBOOL(hAPP(V_Pa_2,B_x))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),V_Pa_2))) # label(fact_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3202 (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]. 9.04/9.02 3203 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> V_x = 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)))) # label(fact_inf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3204 (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]. 9.04/9.02 3205 (all V_u_2 all V_M_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_M_2,B_x)),V_u_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2),V_M_2)),V_u_2))))) # label(fact_SUP__le__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3206 (all T_d all V_f_2 all V_b_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (V_b_2 = hAPP(V_f_2,V_x_2) -> hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))))) # label(fact_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3207 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> (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_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_trancl__trans) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3208 (all V_xs_2 all V_x_2 all V_n_2 all T_c c_Nat_Onat_Onat__case(tc_List_Olist(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Odrop(T_c)),V_xs_2),V_n_2) = hAPP(hAPP(c_List_Odrop(T_c),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_drop__Cons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3209 (all V_K_2 all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Osemiring__1(T_d) & class_Groups_Oordered__ab__semigroup__add(T_d) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,B_i)),V_K_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_d),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))),V_K_2)))))) # label(fact_setsum__bounded) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3210 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Oone__class_Oone(T_a) != c_Groups_Ozero__class_Ozero(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3211 (all V_ts_2 all V_t_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),V_t_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),V_t_2),V_ts_2))))) # label(fact_hoare__derivs_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3212 (all V_l_2 all V_k_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3213 (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]. 9.04/9.02 3214 (all V_m 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)))) = 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)))))) # label(fact_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3215 (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(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__minus) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3216 (all V_g_2 all V_A_2 all V_F_2 all V_z_2 all V_f_2 all T_d all T_c (c_Big__Operators_Ocomm__monoid__big(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2),V_z_2),V_A_2) = hAPP(hAPP(V_F_2,V_g_2),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> V_z_2 = hAPP(hAPP(V_F_2,V_g_2),V_A_2)))) # label(fact_comm__monoid__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3217 (all V_n all V_m 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) # label(fact_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3218 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Set_Ovimage(T_c,T_d,V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3219 (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]. 9.04/9.02 3220 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))),V_xs_2) = c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)) # label(fact_dropWhile__eq__drop) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3221 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2))) # label(fact_Int__lower2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3222 (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_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]. 9.04/9.02 3223 (all T_c c_Relation_OId__on(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) # label(fact_Id__on__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3224 (all T_d all T_c (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(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),c_Orderings_Otop__class_Otop(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))))))) # label(fact_finite__Prod__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3225 (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]. 9.04/9.02 3226 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c),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_c))) <-> c_Groups_Ozero__class_Ozero(T_c) = V_y_2 & V_x_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3227 (all V_a_2 all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_2)),V_r_2))))) # label(fact_refl__onD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3228 (all V_B_2 all V_a_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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_d),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_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3229 (all V_xs_2 all V_a_2 all V_f_2 all T_d all T_c c_List_Ofoldr(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_c),V_f_2),hAPP(c_List_Orev(T_d),V_xs_2),V_a_2) = hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),V_xs_2)) # label(fact_foldl__foldr) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3230 (all V_b all V_a 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)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_b))) # label(fact_zmod__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3231 (all V_ys_2 all V_y_2 all V_xs_2 all V_Pa_2 all T_c (-hBOOL(hAPP(V_Pa_2,V_y_2)) & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) <-> c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))) # label(fact_dropWhile__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3232 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),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_C_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))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))) # label(fact_Times__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3233 (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]. 9.04/9.02 3234 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d hAPP(hAPP(V_f1_2,V_a_2),V_b_2) = c_Nitpick_Opair__box_Opair__box__rec(T_d,T_e,T_c,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_d,T_e,V_a_2,V_b_2))) # label(fact_pair__box_Orecs) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3235 (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]. 9.04/9.02 3236 (all V_r_2 all T_c c_Relation_Osym(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_sym__Un__converse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3237 (all T_c (class_Groups_Oab__semigroup__mult(T_c) -> c_Finite__Set_Ofun__left__comm(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)))) # label(fact_fun__left__comm) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3238 (all V_v all V_b all V_a all T_b (class_Int_Onumber(T_b) & class_Rings_Oring(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]. 9.04/9.02 3239 (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),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))) & (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)))) # label(fact_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3240 (all V_ns_2 all V_m_2 (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) <-> V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & (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)))) # label(fact_sum__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3241 (all V_z all V_y all V_x 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)) = 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))) # label(fact_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3242 (all V_y_2 all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),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))))),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))))) # label(fact_card__Diff2__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3243 (all V_r_2 all T_c c_List_Olenlex(T_c,V_r_2) = hAPP(hAPP(c_Relation_Oinv__image(tc_prod(tc_Nat_Onat,tc_List_Olist(T_c)),tc_List_Olist(T_c)),c_Wellfounded_Olex__prod(tc_Nat_Onat,tc_List_Olist(T_c),c_Wellfounded_Oless__than,c_List_Olex(T_c,V_r_2))),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_prod(tc_Nat_Onat,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_List_Olist(T_c),tc_prod(tc_Nat_Onat,tc_List_Olist(T_c))),tc_List_Olist(T_c)),c_Product__Type_OPair(tc_Nat_Onat,tc_List_Olist(T_c))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),c_COMBI(tc_List_Olist(T_c))))) # label(fact_lenlex__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3244 (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]. 9.04/9.02 3245 (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]. 9.04/9.02 3246 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_fa_2 all T_c c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Hoare__Mirabelle_Otriple_Otriple__size(T_c,V_fa_2,hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_fun1_2),V_com_2),V_fun2_2))) # label(fact_triple_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3247 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = 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)))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3248 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = V_x)) # label(fact_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3249 (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),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_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3250 (all T_c (class_Orderings_Olinorder(T_c) -> c_Finite__Set_Ofun__left__comm(T_c,tc_List_Olist(T_c),c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c))))) # label(fact_fun__left__comm__insort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3251 (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]. 9.04/9.02 3252 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I20_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3253 (all V_r_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel(T_c,V_r_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel__subset__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3254 (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)) -> (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) <-> V_n_2 = V_m_2)))) # label(fact_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3255 (all V_r_2 all T_c ((c_Nitpick_Ounknown(tc_HOL_Obool) | hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2))) & c_Wellfounded_Oacyclic(T_c,V_r_2) <-> c_Nitpick_Owf_H(T_c,V_r_2))) # label(fact_Nitpick_Owf_H__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3256 (all T_c (class_Finite__Set_Ofinite(T_c) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_finite__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3257 (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_a)))) # label(fact_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3258 (all V_g_2 all V_h_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_g_2),V_h_2)),V_A_2))))) # label(fact_setprod_Oreindex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3259 (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]. 9.04/9.02 3260 (all V_f_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(V_f_2,V_m_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m_2)),V_n_2)))))) # label(fact_setsum__head__upt__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3261 (all V_C_2 all V_g_2 all T_e all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> (c_Fun_Obij__betw(T_d,T_e,V_g_2,V_B_2,V_C_2) -> c_Fun_Obij__betw(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_g_2),V_f_2),V_A_2,V_C_2)))) # label(fact_bij__betw__trans) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3262 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Orev(T_c),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct__rev) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3263 (all V_B_2 all V_r_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Relation_OImage(T_d,T_c,V_r_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),c_Set_Oinsert(T_d)),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) = hAPP(c_Relation_OImage(T_d,T_c,V_r_2),V_B_2)) # label(fact_Image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3264 (all V_f_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_d))) -> c_Groups_Oone__class_Oone(T_d) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_f_2),c_Groups_Oone__class_Oone(T_d)),V_S_2))))) # label(fact_fold__image__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3265 (all V_A_2 all V_f_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2))),V_A_2))) # label(fact_image__vimage__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3266 (all V_r_2 all T_c c_Relation_Orefl__on(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) # label(fact_refl__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3267 (all V_z_2 all V_g_2 all V_f_2 all T_d all T_c V_z_2 = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2),V_z_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_fold__image__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3268 (all V_B_2 all V_A_2 all T_c all T_d c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),c_HOL_OEx(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d)),c_fequal)),V_B_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_UN__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3269 (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]. 9.04/9.02 3270 (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)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2)) -> c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,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_OatLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3271 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_SigmaD1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3272 (all V_A_2 all V_y_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(c_COMBK(T_c,T_d),V_y_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_d),V_A_2))),V_y_2))) # label(fact_setsum__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3273 (all V_Pa_2 all V_Qa_2 (hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)) | -hBOOL(V_Qa_2))) # label(help_c__fimplies__2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3274 (all V_B_2 all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(c_Set_OCollect(T_c),hAPP(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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2)))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_f_2)),c_Groups_Ozero__class_Ozero(T_d))),V_A_2)))) # label(fact_setsum__restrict__set_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3275 (all V_x_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2) <-> V_x_2 = c_Groups_Oone__class_Oone(T_c) | V_x_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c))))) # label(fact_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3276 (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_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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a)),V_b)))) # label(fact_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3277 (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]. 9.04/9.02 3278 (all V_B_2 all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))),V_f_2)),c_Groups_Ozero__class_Ozero(T_d))),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_setsum__restrict__set) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3279 (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]. 9.04/9.02 3280 (all V_f_2 all T_d all V_A_2 all T_c (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> (exists B_x (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(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(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),hAPP(V_f_2,B_x)))))) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))))))) # label(fact_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3281 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> hAPP(hAPP(V_f_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)))))) = hAPP(V_F_2,V_A_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)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = hAPP(V_F_2,V_A_2)))))) # label(fact_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3282 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3283 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olistrel(T_c,V_r_2))) -> -(all B_x all B_xs (hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) = V_xs_2 -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),V_y_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),B_xs),V_ys_2)),c_List_Olistrel(T_c,V_r_2)))))))) # label(fact_listrel__Cons2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3284 (all T_a all V_xs (c_List_Olist_ONil(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)),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]. 9.04/9.02 3285 (all V_R_2 all T_d all T_e all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_R_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool)))) # label(fact_rel__comp__empty2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3286 (all V_Qa_2 all V_Pa_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_c),V_Pa_2))),hAPP(c_Set_OCollect(T_c),V_Qa_2)) = hAPP(c_Set_OCollect(T_c),hAPP(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),V_Pa_2)),V_Qa_2))) # label(fact_Collect__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3287 (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]. 9.04/9.02 3288 (all V_ys_2 all V_xs_2 all T_d all T_c hAPP(c_List_Oset(tc_prod(T_c,T_d)),c_Enum_Oproduct(T_c,T_d,V_xs_2,V_ys_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_List_Oset(T_d),V_ys_2)))) # label(fact_product__list__set) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 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),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]. 9.04/9.02 3290 (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)),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_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3291 (all V_A_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)),V_A_2))))))) # label(fact_fold1__in) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3292 (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]. 9.04/9.02 3293 (all V_f_2 all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),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_Nat_Onat),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,tc_Nat_Onat),V_f_2),V_B_2))))) # label(fact_setsum__diff__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3294 (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),V_b) = c_Groups_Ozero__class_Ozero(T_a) -> V_b = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_minus__unique) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3295 (all V_b_2 all V_B_2 all V_A_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_INF1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3296 (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]. 9.04/9.02 3297 (all V_Qa_2 all V_Pa_2 (-hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)) | hBOOL(V_Pa_2))) # label(help_c__fconj__2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3298 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),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)))) # label(fact_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3299 (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__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_Odivmod__int(V_a,V_b) = c_Divides_OposDivAlg(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)) -> (V_a = 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)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Divides_Odivmod__int(V_a,V_b)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_a -> 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)))) & (-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_Odivmod__int(V_a,V_b) = c_Divides_OnegDivAlg(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))))))) # label(fact_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3300 (all V_A_2 all T_c hAPP(c_Relation_ORange(T_c,T_c),c_Relation_OId__on(T_c,V_A_2)) = V_A_2) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3301 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> c_Finite__Set_Ofun__left__comm__idem(T_e,T_d,hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,T_d),T_e),V_f_2),V_g_2)))) # label(fact_fun__left__comm__idem_Ofun__left__comm__idem__apply) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3302 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),V_y)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_x)) | V_x = V_y)) # label(fact_zless__linear) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3303 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) <-> c_Relation_Orefl__on(T_c,V_A_2,V_r_2) & c_Relation_Otrans(T_c,V_r_2) & c_Relation_Osym(T_c,V_r_2))) # label(fact_equiv__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.02 3304 (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]. 9.04/9.02 3305 (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]. 9.04/9.02 3306 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> -(hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))))) # label(fact_IntE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3307 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_l_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_u_2))))) # label(fact_ivl__disj__un_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3308 (all V_B_2 all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_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_subset__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3309 (all V_b all V_a 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)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) # label(fact_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3310 (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_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) <-> 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]. 9.04/9.03 3311 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_d all T_c (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(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__zip__leftD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3312 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3313 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(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(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2))))))))) # label(fact_Sup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3314 (all V_z all V_y all V_x 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_x),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__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_sup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3315 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3316 (all V_g_2 all V_f_2 all V_Pa_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(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_Set_OCollect(T_c),V_Pa_2))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),V_Pa_2)),V_f_2)),V_g_2)),V_A_2)))) # label(fact_setsum__cases) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3317 (all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_neg__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3318 (all V_xs_2 all V_x_2 all T_c hAPP(c_List_Oset(T_c),c_List_Oinsert(T_c,V_x_2,V_xs_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_List_Oset__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3319 (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 3320 (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]. 9.04/9.03 3321 (all V_Pa_2 all V_f_2 all T_c all T_d c_New__Random__Sequence_Oneg__bind(T_d,T_c,V_Pa_2,hAPP(c_Fun_Ocomp(T_c,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_c)))))),T_d,c_New__Random__Sequence_Oneg__single(T_c)),V_f_2)) = c_New__Random__Sequence_Oneg__map(T_d,T_c,V_f_2,V_Pa_2)) # label(fact_neg__map__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3322 (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]. 9.04/9.03 3323 (all V_l_2 all V_k_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3324 (all V_xs_2 all T_c 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_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Otake(T_c),V_m_2),V_xs_2))),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2)))))) # label(fact_set__take__subset__set__take) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3325 (all V_A_2 all V_I_2 all T_d all V_B_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_I_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(V_A_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_INT__subset__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3326 (all V_n_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),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_c),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]. 9.04/9.03 3327 (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]. 9.04/9.03 3328 (all V_r_2 all T_c c_Transitive__Closure_Otrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c))) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3329 (all V_xs_2 all V_n_2 all T_c (hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2) = c_List_Olist_ONil(T_c) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_take__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3330 (all V_c_2 all V_b_2 all V_Pa_2 all V_Ga_2 all T_c c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Com_Ostate,tc_HOL_Obool),tc_fun(tc_Com_Ostate,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Com_Ostate,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Com_Ostate,tc_HOL_Obool),tc_fun(tc_Com_Ostate,tc_HOL_Obool)),T_c),c_COMBS(tc_Com_Ostate,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Com_Ostate,tc_HOL_Obool),tc_fun(tc_Com_Ostate,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Com_Ostate),c_fconj)),V_Pa_2))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Com_Ostate),c_fNot),V_b_2))),c_Com_Ocom_OWhile(V_b_2,V_c_2)),V_Pa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))) # label(fact_LoopF) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3331 (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_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_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]. 9.04/9.03 3332 (all V_y_2 all V_x_2 all V_f_2 all T_c all T_d all T_e hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),hAPP(V_f_2,V_y_2)) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_e),V_x_2),V_y_2))) # label(fact_apsnd__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3333 (all V_f_2 all T_d all T_e all T_c c_Product__Type_Oapfst(T_c,T_e,T_d,V_f_2) = c_Product__Type_Omap__pair(T_c,T_e,T_d,T_d,V_f_2,c_Fun_Oid(T_d))) # label(fact_apfst__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3334 (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))) -> 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_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))) & (-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_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)) = 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)))))) # label(fact_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3335 (all V_Pa_2 all V_f_2 all T_c all T_d c_New__Random__Sequence_Opos__map(T_d,T_c,V_f_2,V_Pa_2) = c_New__Random__Sequence_Opos__bind(T_d,T_c,V_Pa_2,hAPP(c_Fun_Ocomp(T_c,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_c))))),T_d,c_New__Random__Sequence_Opos__single(T_c)),V_f_2))) # label(fact_pos__map__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3336 (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]. 9.04/9.03 3337 (all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_c_2),c_Relation_OId__on(T_c,V_A_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> V_c_2 != hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x))))) # label(fact_Id__onE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3338 (all V_A_2 all V_B_2 all V_f_2 all T_e all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),tc_fun(tc_fun(T_e,tc_HOL_Obool),T_c),T_d),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_c)),V_f_2)),V_B_2)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_c),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_c),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_c),T_e),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c)),hAPP(c_COMBC(T_d,T_e,T_c),V_f_2))),V_A_2)),V_B_2))) # label(fact_setsum__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3339 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3340 (all V_C_2 all V_J_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_J_2),V_C_2))) # label(fact_Sigma__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3341 (all V_k_2 all V_i_2 all V_r_2 all V_f_2 all T_c ((all B_i 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,hAPP(c_Nat_OSuc,B_i))),hAPP(V_f_2,B_i))),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) -> 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,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_c,V_r_2))))) # label(fact_sequence__trans) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3342 (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,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_c),V_b)),V_b))) # label(fact_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3343 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) = V_B_2 -> hAPP(c_Set_OPow(T_c),V_B_2) = hAPP(c_Set_Oimage(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),c_Set_Oimage(T_d,T_c,V_f_2)),hAPP(c_Set_OPow(T_d),V_A_2)))) # label(fact_image__Pow__surj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3344 (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 3345 (all V_A_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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_c),V_f_2),V_A_2))) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3346 (all V_A_2 all V_f_2 all T_d all T_c (class_Fields_Olinordered__field(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2) = hAPP(c_Groups_Oabs__class_Oabs(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_f_2),V_A_2)))) # label(fact_abs__setprod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3347 (all V_A_2 all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_HOL_Obool,tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),c_fequal))),V_A_2)) = c_Relation_OId__on(T_c,V_A_2)) # label(fact_Id__on__def_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3348 (all T_c all V_ys_2 all V_xs_2 (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),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_c)),V_xs_2))) -> hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i) = hAPP(hAPP(c_List_Onth(T_c),V_ys_2),B_i))) <-> V_xs_2 = V_ys_2)) # label(fact_list__eq__iff__nth__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3349 (all V_g_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_setprod_Oempty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3350 (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]. 9.04/9.03 3351 (all V_xs_2 all V_x_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_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_set__removeAll) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3352 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2) = c_Groups_Ozero__class_Ozero(T_c) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3353 (all V_B_2 all T_c 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)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_B_2)) # label(fact_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3354 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_A_2)))) # label(fact_finite__UnionD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3355 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_neg__le__0__iff__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3356 (all V_x_2 all V_Pa_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_Pa_2,hAPP(c_Int_OAbs__Integ,B_y))))) -> hBOOL(hAPP(V_Pa_2,V_x_2)))) # label(fact_Abs__Integ__induct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3357 (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)) -> c_Divides_OnegDivAlg(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_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3358 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs)) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_butlast__drop) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3359 (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]. 9.04/9.03 3360 (all V_c_2 all T_c all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> ((hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))) -> -(-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)))))) # label(fact_equalityCE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3361 (all V_a_2 all T_c c_FunDef_Ois__measure(T_c,V_a_2)) # label(fact_is__measure_Oequations) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3362 (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_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) -> V_b = V_c))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3363 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d c_New__Random__Sequence_Oneg__decr__bind(T_d,T_c,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Oneg__decr__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,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_c)))),hAPP(hAPP(c_COMBC(T_d,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_c))))),hAPP(hAPP(c_COMBC(T_d,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_c)))))),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]. 9.04/9.03 3364 (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]. 9.04/9.03 3365 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Orotate1(T_c),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct1__rotate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3366 (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))) -> (V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> 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]. 9.04/9.03 3367 (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]. 9.04/9.03 3368 (all V_a_2 all V_w_2 all V_b_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) & class_Int_Onumber(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_b_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_a_2))))) # label(fact_divide__less__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3369 (all V_fs_2 all T_c 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_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_c,tc_Nat_Onat)),V_f_2),V_fs_2)))))) # label(fact_measures__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3370 (all V_ys_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = V_xs_2 <-> c_List_Olist_ONil(T_c) = V_ys_2)) # label(fact_append__self__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3371 (all V_Pa_2 all V_a_2 all V_n_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_a_2)))) & hBOOL(hAPP(V_Pa_2,B_x)))) <-> hBOOL(hAPP(V_Pa_2,V_a_2)) & V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Bex__set__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3372 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_neg__0__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3373 (all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Otrancl(T_c,V_R_2))) -> (exists B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_z)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_z),V_y_2)),V_R_2)))))) # label(fact_tranclD2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3374 (all V_A_2 all V_B_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Un__upper2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3375 (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]. 9.04/9.03 3376 (all T_c c_Set_OBall(T_c) = c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_HOL_Obool)) # label(fact_INFI__bool__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3377 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c all T_e hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_e,T_d),V_x_2),V_y_2)) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,V_x_2)),V_y_2)) # label(fact_apfst__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3378 (all V_a_2 all V_A_2 all V_r_2 all T_c (c_Fun_Oinj__on(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),T_c),c_Equiv__Relations_Oquotient(T_c)),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))))),V_r_2),V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2)),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_r_2)) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),V_r_2)))) # label(fact_quotient__diff1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3379 (all V_Y_2 all T_c hAPP(c_Set_Oimage(T_c,T_c,c_Fun_Oid(T_c)),V_Y_2) = V_Y_2) # label(fact_image__id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3380 (all V_r_2 all T_c hAPP(c_Relation_OImage(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Olistrel(T_c,V_r_2)),hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))) # label(fact_listrel__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3381 (all V_m 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_m)),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_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3382 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> -(hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))))) # label(fact_DiffE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3383 (all V_xs_2 all T_c c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Orev(T_c)),hAPP(c_List_Orev(tc_List_Olist(T_c)),V_xs_2))) = hAPP(c_List_Orev(T_c),c_List_Oconcat(T_c,V_xs_2))) # label(fact_rev__concat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3384 (all V_M_2 all V_A_2 all V_i_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_M_2,V_i_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_M_2)))))) # label(fact_le__SUPI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3385 (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]. 9.04/9.03 3386 (all V_n_2 all T_c hAPP(hAPP(c_Nat_Ocompow(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_n_2),c_List_Orotate1(T_c)) = c_List_Orotate(T_c,V_n_2)) # label(fact_rotate__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3387 (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]. 9.04/9.03 3388 (all V_A_2 all V_c_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))))) # label(fact_ComplI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3389 (all V_k all V_n all V_m 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)) = 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)) # label(fact_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3390 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))))) # label(fact_rtrancl__listrel1__if__listrel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3391 (all V_f_2 all V_Pa_2 all V_xs_2 all T_c all T_d (class_Groups_Omonoid__add(T_d) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> (-hBOOL(hAPP(V_Pa_2,B_x)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_d)))) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))))) # label(fact_listsum__map__filter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3392 (all V_S_2 all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) -> (c_Wellfounded_Owf(T_c,V_S_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_S_2)),V_R_2)) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)))))) # label(fact_wf__union__compatible) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3393 (all V_B_2 all V_A_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_d),V_A_2)),hAPP(c_Relation_ODomain(T_c,T_d),V_B_2))),hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Domain__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3394 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Predicate_ORangeP(T_c,T_d,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_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Relation_ORange(T_c,T_d),V_r_2))))) # label(fact_RangeP__Range__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3395 (all V_g_2 all V_h_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (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(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_g_2),V_h_2)),V_A_2))))) # label(fact_setsum_Oreindex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3396 (all V_A_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))))) # label(fact_trancl__subset__Sigma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3397 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2))))))) # label(fact_Min_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3398 (all V_A_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),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),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_Compl__Times__UNIV2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3399 (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]. 9.04/9.03 3400 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_c),c_List_Olex(T_c,V_r_2)))) # label(fact_wf__lex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3401 (all V_p all 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) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m -> 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]. 9.04/9.03 3402 (all V_a_2 all V_A_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_A_2)),V_a_2)))))) # label(fact_fold1__belowI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3403 (all V_n 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)),V_n)) # label(fact_mult__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3404 (all V_R_2 all V_d_2 all V_Qa_2 all V_c_2 all V_Pa_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Qa_2),V_d_2),V_R_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),c_Com_Ocom_OSemi(V_c_2,V_d_2)),V_R_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))))) # label(fact_Comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3405 (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]. 9.04/9.03 3406 (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]. 9.04/9.03 3407 (all V_r_2 all V_I_2 all T_f all V_s_2 all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_s_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool)),V_I_2),V_r_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_f),hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_s_2)),V_r_2))) # label(fact_rel__comp__UNION__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3408 (all V_Y_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(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)) != V_Y_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_Y_2)),c_FunDef_Omax__strict))))) # label(fact_smax__emptyI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3409 (all V_Pa_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_Pa_2,hAPP(c_Code__Numeral_Onat__of,B_x)))) -> hBOOL(hAPP(V_Pa_2,V_y_2))))) # label(fact_nat__of__induct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3410 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2))) -> (exists B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),V_R_2)))))) # label(fact_rel__pow__Suc__D2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3411 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_d_2)) -> V_b_2 = V_d_2))))) # label(fact_atLeastLessThan__inj_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3412 (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]. 9.04/9.03 3413 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2)))))) # label(fact_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3414 (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]. 9.04/9.03 3415 (all V_Ga_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_Ga_2))))) # label(fact_finite__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3416 (all V_f_2 all V_n_2 all T_c hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),hAPP(c_Nat_OSuc,V_n_2)),V_f_2) = hAPP(c_Fun_Ocomp(T_c,T_c,T_c,V_f_2),hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2))) # label(fact_funpow_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3417 (all T_c c_List_Olistset(T_c,c_List_Olist_ONil(tc_fun(T_c,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))) # label(fact_listset_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3418 (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]. 9.04/9.03 3419 (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(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),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))))) # label(fact_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3420 (all V_Pa_2 all T_c c_List_Olist_ONil(T_c) = hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Olist_ONil(T_c))) # label(fact_filter_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3421 (all V_l all V_k c_Int_OBit0(V_l) != c_Int_OBit1(V_k)) # label(fact_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3422 (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)) -> 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)) -> 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(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))))) & (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]. 9.04/9.03 3423 (all V_xs_2 all V_f_2 all V_Pa_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)) = c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))) # label(fact_filter__sort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3424 (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_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b))))) # label(fact_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3425 (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),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)) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3426 (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(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),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))) & (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),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]. 9.04/9.03 3427 (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]. 9.04/9.03 3428 (all V_r_2 all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2)) = V_r_2) # label(fact_converse__converse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3429 (all V_w all V_v 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)) = 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)))) # label(fact_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3430 (all V_f_2 all T_c 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(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_i_2)) = hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),V_f_2),c_List_Oupt(V_m_2,V_n_2))),V_i_2))) # label(fact_nth__map__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3431 (all V_n all V_m all V_k all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),c_Divides_Odiv__class_Omod(T_a,V_m,V_n))) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),V_n)) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_k),V_m)))))) # label(fact_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3432 (all V_y_2 all V_x_2 all T_c (class_Rings_Ocomm__ring__1(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),V_y_2))))) # label(fact_dvd__minus__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3433 (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]. 9.04/9.03 3434 (all V_A_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_A_2)))) # label(fact_Compl__Times__UNIV1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3435 (all V_p_2 all V_f_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (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_c),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_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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_c),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]. 9.04/9.03 3436 (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_d all T_c (c_Finite__Set_Ofolding__image(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),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_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),V_f_2),V_g_2)),V_h_2)),V_A_2)))) # label(fact_folding__image_Odistrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3437 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2))))))) # label(fact_Max_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3438 (all V_C_2 all V_J_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_J_2),V_C_2))) # label(fact_Sigma__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3439 (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]. 9.04/9.03 3440 (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__eq2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3441 (all V_f_2 all T_c c_Wellfounded_Owf(T_c,hAPP(c_Wellfounded_Omeasure(T_c),V_f_2))) # label(fact_wf__measure) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3442 (all V_R_2 all T_c (c_Predicate_Oreflp(T_c,V_R_2) -> (c_Predicate_Osymp(T_c,V_R_2) -> (c_Predicate_Otransp(T_c,V_R_2) -> c_Equiv__Relations_Oequivp(T_c,V_R_2))))) # label(fact_equivpI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3443 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_d,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]. 9.04/9.03 3444 (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]. 9.04/9.03 3445 (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]. 9.04/9.03 3446 (all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_ComplD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3447 (all V_f_2 all V_r_2 all T_d all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),V_r_2),V_f_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))),V_r_2)))) # label(fact_in__inv__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3448 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_2) = c_Transitive__Closure_Otrancl(T_c,c_Transitive__Closure_Ortrancl(T_c,V_R_2))) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3449 (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_H,V_c) = c_Divides_Odiv__class_Omod(T_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) = 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__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3450 (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_x) = V_x)) # label(fact_sup__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3451 (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]. 9.04/9.03 3452 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Fun_Oinj__on(tc_List_Olist(T_c),tc_List_Olist(T_d),hAPP(c_List_Omap(T_c,T_d),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))))) # label(fact_inj__mapI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.03 3453 (all T_c (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_Option_Ooption(T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_Option_Ooption(T_c),tc_HOL_Obool)))))) # label(fact_finite__option__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3454 (all V_xs_2 all V_f_2 all T_c all T_d c_List_Omaps(T_d,T_c,V_f_2,V_xs_2) = c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(T_d,tc_List_Olist(T_c)),V_f_2),V_xs_2))) # label(fact_maps__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3455 (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_w = V_z))) # label(fact_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3456 (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 3457 (all T_c all V_A_2 all T_d 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_d),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)))))) # label(fact_image__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3458 (all V_x all T_a (class_Orderings_Olinorder(T_a) -> V_x = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_x))) # label(fact_Max_Oidem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3459 (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]. 9.04/9.04 3460 (all V_y_2 all V_x_2 all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_d,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]. 9.04/9.04 3461 (all V_x all 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))) & (V_y = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Nitpick_Onat__gcd(V_x,V_y) = V_x))) # label(fact_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3462 (all V_a_2 all T_c -hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_emptyE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3463 (all V_Pa_2 all T_c c_List_Opartition(T_c,V_Pa_2,c_List_Olist_ONil(T_c)) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_List_Olist_ONil(T_c))) # label(fact_partition_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3464 (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]. 9.04/9.04 3465 (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_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))))) # label(fact_pred__nat__trancl__eq__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3466 (all V_x_2 all V_w_2 all T_c (class_Int_Onumber(T_c) -> (V_x_2 = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) <-> V_x_2 = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))) # label(fact_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3467 (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)) -> 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))) = c_SMT_Oz3div(V_k,V_l)))) # label(fact_z3div__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3468 (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]. 9.04/9.04 3469 (all V_w_2 all V_x_2 all V_z_2 all V_y_2 all T_c (class_Fields_Ofield(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_y_2 -> (V_z_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_z_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_w_2),V_y_2) <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_x_2),V_y_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_w_2),V_z_2)))))) # label(fact_frac__eq__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3470 (all V_xs_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (c_List_Olist_ONil(T_c) != V_xs_2 -> hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c),hAPP(c_List_Ohd(T_c),V_xs_2)),hAPP(c_List_Otl(T_c),V_xs_2))))) # label(fact_fold1__set) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3471 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_x_2)),hAPP(V_F_2,V_A_2))))) # label(fact_folding_Ocommute__comp_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3472 (all T_c hAPP(c_Set_OPow(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))) # label(fact_Pow__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3473 (all V_x_2 all V_r_2 all V_v_2 all V_u_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_u_2),V_v_2)),c_List_Olexord(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_u_2)),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_v_2))),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__append__leftI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3474 (all V_xs_2 all V_r_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_COMBK(T_c,T_d),V_r_2)),V_xs_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2))),V_r_2))) # label(fact_listsum__triv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3475 (all V_q 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_Power_Opower__class_Opower(T_a),V_x),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3476 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3477 (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]. 9.04/9.04 3478 (all V_xq_2 all V_g_2 all V_f_2 all T_d all T_c c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_c,T_d,V_f_2,V_g_2,V_xq_2) = c_Option_Ooption_Ooption__case(T_c,tc_prod(T_d,tc_Lazy__Sequence_Olazy__sequence(T_d)),V_f_2,hAPP(c_Product__Type_Oprod_Oprod__case(T_d,tc_Lazy__Sequence_Olazy__sequence(T_d),T_c),V_g_2),hAPP(c_Lazy__Sequence_Oyield(T_d),V_xq_2))) # label(fact_seq__case) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3479 (all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_c,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_prod(T_c,T_c),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),T_c),c_Set_Oinsert(tc_prod(T_c,T_c))),hAPP(hAPP(c_COMBS(T_c,T_c,tc_prod(T_c,T_c)),c_Product__Type_OPair(T_c,T_c)),c_COMBI(T_c)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))) = c_Relation_OId__on(T_c,V_A_2)) # label(fact_Id__on__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3480 (all V_seed_2 all V_size_2 all V_nrandom_2 all T_c c_New__Random__Sequence_Opos__empty(T_c,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__empty(T_c)) # label(fact_pos__empty__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3481 (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_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_d),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__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3482 (all V_b_2 all V_a_2 all V_Pa_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2)) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (all B_d (V_a_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d) -> hBOOL(hAPP(V_Pa_2,B_d)))) <-> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2))))) # label(fact_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3483 (all V_xs_2 all V_f_2 all T_c all T_d c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_d),tc_List_Olist(T_c)),hAPP(c_List_Omap(T_d,T_c),V_f_2)),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Oconcat(T_d,V_xs_2))) # label(fact_map__concat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3484 (all V_b_2 all V_a_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),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]. 9.04/9.04 3485 (all V_xs_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Ominus__class_Ominus(T_c)),V_f_2)),V_g_2)),V_xs_2)))) # label(fact_listsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3486 (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]. 9.04/9.04 3487 (all V_Qa_2 all V_Pa_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_c),V_Pa_2)),hAPP(c_Set_OCollect(T_c),V_Qa_2)) = hAPP(c_Set_OCollect(T_c),hAPP(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),V_Pa_2)),V_Qa_2))) # label(fact_Collect__conj__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3488 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),V_x_2)))))) # label(fact_Min__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3489 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2) = c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2))))) # label(fact_Inf__fin__Inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3490 (all V_y_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(c_COMBK(T_d,T_c),V_y_2)),V_A_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_d),V_y_2),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_setprod__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3491 (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 3492 (all V_xss_2 all T_c hAPP(c_List_Ofoldl(tc_List_Olist(T_c),tc_List_Olist(T_c),c_List_Oappend(T_c),c_List_Olist_ONil(T_c)),V_xss_2) = c_List_Oconcat(T_c,V_xss_2)) # label(fact_concat__conv__foldl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3493 (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]. 9.04/9.04 3494 (all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_List_Omember(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_y_2)) <-> V_y_2 = V_x_2 | hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_y_2)))) # label(fact_member__rec_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3495 (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]. 9.04/9.04 3496 (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_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_us) -> (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]. 9.04/9.04 3497 (all V_b 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)),V_b)) # label(fact_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3498 (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 3499 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_d_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_a_2))))) # label(fact_atLeastLessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3500 (all V_A_2 all V_a_2 all T_c ((exists B_a exists B_l (hAPP(hAPP(c_List_Olist_OCons(T_c),B_a),B_l) = V_a_2 & hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_l),c_List_Olists(T_c,V_A_2))))) | V_a_2 = c_List_Olist_ONil(T_c) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_a_2),c_List_Olists(T_c,V_A_2))))) # label(fact_lists_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3501 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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)))) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,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))))))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_2)))) # label(fact_Sup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3502 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) & class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2))))) # label(fact_less__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3503 (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]. 9.04/9.04 3504 (all V_a_2 all V_D_2 all V_C_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_D_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_C_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_D_2))))) # label(fact_insert__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3505 (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]. 9.04/9.04 3506 (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]. 9.04/9.04 3507 (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]. 9.04/9.04 3508 (all V_A_2 all T_d all T_c c_Fun_Oinj__on(tc_prod(T_c,T_d),tc_prod(T_d,T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,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_A_2)) # label(fact_swap__inj__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3509 (all V_b_2 all V_a_2 all V_g_2 all V_f_2 all T_d all T_f all T_c all T_e hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2)) = hAPP(c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2),hAPP(hAPP(c_Product__Type_OPair(T_e,T_f),V_a_2),V_b_2))) # label(fact_map__pair__simp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3510 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_min__max_OSup__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3511 (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]. 9.04/9.04 3512 (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]. 9.04/9.04 3513 (all V_f_2 all V_r_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),V_r_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool)),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_c),c_COMBB(T_d,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2))),V_f_2)),B_x)))) <-> c_Equiv__Relations_Ocongruent(T_c,T_d,V_r_2,V_f_2))) # label(fact_congruent__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3514 (all V_B_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,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]. 9.04/9.04 3515 (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]. 9.04/9.04 3516 (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]. 9.04/9.04 3517 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,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]. 9.04/9.04 3518 (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]. 9.04/9.04 3519 (all V_z_2 all T_c (class_Int_Oring__char__0(T_c) -> (V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)))) # label(fact_of__int__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3520 (all V_lazy__sequence_2 all V_a_2 all V_fa_2 all T_c 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_c,V_fa_2),V_lazy__sequence_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_fa_2),c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3521 (all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Union__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3522 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__Diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3523 (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]. 9.04/9.04 3524 (all V_list_H all V_a_H all T_a c_List_Olist_ONil(T_a) != hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H)) # label(fact_list_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3525 (all V_xs 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)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Obutlast(T_a,V_xs))) # label(fact_length__butlast) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3526 (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_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)))) # label(fact_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3527 (all V_xs_2 all T_c (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xs_2))) -> c_List_OtakeWhile(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_xs_2) = c_List_Otranspose(T_c,c_List_Otranspose(T_c,V_xs_2)))) # label(fact_transpose__transpose) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3528 (all V_r_2 all V_x_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),c_List_Olist_ONil(T_c))),c_List_Olexord(T_c,V_r_2)))) # label(fact_lexord__Nil__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3529 (all V_n all V_m ((V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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) != V_m -> 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))))) # label(fact_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3530 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3531 (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_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_sup__assoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3532 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_List_Ohd(T_a),V_xs))) # label(fact_last__rev) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3533 (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]. 9.04/9.04 3534 (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]. 9.04/9.04 3535 (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_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)) = V_b))) # label(fact_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3536 (all V_f_2 all V_xq_2 all T_c all T_d c_Lazy__Sequence_Oflat(T_c,c_Lazy__Sequence_Omap(T_d,tc_Lazy__Sequence_Olazy__sequence(T_c),V_f_2,V_xq_2)) = hAPP(c_Lazy__Sequence_Obind(T_d,T_c,V_xq_2),V_f_2)) # label(fact__05) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3537 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),c_List_Obutlast(T_c,V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__butlastD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3538 (all T_c c_Relation_Orefl__on(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)))) # label(fact_refl__on__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3539 (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]. 9.04/9.04 3540 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Oab__group__add(T_d) -> (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(hAPP(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),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_setsum__diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3541 (all V_s_2 all V_r_2 all T_d all T_e all T_c hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_HOL_Obool,T_d),c_HOL_OEx(T_e))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),tc_fun(tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool),T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_c),c_COMBS(T_e,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(T_e,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_e),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_c),c_COMBC(T_e,tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_c,T_e)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_e),tc_fun(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_c,T_e)))),c_Product__Type_OPair(T_c,T_e)))),V_r_2))))),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_d),c_COMBC(T_e,tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_e,T_d)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_d),hAPP(c_COMBB(tc_prod(T_e,T_d),tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_e,T_d)))),hAPP(c_COMBC(T_e,T_d,tc_prod(T_e,T_d)),c_Product__Type_OPair(T_e,T_d))))),V_s_2))))) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_r_2),V_s_2)) # label(fact_rel__comp__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3542 (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))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> (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]. 9.04/9.04 3543 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(tc_List_Olist(T_c),c_List_Olenlex(T_c,V_r_2)))) # label(fact_wf__lenlex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3544 (all V_g_2 all V_x_2 all T_e all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),V_g_2) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),V_g_2)))) # label(fact_folding__idem_Oidem__left__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3545 (all V_xs_2 all V_Pa_2 all T_c ((exists B_x ((all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_y)) -> B_x = B_y)) & hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))))) <-> c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex1__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3546 (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]. 9.04/9.04 3547 (all V_n_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)) # label(fact_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3548 (all V_f_2 all T_d all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(T_d,hAPP(hAPP(c_Relation_Oinv__image(T_c,T_d),V_r_2),V_f_2)))) # label(fact_wf__inv__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3549 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_Set_OCollect(T_c),hAPP(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)),hAPP(c_List_Oset(T_c),V_xs_2)))),V_Pa_2))) # label(fact_set__filter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3550 (all V_r_2 all T_c c_Relation_Otrans(T_c,c_Transitive__Closure_Otrancl(T_c,V_r_2))) # label(fact_trans__trancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3551 (all V_r_2 all T_c hAPP(c_Relation_OField(T_c),V_r_2) = hAPP(c_Relation_OField(T_c),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2))) # label(fact_Field__converse) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3552 (all V_r_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_r_2),c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = c_Transitive__Closure_Otrancl(T_c,V_r_2)) # label(fact_trancl__unfold__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3553 (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]. 9.04/9.04 3554 (all V_B_2 all V_C_2 all V_a_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3555 (all V_x all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> 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]. 9.04/9.04 3556 (all V_r_2 all T_c ((all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2)) -> (all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_z)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_z)),V_r_2)))))) <-> c_Relation_Otrans(T_c,V_r_2))) # label(fact_trans__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3557 (all V_B_2 all T_d all T_c 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_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2I1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3558 (all V_R_2 all T_d all T_c V_R_2 = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_d),c_Relation_OId(T_c)),V_R_2)) # label(fact_Id__O__R) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3559 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omin(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_min__max_OInf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3560 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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_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)) = 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]. 9.04/9.04 3561 (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]. 9.04/9.04 3562 (all V_r_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,c_Wellfounded_Oacc(T_c,V_r_2)))))) # label(fact_lists__accI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3563 (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]. 9.04/9.04 3564 (all V_A_2 all V_B_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2)),V_A_2)) # label(fact_Int__Union2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3565 (all V_y_2 all V_n_2 all V_x_2 all V_m_2 all T_c ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m_2 -> V_x_2 = V_y_2) & V_n_2 = V_m_2 <-> c_List_Oreplicate(T_c,V_m_2,V_x_2) = c_List_Oreplicate(T_c,V_n_2,V_y_2))) # label(fact_replicate__eq__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3566 (all V_r_2 all T_c hAPP(c_Relation_OField(T_c),V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_c),V_r_2)),hAPP(c_Relation_ORange(T_c,T_c),V_r_2))) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3567 (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]. 9.04/9.04 3568 (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]. 9.04/9.04 3569 (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__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2))))) # label(fact_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3570 (all V_f2_2 all V_f1_2 all T_c V_f1_2 = c_Nat_Onat_Onat__case(T_c,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__case__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3571 (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]. 9.04/9.04 3572 (all V_vs_2 all V_us_2 all V_Pa_2 all V_ys_2 all T_d all V_xs_2 all T_c (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) -> (c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_us_2),hAPP(hAPP(c_List_Oappend(T_d),V_ys_2),V_vs_2)) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) & c_List_Olist__all2(T_c,T_d,V_Pa_2,V_us_2,V_vs_2)))) # label(fact_list__all2__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3573 (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]. 9.04/9.04 3574 (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_Otake(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),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))))) # label(fact_take__add) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3575 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2)))) # label(fact_Inter__lower) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3576 (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(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_b) = V_a)) # label(fact_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3577 (all V_S_2 all V_R_2 all T_c (V_R_2 = V_S_2 <-> hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2) = hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2))) # label(fact_pred__equals__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3578 (all V_m_2 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_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)) = c_Divides_Odivmod__nat(V_m_2,V_n_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Divides_Odivmod__nat(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)))) # label(fact_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3579 (all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_A_2) <-> c_Equiv__Relations_Oequivp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_A_2)))) # label(fact_equivp__equiv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3580 (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]. 9.04/9.04 3581 (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_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) # label(fact_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3582 (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]. 9.04/9.04 3583 (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]. 9.04/9.04 3584 (all V_n_2 all T_c all V_x_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Oreplicate(T_c,V_n_2,V_x_2)) = c_List_Oreplicate(T_c,V_n_2,V_x_2)) & (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist_ONil(T_c) = hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Oreplicate(T_c,V_n_2,V_x_2))))) # label(fact_filter__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3585 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_finite__Diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3586 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,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_c,T_d,V_f_2),V_z_2),V_A_2))))) # label(fact_fun__left__comm_Ofold__fun__comm) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3587 (all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 all V_com2_2 all V_com1_2 c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OSemi(V_com1_2,V_com2_2)) # label(fact_com_Osimps_I44_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3588 (all V_n all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> 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]. 9.04/9.04 3589 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3590 (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]. 9.04/9.04 3591 (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)),V_x) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_compl__inf__bot) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3592 (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,c_Groups_Ozero__class_Ozero(T_a),V_a))) # label(fact_mod__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3593 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(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__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3594 (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),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_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))) # label(fact_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3595 (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)) <-> (exists B_m hAPP(c_Nat_OSuc,B_m) = V_n_2))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3596 (all V_B_2 all V_A_2 all T_d all T_c (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)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) | c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = V_B_2)) # label(fact_Times__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3597 (all V_B_2 all V_A_2 all T_c all T_d hAPP(c_Set_Oimage(tc_prod(T_d,T_c),tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,tc_prod(T_c,T_d)),hAPP(c_COMBC(T_c,T_d,tc_prod(T_c,T_d)),c_Product__Type_OPair(T_c,T_d)))),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_B_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_A_2))) # label(fact_swap__product) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3598 (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]. 9.04/9.04 3599 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_A_2))) # label(fact_empty__subsetI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3600 (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),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_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_zadd__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3601 (all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Ocomm__semiring__1(T_d) & class_Rings_Ono__zero__divisors(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Groups_Ozero__class_Ozero(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2) <-> (exists B_x (c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,B_x) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))))) # label(fact_setprod__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3602 (all V_lazy__sequence_2 all V_a_2 all V_f2_2 all V_f1_2 all T_d all T_c hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2) = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_c,T_d,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_d,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3603 (all V_B_2 all V_C_2 all V_A_2 all T_c (hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),V_C_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) -> (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_B_2),V_C_2) -> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)))) # label(fact_union__comp__emptyL) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3604 (all V_com_2 all V_fun_2 c_Com_Ocom_Ocom__size(c_Com_Ocom_OWhile(V_fun_2,V_com_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_com_Osize_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3605 (all V_z_2 all T_c (class_Rings_Osemiring__1(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_z_2 <-> c_Int_Oiszero(T_c,V_z_2)))) # label(fact_iszero__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3606 (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]. 9.04/9.04 3607 (all V_l_2 all V_k_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3608 (all V_x_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) = hAPP(c_List_Orev(T_c),V_xs_2) <-> V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_rev__singleton__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3609 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (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)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_e),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e)),V_f_2)),V_B_2)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(T_c,T_d),T_e),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)))))) # label(fact_setsum__Sigma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3610 (all V_Ga_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_Ga_2)))))) # label(fact_finite__UnI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3611 (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]. 9.04/9.04 3612 (all V_y all V_x c_Nitpick_Onat__lcm(V_x,V_y) = 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))) # label(fact_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.04 3613 (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]. 9.04/9.04 3614 (all V_x_2 all V_A_2 all T_c (class_Lattices_Odistrib__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2))))),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_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_inf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3615 (all T_c c_Transitive__Closure_Ortrancl(T_c,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) = c_Relation_OId(T_c)) # label(fact_rtrancl__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3616 (all V_v_2 all V_u_2 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)),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)) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2))) # label(fact_intrel__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3617 (all V_c_2 all V_z_2 all T_e all T_d all T_c all V_p_2 ((all B_a all B_b (hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b) = V_p_2 -> hBOOL(hAPP(hAPP(c_member(T_e),V_z_2),hAPP(hAPP(V_c_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(c_member(T_e),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_fun(T_e,tc_HOL_Obool)),V_c_2),V_p_2))))) # label(fact_mem__splitI2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3618 (all V_xs_2 all V_x_2 all V_n_2 all T_c c_Nat_Onat_Onat__case(tc_List_Olist(T_c),c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_c),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Otake(T_c)),V_xs_2)),V_n_2) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_take__Cons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3619 (all T_c (class_Nat_Osemiring__char__0(T_c) -> c_Fun_Oinj__on(tc_Nat_Onat,T_c,c_Nat_Osemiring__1__class_Oof__nat(T_c),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_inj__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3620 (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))) -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys) != V_xs)) # label(fact_impossible__Cons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3621 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) -> (V_x_2 = V_y_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3622 (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_ys),B_i) = hAPP(hAPP(c_List_Onth(T_a),V_xs),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]. 9.04/9.05 3623 (all T_c all V_c_2 all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) -> (V_c_2 = c_Hilbert__Choice_OEps(T_c,V_Pa_2) -> hBOOL(hAPP(V_Pa_2,V_c_2))))) # label(fact_exE__some) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3624 (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]. 9.04/9.05 3625 (all V_r_2 all V_b_2 all V_a_2 all T_c hAPP(c_Relation_OField(T_c),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(c_Relation_OField(T_c),V_r_2))) # label(fact_Field__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3626 (all V_B_2 all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3627 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3628 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,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_d,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_c,T_d,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_Ovimage(T_c,T_d,V_f_2),V_B_2)),V_A_2))))) # label(fact_vimage__subsetI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3629 (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]. 9.04/9.05 3630 (all V_xs_2 all V_c_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))),V_c_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Otimes__class_Otimes(T_c)),V_f_2)),V_c_2)),V_xs_2)))) # label(fact_listsum__mult__const) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3631 (all V_n_2 all V_m_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),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]. 9.04/9.05 3632 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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))) -> hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2) = hAPP(c_Set_Oimage(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),V_A_2))) # label(fact_bij__vimage__eq__inv__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3633 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(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_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_set__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3634 (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]. 9.04/9.05 3635 (all V_A_2 all V_f_2 all T_c hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_c),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_c,tc_Nat_Onat),V_f_2),V_A_2))) # label(fact_int__setsum) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3636 (all 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))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_A_2)))) # label(fact_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3637 (all V_xs_2 all T_c (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_c)),V_xs_2))) <-> c_List_Olist_ONil(T_c) != V_xs_2)) # label(fact_length__greater__0__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3638 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_d,T_e,V_g_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool))) -> c_Fun_Obij__betw(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_g_2),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)))))) # label(fact_bij__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3639 (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]. 9.04/9.05 3640 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_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]. 9.04/9.05 3641 (all V_r_2 all T_c c_Relation_Ototal__on(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),V_r_2)) # label(fact_total__on__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3642 (all V_zs_2 all V_xs_2 all V_ys_2 all T_c (V_ys_2 = V_zs_2 <-> hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_c),V_zs_2),V_xs_2))) # label(fact_append__same__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3643 (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_b),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),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_a = V_b))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3644 (all V_S_2 all V_g_2 all V_f_2 all V_Pa_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBS(T_d,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_d),c_If(T_c)),V_Pa_2)),V_f_2)),V_g_2)),V_S_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_S_2),hAPP(c_Set_OCollect(T_d),V_Pa_2)))),hAPP(c_Set_Oimage(T_d,T_c,V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_S_2),hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d),c_fNot),V_Pa_2)))))) # label(fact_if__image__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3645 (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),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]. 9.04/9.05 3646 (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]. 9.04/9.05 3647 (all V_m 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),c_Groups_Oone__class_Oone(T_a)),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3648 (all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_abs__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3649 (all V_xs_2 all V_a_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> c_List_Olist_ONil(T_c) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_a_2),V_xs_2))) # label(fact_insort__not__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3650 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d))),V_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))))) # label(fact_SUP__UN__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3651 (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))) -> (V_y_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Nitpick_Onat__gcd(V_x_2,V_y_2) = V_x_2) & (V_y_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Nitpick_Onat__gcd(V_x_2,V_y_2) = c_Nitpick_Onat__gcd(V_y_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x_2,V_y_2))))) # label(fact_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3652 (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]. 9.04/9.05 3653 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3654 (all V_j_2 all V_i_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)))) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_i_2),V_j_2)) # label(fact_atLeastLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3655 (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]. 9.04/9.05 3656 (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]. 9.04/9.05 3657 (all V_l_2 all V_k_2 all T_c (class_Orderings_Oorder(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_k_2),V_l_2))))) # label(fact_greaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3658 (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]. 9.04/9.05 3659 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_OremoveAll(T_c,V_x_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_removeAll__filter__not) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3660 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))) # label(fact_minus__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3661 (all V_B_2 all V_f_2 all T_d all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_a_2)),V_B_2)))) # label(fact_vimage__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3662 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,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)))))) = hAPP(c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,T_c),V_A_2),V_x_2))) # label(fact_Sup__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3663 (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]. 9.04/9.05 3664 (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]. 9.04/9.05 3665 (all V_Pa_2 all T_c (class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_c),V_Pa_2)) <-> (all B_x1 hBOOL(hAPP(V_Pa_2,B_x1)))))) # label(fact_enum__all) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3666 (all V_a_2 all V_x_2 all V_A_2 all V_b_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c),V_b_2,V_A_2),V_x_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c),V_a_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_b_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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))),V_x_2))))))) # label(fact_fold__graph__permute__diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3667 (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]. 9.04/9.05 3668 (all V_y_2 all V_x_2 all T_d all T_c -hBOOL(hAPP(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),V_x_2),V_y_2))) # label(fact_bot2E) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3669 (all V_x_2 all V_Pa_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_Pa_2,hAPP(c_Code__Numeral_Oof__nat,B_y))))) -> hBOOL(hAPP(V_Pa_2,V_x_2)))) # label(fact_of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3670 (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_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = 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)))) # label(fact_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3671 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (V_f_2 = c_Fun_Ofun__upd(T_c,T_d,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]. 9.04/9.05 3672 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Olist__all(tc_Int_Oint,V_Pa_2,c_List_Oupto(V_i_2,V_j_2)) <-> c_List_Oall__interval__int(V_Pa_2,V_i_2,V_j_2))) # label(fact_list__all__iff__all__interval__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3673 (all V_k_2 all V_N_2 all T_c (class_Groups_Olinordered__ab__semigroup__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(T_c),V_k_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_N_2))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_k_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_N_2)))))) # label(fact_add__Min__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3674 (all V_f_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hAPP(c_Finite__Set_Ocard(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2) <-> c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2)))) # label(fact_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3675 (all V_ys_2 all V_xs_2 all V_y_2 all V_x_2 all T_d all T_c (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(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) -> -(hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> -hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_List_Oset(T_d),V_ys_2)))))) # label(fact_in__set__zipE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3676 (all T_c c_Relation_Orefl__on(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Relation_OId(T_c))) # label(fact_refl__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3677 (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]. 9.04/9.05 3678 (all V_k_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThan(T_c,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]. 9.04/9.05 3679 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_d,T_c),V_A_2),V_x_2) = c_Complete__Lattice_OInf__class_OInf(T_c,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)))))))) # label(fact_Inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3680 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,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(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_B_2))) # label(fact_image__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3681 (all T_c all V_ys_2 all V_xs_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_c)),hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),c_fequal),B_x)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) <-> V_ys_2 = V_xs_2)) # label(fact_list__eq__iff__zip__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3682 (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_H = V_q)))) # label(fact_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3683 (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]. 9.04/9.05 3684 (all T_a (class_Rings_Osemiring__1(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)))) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_zero__power2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3685 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OMin),V_k)) # label(fact_mult__Min) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3686 (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]. 9.04/9.05 3687 (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]. 9.04/9.05 3688 (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]. 9.04/9.05 3689 (all V_x_2 all V_A_2 all T_c (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(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),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_c),V_A_2))))) # label(fact_card_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3690 (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]. 9.04/9.05 3691 (all V_Pa_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_Pa_2,B_x)))) <-> (all B_x hBOOL(hAPP(V_Pa_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))))) # label(fact_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3692 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2))))) # label(fact_min__max_OInf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3693 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))))) # label(fact_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3694 (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]. 9.04/9.05 3695 (all V_x_2 all V_A_2 all T_c (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(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Finite__Set_Ocard(T_c),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_card__Diff__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3696 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),c_List_OtakeWhile(T_d,V_Pa_2,V_ys_2)) = c_List_OtakeWhile(tc_prod(T_c,T_d),hAPP(c_Fun_Ocomp(T_d,tc_HOL_Obool,tc_prod(T_c,T_d),V_Pa_2),c_Product__Type_Osnd(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2))) # label(fact_zip__takeWhile__snd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3697 (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]. 9.04/9.05 3698 (all V_x_2 all V_A_2 all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2)) -> (c_Fun_Oinj__on(T_e,T_c,V_g_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2)))) -> hAPP(hAPP(c_Fun_Ocomp(T_c,T_e,T_d,c_Hilbert__Choice_Oinv__into(T_e,T_c,V_A_2,V_g_2)),c_Hilbert__Choice_Oinv__into(T_c,T_d,hAPP(c_Set_Oimage(T_e,T_c,V_g_2),V_A_2),V_f_2)),V_x_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_e,T_d,V_A_2,hAPP(c_Fun_Ocomp(T_c,T_d,T_e,V_f_2),V_g_2)),V_x_2))))) # label(fact_inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3699 (all V_ys_2 all V_xs_2 all T_c (c_List_Oconcat(T_c,V_ys_2) = c_List_Oconcat(T_c,V_xs_2) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_ys_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),B_x),hAPP(c_List_Oset(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_List_Ozip(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),B_x)))) -> V_xs_2 = V_ys_2)))) # label(fact_concat__injective) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3700 (all V_r_2 all T_c c_Transitive__Closure_Otrancl(T_c,V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)),V_r_2))) # label(fact_trancl__unfold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3701 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,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)),V_A_2),V_B_2)) & V_B_2 != V_A_2)) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3702 (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]. 9.04/9.05 3703 (all V_com_2 all V_fun_2 all V_loc_2 all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) != c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2)) # label(fact_com_Osimps_I37_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3704 (all V_xs_2 all V_g_2 all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olinorder__class_Osorted(T_c,hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,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]. 9.04/9.05 3705 (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]. 9.04/9.05 3706 (all V_A_2 all T_c all T_d c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_UN__empty2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3707 (all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Olist_ONil(T_d)) = c_List_Olist_ONil(T_c)) # label(fact_map_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3708 (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]. 9.04/9.05 3709 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)))) # label(fact_length__insort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3710 (all T_d 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_b_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))))) # label(fact_range__eqI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3711 (all T_c all V_R_2 ((exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) -> (c_Predicate_Osymp(T_c,V_R_2) -> (c_Predicate_Otransp(T_c,V_R_2) -> c_Equiv__Relations_Opart__equivp(T_c,V_R_2))))) # label(fact_part__equivpI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3712 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))))) # label(fact_INT__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3713 (all V_A_2 all T_c c_Relation_Osingle__valued(T_c,T_c,c_Relation_OId__on(T_c,V_A_2))) # label(fact_single__valued__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3714 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2) = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_min__max_Ole__iff__sup) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3715 (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_b),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) -> V_b = V_c))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3716 (all V_a_2 all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_fun(T_e,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_c),V_a_2),V_A_2))))) # label(fact_INF2__E) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3717 (all T_c c_Relation_Osingle__valued(T_c,T_c,c_Relation_OId(T_c))) # label(fact_single__valued__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3718 (all V_y_2 all V_x_2 all V_z_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_y_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_x_2))))) # label(fact_less__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3719 (all V_R_2 all V_T_2 all V_S_2 all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)),V_S_2),V_T_2)),V_R_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_S_2),V_R_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_T_2),V_R_2))) # label(fact_rel__comp__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3720 (all V_x all T_a (class_Fields_Ofield(T_a) & class_Int_Onumber__ring(T_a) -> 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))) = V_x)) # label(fact_divide__Numeral1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3721 (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 3722 (all V_q all V_p 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)) = c_Divides_Odivmod__int(V_p,V_q)) # label(fact_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3723 (all V_k_2 all V_n_2 all V_Pa_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hBOOL(hAPP(V_Pa_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 (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(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) -> hBOOL(hAPP(V_Pa_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 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),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(V_Pa_2,B_j))))) <-> hBOOL(hAPP(V_Pa_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]. 9.04/9.05 3724 (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]. 9.04/9.05 3725 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> (c_Relation_Oantisym(T_c,V_s_2) -> c_Relation_Oantisym(T_c,V_r_2)))) # label(fact_antisym__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3726 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__1(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),c_Int_Oring__1__class_OInts(T_c))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_b_2)),c_Int_Oring__1__class_OInts(T_c))))))) # label(fact_Ints__add) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3727 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2))))) # label(fact_bij__betw__finite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3728 (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]. 9.04/9.05 3729 (all V_y_2 all V_x_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,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]. 9.04/9.05 3730 (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]. 9.04/9.05 3731 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) & class_Int_Onumber(T_c) -> ((V_c_2 = c_Groups_Ozero__class_Ozero(T_c) -> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = c_Groups_Ozero__class_Ozero(T_c)) & (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))) # label(fact_eq__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3732 (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]. 9.04/9.05 3733 (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]. 9.04/9.05 3734 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),hAPP(c_List_Orev(T_a),V_xs)) = c_List_Olast(T_a,V_xs))) # label(fact_hd__rev) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3735 (all V_l_2 all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_k_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)) & (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_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))) & (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]. 9.04/9.05 3736 (all V_ys all V_xs all T_a 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)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Osplice(T_a,V_xs,V_ys))) # label(fact_length__splice) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3737 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) & hAPP(c_Set_Oimage(T_c,T_d,V_f_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_c,T_d,V_f_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))))) # label(fact_bij__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3738 (all V_a_2 all T_c (class_Int_Oring__char__0(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),c_Int_Oring__1__class_OInts(T_c))) -> c_Groups_Ozero__class_Ozero(T_c) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),c_Groups_Oone__class_Oone(T_c)),V_a_2)),V_a_2)))) # label(fact_Ints__odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3739 (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]. 9.04/9.05 3740 (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]. 9.04/9.05 3741 (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]. 9.04/9.05 3742 (all V_A_2 all V_b_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((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_c),V_b_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)))))) # label(fact_le__Inf__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3743 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),c_Predicate_OPowp(T_c,V_A_2)),c_Predicate_OPowp(T_c,V_B_2))))) # label(fact_Powp__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3744 (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]. 9.04/9.05 3745 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> c_Fun_Obij__betw(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_B_2,V_A_2))) # label(fact_bij__betw__inv__into) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3746 (all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_vimage__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3747 (all V_r_2 all V_A_2 all T_d all V_S_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) -> c_Relation_Orefl__on(T_d,hAPP(V_A_2,B_x),hAPP(V_r_2,B_x)))) -> c_Relation_Orefl__on(T_d,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_S_2),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),V_S_2),V_r_2)))) # label(fact_refl__on__UNION) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3748 (all V_u_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_u_2)) = hAPP(c_Nat_OSuc,V_u_2)) # label(fact_card__atMost) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3749 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) | hBOOL(hAPP(V_Pa_2,V_x_2)) <-> c_List_Olist__ex(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) # label(fact_list__ex__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3750 (all V_A_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(c_COMBK(T_c,T_d),c_Groups_Oone__class_Oone(T_c))),V_A_2) = c_Groups_Oone__class_Oone(T_c))) # label(fact_setprod__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3751 (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]. 9.04/9.05 3752 (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]. 9.04/9.05 3753 (all V_xs_2 all T_c (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremdups(T_c,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) <-> V_xs_2 = c_List_Oremdups(T_c,V_xs_2))) # label(fact_length__remdups__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.05 3754 (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]. 9.04/9.05 3755 (all V_r_2 all V_x_2 all V_y_2 all T_c c_Transitive__Closure_Otrancl(T_c,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),V_r_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,V_r_2)),hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,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_c),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_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),hAPP(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_c),c_member(tc_prod(T_c,T_c))),hAPP(hAPP(c_COMBC(T_c,T_c,tc_prod(T_c,T_c)),c_Product__Type_OPair(T_c,T_c)),V_y_2))),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),hAPP(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_c),c_member(tc_prod(T_c,T_c))),hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2))),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))))) # label(fact_trancl__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3756 (all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) <-> hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3757 (all V_a_2 all T_c 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_c),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))) # label(fact_singleton__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3758 (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]. 9.04/9.06 3759 (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_m_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) # label(fact_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3760 (all V_l_2 all V_k_2 (V_k_2 = V_l_2 <-> c_Int_OBit1(V_l_2) = c_Int_OBit1(V_k_2))) # label(fact_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3761 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3762 (all V_M_2 all T_c all V_A_2 all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),hAPP(c_COMBK(T_d,T_c),V_M_2)) = V_M_2))) # label(fact_INF__const) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3763 (all V_z hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)) = V_z) # label(fact_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3764 (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]. 9.04/9.06 3765 (all V_c_2 all V_b_2 all V_a_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))) = c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_b_2) <-> V_b_2 = V_a_2 & V_b_2 = V_c_2)) # label(fact_dvd_OatLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3766 (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]. 9.04/9.06 3767 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)) = c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2))) # label(fact_listrel__rtrancl__eq__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3768 (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]. 9.04/9.06 3769 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c),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_c) != V_x_2 | V_y_2 != c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3770 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> (V_m != V_n -> 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]. 9.04/9.06 3771 (all V_size_2 all V_g_2 all T_c 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))) -> c_Random__Sequence_ORandom(T_c,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_c,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_c)))),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_c)))),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_c)))),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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_prod(T_c,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_c)))),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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random__Sequence_ORandom(T_c,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_c)))),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c))))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)))),tc_prod(T_c,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_c)))),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_c)))),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_c))))),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_c)))),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_c)))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c))))),tc_prod(T_c,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),c_DSequence_Ounion(T_c)),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_prod(T_c,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm))),c_DSequence_Osingle(T_c)),c_Product__Type_Ofst(T_c,tc_fun(tc_Product__Type_Ounit,tc_Code__Evaluation_Oterm)))))))) & (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_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_c)) = c_Random__Sequence_ORandom(T_c,V_g_2,V_nrandom_2,V_size_2)))) # label(fact_Random_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3772 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> 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_d,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_folding__image__simple_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3773 (all V_g2_2 all V_g1_2 all T_h all V_f2_2 all V_f1_2 all T_g all T_f all T_d all T_e all T_c c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,hAPP(c_Fun_Ocomp(T_g,T_e,T_c,V_f1_2),V_f2_2),hAPP(c_Fun_Ocomp(T_h,T_f,T_d,V_g1_2),V_g2_2)) = hAPP(c_Fun_Ocomp(tc_prod(T_g,T_h),tc_prod(T_e,T_f),tc_prod(T_c,T_d),c_Product__Type_Omap__pair(T_g,T_e,T_h,T_f,V_f1_2,V_g1_2)),c_Product__Type_Omap__pair(T_c,T_g,T_d,T_h,V_f2_2,V_g2_2))) # label(fact_map__pair__compose) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3774 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,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)))))),hAPP(V_f_2,V_x_2))))) # label(fact_folding_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3775 (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]. 9.04/9.06 3776 (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]. 9.04/9.06 3777 (all V_B_2 all V_A_2 all T_e all V_r_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Relation_OImage(T_d,T_c,V_r_2)),V_B_2))) # label(fact_Image__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3778 (all V_A_2 all T_c (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_card__infinite) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3779 (all V_c_2 all T_d all T_c all V_A_2 ((V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_c_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2))) & (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_INT__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3780 (all V_a_2 all V_c_2 all V_b_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2))))) # label(fact_divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3781 (all V_y_2 all V_x_2 all V_c_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Ocurry(T_d,T_e,T_c,V_c_2),V_x_2),V_y_2) = hAPP(V_c_2,hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2))) # label(fact_curry__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3782 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3783 (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))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = V_q))) # label(fact_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3784 (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]. 9.04/9.06 3785 (all V_zs_2 all V_ys_2 all T_d all V_xs_2 all T_c (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) -> (V_xs_2 = hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_c),c_Product__Type_Ofst(T_c,T_d)),V_zs_2) & hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_d),c_Product__Type_Osnd(T_c,T_d)),V_zs_2) = V_ys_2 <-> V_zs_2 = hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) # label(fact_zip__eq__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3786 (all V_xs 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_On__lists(T_a,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_xs)) # label(fact_n__lists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3787 (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),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3788 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) & V_x_2 != V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3789 (all V_y all V_j all V_x all V_i all T_b all T_a 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)) = 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))) # label(fact_zip__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3790 (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))) -> 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))) = 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))))) # label(fact_upt__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3791 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2)) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3792 (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) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = c_Orderings_Otop__class_Otop(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = V_y)))) # label(fact_compl__unique) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3793 (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_number__of__add) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3794 (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]. 9.04/9.06 3795 (all V_A_2 all V_f_2 all T_d all T_c (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3796 (all V_n_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),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_c),V_f_2),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_m_2,V_n_2))))) # label(fact_setsum__set__upt__conv__listsum__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3797 (all V_x_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Olinorder__class_Oinsort__insert__key(T_d,T_c,V_f_2,V_x_2,V_xs_2)))))) # label(fact_sorted__insort__insert__key) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3798 (all V_w all T_b (class_Int_Onumber(T_b) & class_Groups_Omonoid__mult(T_b) -> 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)) = 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)))))) # label(fact_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3799 (all V_B_2 all T_d all V_A_2 all T_c (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)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_Nat_Onat,T_c),c_Finite__Set_Ocard(T_d)),V_B_2)),V_A_2) = hAPP(c_Finite__Set_Ocard(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2))))) # label(fact_card__SigmaI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3800 (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]. 9.04/9.06 3801 (all V_n 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)),V_n) = 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)) # label(fact_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3802 (all V_n_2 all V_m_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3803 (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]. 9.04/9.06 3804 (all V_xs_2 all V_p_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_p_2),V_xs_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_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat),V_p_2),hAPP(c_List_Onth(T_c),V_xs_2)))))) # label(fact_length__filter__conv__card) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3805 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3806 (all V_w all V_z2 all V_z1 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) = 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))) # label(fact_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3807 (all V_n all V_m hAPP(hAPP(c_Orderings_Oord__class_Omax(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_Omax(tc_Nat_Onat),V_m),V_n))) # label(fact_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3808 (all V_k_2 all V_n_2 all V_Pa_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_Pa_2,B_j)))))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> hBOOL(hAPP(V_Pa_2,V_n_2))) <-> hBOOL(hAPP(V_Pa_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]. 9.04/9.06 3809 (all V_v ((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(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = 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))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) # label(fact_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3810 (all V_p_2 all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),V_p_2),V_r_2)) -> c_Wellfounded_OwfP(T_c,V_p_2)))) # label(fact_wfP__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3811 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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))) -> c_Groups_Oone__class_Oone(T_d) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setprod_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3812 (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))) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)) = c_Groups_Osgn__class_Osgn(T_a,V_a)))) # label(fact_sgn__neg) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3813 (all V_a2_2 all V_a1_2 all T_c c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_a1_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_a2_2),c_Com_Ocom_OSKIP),V_a2_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool))))) # label(fact_hoare__derivs_Oequations_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3814 (all V_R_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Wellfounded_Oacc(T_c,V_R_2))) -> -(all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_z),V_x_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),B_z),c_Wellfounded_Oacc(T_c,V_R_2))))))) # label(fact_not__acc__down) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3815 (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)) & (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]. 9.04/9.06 3816 (all V_B_2 all V_A_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_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))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,B_x))))))) # label(fact_INT__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3817 (all V_xs_2 all V_Pa_2 all T_c (c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = V_xs_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_takeWhile__eq__all__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3818 (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]. 9.04/9.06 3819 (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]. 9.04/9.06 3820 (all V_R_2 all V_n_2 all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))))) # label(fact_rel__pow__imp__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3821 (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]. 9.04/9.06 3822 (all V_xs_2 all V_Pa_2 all T_c (c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) != c_List_Olist_ONil(T_c) -> -hBOOL(hAPP(V_Pa_2,hAPP(c_List_Ohd(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))))) # label(fact_hd__dropWhile) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3823 (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]. 9.04/9.06 3824 (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_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_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2) -> 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)))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | V_a_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)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2))))) # label(fact_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3825 (all V_y_2 all V_x_2 all V_r_2 all T_c (c_Predicate_Osymp(T_c,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]. 9.04/9.06 3826 (all T_c (class_Lattices_Osemilattice__inf(T_c) & class_Orderings_Olinorder(T_c) -> c_Orderings_Oord__class_Omin(T_c) = c_Lattices_Osemilattice__inf__class_Oinf(T_c))) # label(fact_inf__min) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3827 (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),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_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_a))) # label(fact_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3828 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) -> V_x_2 = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_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)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,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)))))) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3829 (all V_x_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),B_x)))) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))))) # label(fact_sorted_OCons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3830 (all T_c hAPP(c_Relation_Oinv__image(tc_Nat_Onat,T_c),c_Wellfounded_Oless__than) = c_Wellfounded_Omeasure(T_c)) # label(fact_measure__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3831 (all V_B_2 all V_A_2 all V_C_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),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_OSUPR(T_d,tc_fun(T_c,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]. 9.04/9.06 3832 (all V_z_2 all V_x_2 all V_y_2 all V_w_2 all T_c (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_c) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_w_2),V_y_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_z_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_w_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_y_2)) <-> V_x_2 = V_w_2 | V_z_2 = V_y_2))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3833 (all V_r all V_q 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)) = hAPP(c_Divides_OnegateSnd,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) # label(fact_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3834 (all V_n all V_m (V_n = 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__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3835 (all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) -> c_Wellfounded_OwfP(T_c,c_FunDef_Oin__rel(T_c,T_c,V_R_2)))) # label(fact_wf__in__rel) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3836 (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]. 9.04/9.06 3837 (all V_j_2 all V_i_2 all V_xs_2 all T_c (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),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_c))),c_List_Otranspose(T_c,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_c))),hAPP(c_List_Ofilter(tc_List_Olist(T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),c_List_Otranspose(T_c,V_xs_2)),V_i_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),V_j_2)),V_i_2))))) # label(fact_nth__nth__transpose__sorted) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3838 (all V_B_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_Sigma__empty1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3839 (all V_A_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_List_Olists(T_c,V_A_2)))) # label(fact_lists_ONil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3840 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_a (hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_a))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)))))) # label(fact_finite__SigmaI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3841 (all V_xss_2 all T_c hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),T_c),c_List_Ohd(T_c)),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_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_Olist__case(tc_List_Olist(T_c),T_c,c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_c),c_COMBK(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c))))),V_xss_2))) # label(fact_transpose__aux__filter__head) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3842 (all V_xs_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(c_List_Oset(T_c),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)) = hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_set__sort) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3843 (all V_A_2 all T_c c_Relation_Orefl__on(T_c,V_A_2,c_Relation_OId__on(T_c,V_A_2))) # label(fact_refl__on__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3844 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBS(T_d,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_d),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_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_B_2))) # label(fact_UN__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3845 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3846 (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]. 9.04/9.06 3847 (all T_c c_Relation_Osym(T_c,c_Relation_OId(T_c))) # label(fact_sym__Id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3848 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2))))) # label(fact_finite__Pow__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3849 (all V_Pa_2 (--hBOOL(V_Pa_2) | hBOOL(hAPP(c_fNot,V_Pa_2)))) # label(help_c__fNot__2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3850 (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]. 9.04/9.06 3851 (all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_B_2)) -> (c_Fun_Oinj__on(T_d,T_c,V_g_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_d,T_c,V_g_2),V_B_2)),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hAPP(c_Finite__Set_Ocard(T_d),V_B_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))))))) # label(fact_card__bij__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3852 (all V_f_2 all T_c all T_d c_List_Omaps(T_d,T_c,V_f_2,c_List_Olist_ONil(T_d)) = c_List_Olist_ONil(T_c)) # label(fact_maps__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3853 (all V_a1_2 all V_r_2 all T_d all T_c ((exists B_x1 hBOOL(hAPP(hAPP(V_r_2,V_a1_2),B_x1))) <-> hBOOL(hAPP(c_Predicate_ODomainP(T_c,T_d,V_r_2),V_a1_2)))) # label(fact_DomainP_Oequations) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3854 (all V_f_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_fun(T_d,tc_HOL_Obool),T_c,c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(T_d,tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_Set_Oinsert(T_d)),V_f_2)),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))),V_A_2))) # label(fact_image__eq__fold__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3855 (all V_A_2 all T_c hAPP(c_Relation_ODomain(T_c,T_c),c_Relation_OId__on(T_c,V_A_2)) = V_A_2) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3856 (all T_c hAPP(c_List_Omap(T_c,T_c),c_Fun_Oid(T_c)) = c_Fun_Oid(tc_List_Olist(T_c))) # label(fact_List_Omap_Oid) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3857 (all V_Pa_2 all T_c c_List_OtakeWhile(T_c,V_Pa_2,c_List_Olist_ONil(T_c)) = c_List_Olist_ONil(T_c)) # label(fact_takeWhile_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3858 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3859 (all V_xs_2 all V_n_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2))),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_set__take__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3860 (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]. 9.04/9.06 3861 (all V_B_2 all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_2))) <-> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),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))))),V_B_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_psubset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3862 (all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mod__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3863 (all V_y_2 all V_x_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,c_List_Olist_ONil(tc_fun(T_c,tc_Nat_Onat)))))) # label(fact_in__measures_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3864 (all V_com2_2 all V_com1_2 all V_fun_2 all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) # label(fact_com_Osimps_I53_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3865 (all V_x all V_xs all T_a all V_k ((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,V_xs) = c_List_Obutlast(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_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)))) # label(fact_butlast__list__update) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3866 (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]. 9.04/9.06 3867 (all V_k_2 all V_A_2 all T_c (hAPP(c_Nat_OSuc,V_k_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2) <-> (exists B_b exists B_B ((V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = B_B) & hAPP(c_Finite__Set_Ocard(T_c),B_B) = V_k_2 & -hBOOL(hAPP(hAPP(c_member(T_c),B_b),B_B)) & hAPP(hAPP(c_Set_Oinsert(T_c),B_b),B_B) = V_A_2)))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3868 (all V_x_2 all V_A_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (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(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))))) # label(fact_fold1__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3869 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2)),V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) # label(fact_sum__length__filter__compl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3870 (all V_y_2 all V_x_2 all V_S_2 all V_R_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),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_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)) <-> 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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_inf__Int__eq2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3871 (all V_f2_2 all V_f1_2 all T_d all T_c V_f1_2 = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case(T_c,T_d,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_d))) # label(fact_lazy__sequence_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3872 (all V_A_2 all V_b_2 all V_a_2 all V_r_2 all T_c (hAPP(c_Relation_OImage(T_c,T_c,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_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)))))) # label(fact_eq__equiv__class) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3873 (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]. 9.04/9.06 3874 (all V_r_2 all V_xs_2 all T_c -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),c_List_Olist_ONil(T_c))),c_List_Olistrel1(T_c,V_r_2)))) # label(fact_not__listrel1__Nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3875 (all V_B_2 all V_A_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),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_d),V_b_2),hAPP(V_B_2,V_a_2)))))) # label(fact_SigmaE2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3876 (all V_r_H_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_r_H_2)))) # label(fact_wf__Int1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3877 (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]. 9.04/9.06 3878 (all V_list_H all V_a_H all T_a c_List_Olist_ONil(T_a) != hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H)) # label(fact_list_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3879 (all V_ys_2 all V_xs_2 all T_d all T_c hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_prod(T_c,T_d)),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_prod(T_c,T_d),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_c,T_d)),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_c,T_d)),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_c,T_d),tc_fun(tc_Nat_Onat,tc_prod(T_c,T_d)),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_prod(T_c,T_d)),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_prod(T_c,T_d)),c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_d,tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_prod(T_c,T_d)),tc_Nat_Onat),c_Product__Type_OPair(T_c,T_d)),hAPP(c_List_Onth(T_c),V_xs_2))),hAPP(c_List_Onth(T_d),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_c)),V_xs_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2)))))) = hAPP(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2))) # label(fact_set__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3880 (all V_xs all V_x all T_a 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)) = V_x) # label(fact_nth__Cons__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3881 (all V_Pa_2 all V_xs_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_Ball__set__list__all) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3882 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_List_Oappend(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)) = V_xs_2) # label(fact_takeWhile__dropWhile__id) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3883 (all V_lazy__sequence all V_a all T_a 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))) = 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))) # label(fact_lazy__sequence_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3884 (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]. 9.04/9.06 3885 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_UN__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3886 (all V_fs_2 all T_c c_List_Omeasures(T_c,V_fs_2) = hAPP(hAPP(c_Relation_Oinv__image(tc_List_Olist(tc_Nat_Onat),T_c),c_List_Olex(tc_Nat_Onat,c_Wellfounded_Oless__than)),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(tc_fun(T_c,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_Nat_Onat),tc_Nat_Onat),tc_fun(tc_List_Olist(tc_fun(T_c,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),T_c),c_List_Omap(tc_fun(T_c,tc_Nat_Onat),tc_Nat_Onat)),hAPP(c_COMBC(tc_fun(T_c,tc_Nat_Onat),T_c,tc_Nat_Onat),c_COMBI(tc_fun(T_c,tc_Nat_Onat))))),V_fs_2))) # label(fact_measures__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3887 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)))))) # label(fact_card__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3888 (all V_ts_2 all V_ts_H_2 all V_Ga_2 all T_c (c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_H_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)),V_ts_2),V_ts_H_2)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,V_ts_2)))) # label(fact_weaken) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3889 (all V_n all 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) = V_m) # label(fact_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3890 (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]. 9.04/9.06 3891 (all V_v 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)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = 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))) # label(fact_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3892 (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]. 9.04/9.06 3893 (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]. 9.04/9.06 3894 (all V_n_2 all V_m_2 all V_a_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Oone__class_Oone(T_c)),V_a_2)) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_n_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_m_2) <-> V_m_2 = V_n_2)))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3895 (all V_T_2 all V_S_2 all V_R_2 all T_f all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_f,T_d),V_R_2),hAPP(hAPP(c_Relation_Orel__comp(T_f,T_e,T_d),V_S_2),V_T_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_f,T_e),V_R_2),V_S_2)),V_T_2)) # label(fact_O__assoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.06 3896 (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]. 9.04/9.07 3897 (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_n_2),V_m_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3898 (all V_xs_2 all T_c 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_c)),V_xs_2))) -> hAPP(c_Random_Opick(T_c,hAPP(hAPP(c_List_Omap(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,T_c)),hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_c),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_c),V_xs_2),V_l_2))) # label(fact_pick__same) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3899 (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]. 9.04/9.07 3900 (all V_Pa_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) & hBOOL(hAPP(V_Pa_2,V_x_2)))) # label(fact_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3901 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_x_2),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3902 (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_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3903 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_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_fun__left__comm_Ofold__insert__remove) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3904 (all V_z_2 all V_y_2 all V_f_2 all V_r_2 all T_d all T_c (c_Equiv__Relations_Ocongruent(T_c,T_d,V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_z_2)),V_r_2)) -> hAPP(V_f_2,V_z_2) = hAPP(V_f_2,V_y_2)))) # label(fact_congruentD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3905 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olexord(T_c,V_r_2))) <-> (exists B_i (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(hAPP(c_List_Onth(T_c),V_x_2),B_i)),hAPP(hAPP(c_List_Onth(T_c),V_y_2),B_i))),V_r_2)) & hAPP(hAPP(c_List_Otake(T_c),B_i),V_x_2) = hAPP(hAPP(c_List_Otake(T_c),B_i),V_y_2) & 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_c)),V_x_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_y_2)))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_y_2))) & V_x_2 = hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2)),V_y_2))) # label(fact_lexord__take__index__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3906 (all V_Pa_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_Pa_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_Pa_2,B_x)))))) # label(fact_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3907 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) & c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) & hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) <-> c_Fun_Oinj__on(T_c,T_d,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_inj__on__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3908 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3909 (all V_A_2 all V_B_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2)))) # label(fact_subset__inj__on) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3910 (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]. 9.04/9.07 3911 (all V_D_2 all V_B_2 all V_C_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_D_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)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_D_2)))))) # label(fact_Un__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3912 (all V_a_2 all V_c_2 all V_b_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2))))) # label(fact_divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3913 (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]. 9.04/9.07 3914 (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]. 9.04/9.07 3915 (all V_g_2 all V_f_2 all T_c all T_d all T_e hAPP(c_Fun_Ocomp(tc_List_Olist(T_e),tc_List_Olist(T_d),tc_List_Olist(T_c),hAPP(c_List_Omap(T_e,T_d),V_f_2)),hAPP(c_List_Omap(T_c,T_e),V_g_2)) = hAPP(c_List_Omap(T_c,T_d),hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_2),V_g_2))) # label(fact_List_Omap_Ocomp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3916 (all V_A_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,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_d),c_Set_Oinsert(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)) # label(fact_image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3917 (all V_j_2 all V_i_2 all T_c (class_Rings_Oring__1(T_c) -> hAPP(c_Int_Oring__1__class_Oof__int(T_c),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_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_i_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_j_2)))) # label(fact_of__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3918 (all V_w_2 all V_v_2 all T_c (class_Int_Onumber(T_c) & class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)))))) # label(fact_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3919 (all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,V_R_2) -> -((exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))) -> (c_Predicate_Osymp(T_c,V_R_2) -> -c_Predicate_Otransp(T_c,V_R_2))))) # label(fact_part__equivpE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3920 (all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),c_List_Olist_ONil(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_listsum__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3921 (all V_xs_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_List_Oset(T_c),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_card__length) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3922 (all V_l all V_k c_Int_OBit1(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_OBit1(V_k)),c_Int_OBit0(V_l))) # label(fact_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3923 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,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]. 9.04/9.07 3924 (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_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) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3925 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Oappend(T_c),hAPP(V_f_2,V_x_2)),c_List_Omaps(T_d,T_c,V_f_2,V_xs_2)) = c_List_Omaps(T_d,T_c,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2))) # label(fact_maps__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3926 (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]. 9.04/9.07 3927 (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]. 9.04/9.07 3928 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2) = c_Groups_Oone__class_Oone(T_c) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),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)))))))) # label(fact_eq__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3929 (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]. 9.04/9.07 3930 (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]. 9.04/9.07 3931 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3932 (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]. 9.04/9.07 3933 (all V_l_2 all 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)) <-> -(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))))) # label(fact_dvd_OgreaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3934 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_finite__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3935 (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_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__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))))) # label(fact_le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3936 (all V_x_2 all T_c 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_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)),V_A_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c))) & (-hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)) -> c_List_Osublist(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)),V_A_2) = c_List_Olist_ONil(T_c)))) # label(fact_sublist__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3937 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_OtakeWhile(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2),V_xs_2)) = c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_takeWhile__map) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3938 (all V_A_2 all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Relation_OImage(T_c,T_c,V_R_2),V_A_2))) -> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_wfE__pf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3939 (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(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)))) = 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))) -> 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))))) # label(fact_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3940 (all V_s_2 all V_r_2 all T_e all V_xz_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),V_xz_2),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),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_c,T_d),B_x),B_z) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_e),B_x),B_y)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_e,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_e,T_d),B_y),B_z)),V_s_2))))))) # label(fact_rel__compE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3941 (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]. 9.04/9.07 3942 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> hAPP(c_List_OremoveAll(T_a,V_x),V_xs) = c_List_Oremove1(T_a,V_x,V_xs))) # label(fact_distinct__remove1__removeAll) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3943 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)))))) # label(fact_min__max_OSup__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3944 (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]. 9.04/9.07 3945 (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_OBit0(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_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3946 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Diff__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3947 (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]. 9.04/9.07 3948 (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]. 9.04/9.07 3949 (all V_a_2 all V_f_2 all T_e all V_r2_2 all V_A2_2 all T_d all V_r1_2 all V_A1_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A1_2,V_r1_2) -> (c_Equiv__Relations_Oequiv(T_d,V_A2_2,V_r2_2) -> (c_Equiv__Relations_Ocongruent2(T_c,T_d,tc_fun(T_e,tc_HOL_Obool),V_r1_2,V_r2_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_a_2),V_A2_2)) -> c_Equiv__Relations_Ocongruent(T_c,tc_fun(T_e,tc_HOL_Obool),V_r1_2,hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_e,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_d,V_r2_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))),V_f_2))))))) # label(fact_congruent2__implies__congruent__UN) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3950 (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]. 9.04/9.07 3951 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)),V_A_2)))))) # label(fact_Min__in) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3952 (all V_ys_2 all V_xs_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oremove1(T_c,V_x_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),c_List_Oremove1(T_c,V_x_2,V_ys_2))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oremove1(T_c,V_x_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_List_Oappend(T_c),c_List_Oremove1(T_c,V_x_2,V_xs_2)),V_ys_2)))) # label(fact_remove1__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3953 (all V_h_2 all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_B_2 = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hAPP(V_g_2,B_x) = hAPP(hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2),B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e),V_h_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_e),V_g_2),V_A_2)))))) # label(fact_strong__setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3954 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> V_a_2 = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)))))) # label(fact_min__max_Osup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3955 (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]. 9.04/9.07 3956 (all T_d all T_c c_Fun_Oinj__on(tc_fun(T_c,T_d),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_c,T_d)),c_Set_OCollect(tc_prod(T_c,T_d))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_c,T_d)),c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool)),hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBC(T_d,T_d,tc_HOL_Obool),c_fequal)))),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,T_d),tc_HOL_Obool)))) # label(fact_inj__graph) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3957 (all V_n_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Nat_Onat),V_f_2),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)))) # label(fact_setsum__shift__bounds__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3958 (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]. 9.04/9.07 3959 (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]. 9.04/9.07 3960 (all V_A_2 all V_C_2 all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Un__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3961 (all V_f_2 all V_T_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_S_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_T_2))))))) # label(fact_setsum__Un__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3962 (all V_f_2 all V_n_2 all V_m_2 all T_c (class_Groups_Oab__group__add(T_c) -> (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_c),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(T_c)),V_f_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),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))))) & (-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_c),hAPP(hAPP(c_COMBS(tc_Nat_Onat,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(T_c)),V_f_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,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_c)))) # label(fact_setsum__natinterval__difff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3963 (all V_Qa_2 all V_Pa_2 (hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)) | -hBOOL(V_Pa_2))) # label(help_c__fdisj__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3964 (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]. 9.04/9.07 3965 (all V_R_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_R_2)) <-> c_Wellfounded_Owf(T_c,V_R_2))) # label(fact_wf__comp__self) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3966 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2)) # label(fact_Int__absorb1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3967 (all V_g_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_setsum_Oempty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3968 (all V_x_2 all T_d all T_c (class_Orderings_Obot(T_c) -> c_Orderings_Obot__class_Obot(T_c) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_d,T_c)),V_x_2))) # label(fact_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3969 (all V_y_2 all V_xs_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_y_2))),V_xs_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_set__minus__filter__out) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3970 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3971 (all V_k_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OatMost(T_c,V_less__eq_2,V_k_2))))) # label(fact_ord_OatMost__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3972 (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]. 9.04/9.07 3973 (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]. 9.04/9.07 3974 (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]. 9.04/9.07 3975 (all V_x all V_y all T_a (class_Divides_Oring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_y),V_x)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_y) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_x),V_y))))) # label(fact_dvd__neg__div) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3976 (all V_r_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),V_r_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,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_c),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_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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))),B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x)),V_r_2)))) <-> c_Relation_Orefl__on(T_c,V_A_2,V_r_2))) # label(fact_refl__on__def_H) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3977 (all V_f_2 all V_S_2 all V_T_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_T_2),V_S_2))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_T_2)))))) # label(fact_setsum__mono__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3978 (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]. 9.04/9.07 3979 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_DiffI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3980 (all V_Qa_2 all V_c_2 all V_Ga_2 all T_c all V_Pa_2 ((all B_Z all B_s (hBOOL(hAPP(hAPP(V_Pa_2,B_Z),B_s)) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),hAPP(c_COMBK(tc_fun(tc_Com_Ostate,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBC(tc_Com_Ostate,tc_Com_Ostate,tc_HOL_Obool),c_fequal),B_s))),V_c_2),hAPP(c_COMBK(tc_fun(tc_Com_Ostate,tc_HOL_Obool),T_c),hAPP(V_Qa_2,B_Z)))),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))))) -> c_Hoare__Mirabelle_Ohoare__derivs(T_c,V_Ga_2,hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(T_c)),hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_Pa_2),V_c_2),V_Qa_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(T_c),tc_HOL_Obool)))))) # label(fact_escape) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3981 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (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)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_prod(T_c,T_d),T_e),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_e),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_e),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e),T_c),c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e)),V_f_2)),V_B_2)),V_A_2))))) # label(fact_setprod__Sigma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3982 (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]. 9.04/9.07 3983 (all V_y_2 all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_x_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,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]. 9.04/9.07 3984 (all V_t_2 all V_D_2 all V_d_2 all T_c (class_Rings_Odvd(T_c) & class_Rings_Ocomm__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),V_D_2)) -> (all B_x all B_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),B_k),V_D_2))),V_t_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),B_x),V_t_2)))))))) # label(fact_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3985 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_inj__on__Int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3986 (all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_image__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3987 (all V_z_2 all T_d all T_c V_z_2 = hAPP(c_Product__Type_Omap__pair(T_c,T_c,T_d,T_d,c_COMBI(T_c),c_COMBI(T_d)),V_z_2)) # label(fact_map__pair__ident) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3988 (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(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)))) & (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)))) # label(fact_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3989 (all 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)) = hAPP(c_Int_Onat,V_u_2)) # label(fact_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3990 (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]. 9.04/9.07 3991 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d))),V_r_2)),V_x_2)))) # label(fact_INF__INT__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3992 (all V_A_2 all V_a_2 all T_c hAPP(c_Set_OPow(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)),hAPP(c_Set_Oimage(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oinsert(T_c),V_a_2)),hAPP(c_Set_OPow(T_c),V_A_2)))) # label(fact_Pow__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3993 (all V_Ga_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_Ga_2))))) # label(fact_finite__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3994 (all V_k c_Int_Osucc(c_Int_OBit0(V_k)) = c_Int_OBit1(V_k)) # label(fact_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3995 (all V_r_2 all T_c ((all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_x)),V_r_2)) -> B_y = B_x))) <-> c_Relation_Oantisym(T_c,V_r_2))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3996 (all V_f_2 all V_A_2 all T_d all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) & class_Orderings_Olinorder(T_c) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(V_f_2,B_x))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2),V_f_2)))))) # label(fact_less__SUP__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3997 (all V_B_2 all V_A_2 all T_d all V_b_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_d),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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UN__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3998 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_x_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_y_2)))))) # label(fact_greaterThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 3999 (all V_m_2 all T_d all V_p_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_p_2)) -> hAPP(V_m_2,V_x_2) = hAPP(c_Quotient_OBabs(T_c,T_d,V_p_2,V_m_2),V_x_2))) # label(fact_Babs__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4000 (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]. 9.04/9.07 4001 (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]. 9.04/9.07 4002 (all T_1 class_Nat_Osize(tc_List_Olist(T_1))) # label(arity_List__Olist__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4003 (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]. 9.04/9.07 4004 (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]. 9.04/9.07 4005 (all V_f_2 all T_c hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_f_2) = c_Fun_Oid(T_c)) # label(fact_funpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4006 (all T_e all T_d all V_b_2 all V_a_2 all V_c_2 all V_z_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(hAPP(V_c_2,V_a_2),V_b_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,tc_fun(T_c,tc_HOL_Obool)),V_c_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2)))))) # label(fact_mem__splitI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4007 (all T_c c_Wellfounded_OwfP(T_c,hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBK(tc_HOL_Obool,T_c),c_fFalse)))) # label(fact_wfP__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4008 (all V_r_2 all V_y_2 all T_c ((exists B_a exists B_x V_y_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_a),B_x)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),V_y_2)),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__Nil__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4009 (all V_Pa_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2)),hAPP(c_Product__Type_Osnd(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2))),hAPP(c_Product__Type_Ofst(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_Pa_2))) <-> c_FunDef_Oreduction__pair(T_c,V_Pa_2))) # label(fact_reduction__pair__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4010 (all V_B_2 all V_X_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),c_Product__Type_OSigma(T_c,T_d)),V_B_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_X_2)),V_B_2)) # label(fact_Sigma__Union) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4011 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)),V_a_2)))))) # label(fact_min__max_Ofold1__belowI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4012 (all V_B_2 all T_c all T_d c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))),V_B_2)) # label(fact_INT__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4013 (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(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4014 (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]. 9.04/9.07 4015 (all V_S_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_S_2 -> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_S_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),V_S_2)))))) # label(fact_minus__Min__eq__Max) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4016 (all V_y_2 all V_x_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)))) # label(fact_transfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4017 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_A_2))) # label(fact_INTER__eq__Inter__image) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4018 (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]. 9.04/9.07 4019 (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),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)) = 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)))) # label(fact_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4020 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_y_2 = V_x_2 <-> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_y_2)))) # label(fact_lessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4021 (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]. 9.04/9.07 4022 (all V_Pa_2 all T_c ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))))) # label(fact_bex__UNIV) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4023 (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]. 9.04/9.07 4024 (all V_Pa_2 all T_c (class_Enum_Oenum(T_c) -> ((all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_c),V_Pa_2))))) # label(fact_all__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4025 (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),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4026 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,tc_fun(T_c,tc_HOL_Obool),c_Set_Oinsert(T_c)),V_B_2),V_A_2))) # label(fact_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4027 (all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),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_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_c,V_R_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)),c_Wellfounded_Omax__ext(T_c,V_S_2)),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_HOL_Obool)))))),c_Wellfounded_Omax__ext(T_c,V_R_2))))) # label(fact_max__ext__compat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4028 (all V_d_2 all T_c all V_Pa_2 ((-(exists B_x ((all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_x = B_y)) & hBOOL(hAPP(V_Pa_2,B_x)))) -> V_d_2 = c_FunDef_OTHE__default(T_c,V_d_2,V_Pa_2)) & ((exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & (all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_y = B_x)))) -> c_HOL_OThe(T_c,V_Pa_2) = c_FunDef_OTHE__default(T_c,V_d_2,V_Pa_2)))) # label(fact_THE__default__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4029 (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]. 9.04/9.07 4030 (all V_xs_2 all V_f_2 all T_d all T_c c_List_Oremdups(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = c_List_Oremdups(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Oremdups(T_d,V_xs_2)))) # label(fact_remdups__map__remdups) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4031 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (c_Fun_Oinj__on(T_c,T_d,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))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2))) -> (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__sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 9.04/9.07 4032 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__inf__class_Oinf(T_d)),V_f_2)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_d),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2))))) # label(fact_inf__INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4033 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_List_Oset(T_c),hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_set__rev) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4034 (all V_r_2 all V_A_2 all T_c (hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_quotient__is__empty2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4035 (all V_xq_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_c)),V_xq_2) = c_Option_Ooption_Ooption__case(tc_Nat_Onat,tc_prod(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),hAPP(c_COMBK(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),T_c),hAPP(hAPP(c_COMBC(tc_Lazy__Sequence_Olazy__sequence(T_c),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_c)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_c)))),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(c_Lazy__Sequence_Oyield(T_c),V_xq_2))) # label(fact_size__code) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4036 (all V_R_2 all V_p_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),V_p_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(T_c,T_c),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_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2)))))) # label(fact_rtrancl__imp__UN__rel__pow) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4037 (all V_a_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) # label(fact_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4038 (all V_ys all V_xs all V_n all T_a 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)) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_take__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4039 (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]. 9.04/9.08 4040 (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]. 9.04/9.08 4041 (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(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))) & (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)))) # label(fact_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4042 (all V_j_2 all V_i_2 all V_k_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)) = 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))) # label(fact_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4043 (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]. 9.04/9.08 4044 (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]. 9.04/9.08 4045 (all V_c_2 all T_e all T_d all T_c all V_q_2 all V_p_2 (V_p_2 = V_q_2 -> hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_c_2),V_p_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_c_2),V_q_2))) # label(fact_split__weak__cong) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4046 (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)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_x))) # label(fact_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4047 (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]. 9.04/9.08 4048 (all V_b_2 all V_a_2 all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> 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_a_2)),V_r_2))))) # label(fact_symD) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4049 (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]. 9.04/9.08 4050 (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]. 9.04/9.08 4051 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))))))))) # label(fact_min__max_Osup__Inf1__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4052 (all V_A_2 all V_I_2 all T_d all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2)),V_A_2))) # label(fact_Un__INT__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4053 (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]. 9.04/9.08 4054 (all V_b_2 all V_a_2 all V_A_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_b_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),V_b_2))))))) # label(fact_fold__inf__le__inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4055 (all V_ys_2 all T_d all V_xs_2 all T_c (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) -> hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_d),c_Product__Type_Osnd(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)) = V_ys_2)) # label(fact_map__snd__zip) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4056 (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]. 9.04/9.08 4057 (all V_f_2 all V_S_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_Nat_Onat),V_f_2),V_S_2)))))) # label(fact_setprod__pos__nat) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4058 (all V_Qa_2 all V_Pa_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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(V_Pa_2,B_x)) -> hBOOL(hAPP(V_Qa_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OCollect(T_c),V_Pa_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_OCollect(T_c),V_Qa_2))))))) # label(fact_Int__Collect__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4059 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_x) = 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)))) # label(fact_minus1__divide) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4060 (all V_x_2 all V_xs_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_set__subset__Cons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4061 (all V_i_2 all V_f_2 all V_x_2 all T_c all T_d hAPP(c_New__DSequence_Opos__bind(T_d,T_c,V_x_2,V_f_2),V_i_2) = hAPP(c_Lazy__Sequence_Obind(T_d,T_c,hAPP(V_x_2,V_i_2)),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_c)),V_f_2),V_i_2))) # label(fact_New__DSequence_Opos__bind__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4062 (all V_h_2 all V_B_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__mult(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e),V_h_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_e),hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2)),V_B_2)))) # label(fact_setprod__reindex) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4063 (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]. 9.04/9.08 4064 (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]. 9.04/9.08 4065 (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]. 9.04/9.08 4066 (all V_com2_H_2 all V_com1_H_2 all V_com2_2 all V_com1_2 (c_Com_Ocom_OSemi(V_com1_2,V_com2_2) = c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) <-> V_com2_H_2 = V_com2_2 & V_com1_H_2 = V_com1_2)) # label(fact_com_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4067 (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_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))) & (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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)))) # label(fact_drop__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4068 (all V_N_2 all V_M_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_M_2),V_N_2)) -> (V_M_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_N_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_M_2)))))))) # label(fact_Min__antimono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4069 (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]. 9.04/9.08 4070 (all V_com_H_2 all V_fun_H_2 all V_com2_2 all V_com1_2 all V_fun_2 c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2) != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I52_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4071 (all V_r_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,V_r_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_y)),V_r_2)) -> (all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_z)),V_r_2)) -> B_z = B_y)))))) # label(fact_single__valued__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4072 (all V_z_2 all 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__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)))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4073 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_HOL_Obool)),V_A_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4074 (all V_R_2 all T_c V_R_2 = hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_R_2)) # label(fact_rel__pow__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4075 (all V_y_2 all V_x_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_y_2) = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4076 (all V_i_2 all V_Pa_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_Pa_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]. 9.04/9.08 4077 (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]. 9.04/9.08 4078 (all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(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(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),hAPP(hAPP(c_COMBC(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_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_COMBS(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(T_d,T_c)),c_Product__Type_OPair(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),c_Relation_Oinv__image(T_c,T_d))))),c_Relation_Oinv__image(T_c,T_d))) = c_FunDef_Orp__inv__image(T_c,T_d)) # label(fact_rp__inv__image__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4079 (all V_y_2 all V_x_2 all T_c (V_y_2 != V_x_2 <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_c,hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot)),c_fequal)),V_x_2),V_y_2)))) # label(fact_conversep__noteq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4080 (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),hAPP(c_Nat_OSuc,V_m)),V_n)) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4081 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),V_A_2)))))) # label(fact_Max__in) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4082 (all V_B_2 all V_A_2 all T_e all V_r_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Relation_OImage(T_d,T_c,V_r_2)),V_B_2))))) # label(fact_Image__INT__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4083 (all V_w all V_z 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)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w))) # label(fact_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4084 (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]. 9.04/9.08 4085 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),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_c)),V_xs_2))) -> hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i)),hAPP(hAPP(c_List_Onth(T_d),V_ys_2),B_i))))))) # label(fact_list__all2__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4086 (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)),V_w)) = 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)))) # label(fact_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4087 (all V_A_2 all T_c V_A_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_c,tc_HOL_Obool)),V_A_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_UN__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4088 (all V_x_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) -> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Oremove1(T_d,V_x_2,V_xs_2)))))) # label(fact_sorted__map__remove1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4089 (all V_c_2 all T_d all T_c all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> V_c_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_UN__constant) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4090 (all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,T_e),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),T_e,T_d),V_f_2)),c_Product__Type_OPair(T_c,T_d))) = V_f_2) # label(fact_split__eta) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4091 (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),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]. 9.04/9.08 4092 (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]. 9.04/9.08 4093 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hAPP(V_f_2,V_x_2) = V_y_2 -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_y_2) = V_x_2)))) # label(fact_inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4094 (all V_B_2 all T_d all T_c all V_A_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_d,c_Product__Type_Osnd(T_c,T_d)),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)))) & (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_d,c_Product__Type_Osnd(T_c,T_d)),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))) = V_B_2))) # label(fact_snd__image__times) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4095 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = V_a_2))) # label(fact_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4096 (all T_c 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_c,tc_HOL_Obool)),V_A_2),V_B_2),V_x_2))))) # label(fact_inf1I) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4097 (all V_b_2 all V_a_2 all V_A_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),V_b_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_b_2),V_A_2))))))) # label(fact_sup__le__fold__sup) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4098 (all V_Pa_2 all V_B_2 all V_A_2 all T_c ((exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) | (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_B_2)) & hBOOL(hAPP(V_Pa_2,B_x)))) <-> (exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4099 (all V_a2_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Predicate_ORangeP(T_c,T_d,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]. 9.04/9.08 4100 (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]. 9.04/9.08 4101 (all V_r_2 all T_c all T_d hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),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_r_2))) = hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)) # label(fact_converse__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4102 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Lattices_Osemilattice__inf(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_y_2),V_z_2)))))) # label(fact_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4103 (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]. 9.04/9.08 4104 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),c_List_Olistsp(T_c,V_A_2)),c_List_Olistsp(T_c,V_B_2))))) # label(fact_listsp__mono) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4105 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d 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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_UN__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4106 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4107 (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]. 9.04/9.08 4108 (all V_n_2 all V_m_2 c_List_Oupt(hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),c_List_Oupt(V_m_2,V_n_2))) # label(fact_map__Suc__upt) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4109 (all T_c all V_a_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_a_2)) -> hAPP(c_Set_OCollect(T_c),hAPP(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,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_Pa_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) & (-hBOOL(hAPP(V_Pa_2,V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),hAPP(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,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_Pa_2))))) # label(fact_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4110 (all V_v_2 all V_c_2 all V_b_2 all V_a_2 all T_c all T_d all T_e (hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_a_2),V_b_2) = V_c_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]. 9.04/9.08 4111 (all V_y_2 all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_d,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_r_2)),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2))))) # label(fact_conversep__converse__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4112 (all V_r_2 all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Otimes__class_Otimes(T_c)),V_f_2)),V_r_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),V_r_2))) # label(fact_setsum__left__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4113 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_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))) # label(fact_Int__empty__right) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4114 (all V_xs_2 all V_x_2 all T_c (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) <-> hAPP(c_List_Orev(T_c),V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_singleton__rev__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4115 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2))))) # label(fact_min__max_Ole__sup__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4116 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (exists B_x (hAPP(c_List_Oset(T_c),B_x) = V_A_2 & hBOOL(hAPP(c_List_Odistinct(T_c),B_x)) & (all B_y (hBOOL(hAPP(c_List_Odistinct(T_c),B_y)) & c_List_Olinorder__class_Osorted(T_c,B_y) & V_A_2 = hAPP(c_List_Oset(T_c),B_y) -> B_y = B_x)) & c_List_Olinorder__class_Osorted(T_c,B_x)))))) # label(fact_finite__sorted__distinct__unique) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4117 (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]. 9.04/9.08 4118 (all V_s_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_s_2),V_A_2) = hAPP(hAPP(V_F_2,V_A_2),V_s_2)))) # label(fact_folding_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4119 (all V_A_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,T_c,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_COMBK(tc_Nat_Onat,T_c),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_c),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4120 (all V_B_2 all V_b_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(V_B_2,V_a_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)))))) # label(fact_SigmaI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4121 (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]. 9.04/9.08 4122 (all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_bij__imp__bij__inv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4123 (all T_c c_Fun_Oinj__on(T_c,tc_fun(T_c,tc_HOL_Obool),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))),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_inj__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4124 (all V_f_2 all V_B_H_2 all V_A_H_2 all T_d all V_B_2 all V_A_2 all T_c (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_H_2),V_B_H_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_B_2,V_B_H_2) -> c_Fun_Obij__betw(T_c,T_d,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(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_H_2),V_B_H_2))))))) # label(fact_bij__betw__Disj__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4125 (all V_zs_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) & c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_zs_2)) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_zs_2)))))) # label(fact_sorted__many__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4126 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2))) <-> V_a_2 != c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4127 (all V_A_2 all T_c c_List_Olist_ONil(T_c) = c_List_Osublist(T_c,c_List_Olist_ONil(T_c),V_A_2)) # label(fact_sublist__nil) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4128 (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)) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) & (-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))))) # label(fact_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4129 (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]. 9.04/9.08 4130 (all V_f_2 all T_d all T_c (c_Fun_Oid(T_c) = hAPP(c_Fun_Ocomp(T_d,T_c,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),V_f_2) <-> c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_inj__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4131 (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]. 9.04/9.08 4132 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4133 (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]. 9.04/9.08 4134 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) -> c_Wellfounded_Owf(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_wf__converse__trancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4135 (all V_A_2 all V_C_2 all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Int__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4136 (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)) -> 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),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]. 9.04/9.08 4137 (all V_z_2 all V_y_2 all V_x_2 all V_f_2 all T_d all T_c c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_z_2) = c_Fun_Ofun__upd(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_x_2,V_z_2)) # label(fact_fun__upd__upd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4138 (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_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)))) # label(fact_min__max_Oinf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4139 (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]. 9.04/9.08 4140 (all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2))) -> V_y_2 = V_x_2)) # label(fact_rel__pow__0__E) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4141 (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]. 9.04/9.08 4142 (all T_c (class_Rings_Osemiring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Groups_Oone__class_Oone(T_c)),c_Nat_Osemiring__1__class_ONats(T_c))))) # label(fact_Nats__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4143 (all V_n V_n = hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_n)) # label(fact_nat__size) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4144 (all V_z_2 all 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))) & (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))))) # label(fact_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4145 (all V_ys_2 all V_xs_2 all T_c 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_c,V_r_2,V_xs_2,V_ys_2) -> c_List_Olistrelp(T_c,V_r_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))))) # label(fact_listrelp_OCons) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4146 (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]. 9.04/9.08 4147 (all V_k_2 all V_n_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_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),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))) & 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),V_k_2),B_j)))) -> hBOOL(hAPP(V_Pa_2,B_i))))) & (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 (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),B_j),V_k_2)) & 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_Pa_2,B_i))))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4148 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> V_A_2 = hAPP(c_Set_Oimage(T_d,T_c,c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)))) # label(fact_the__inv__into__onto) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4149 (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]. 9.04/9.08 4150 (all V_r_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),V_r_2) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) # label(fact_Range__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4151 (all V_xs all T_a all V_ys (c_List_Olist_ONil(T_a) != V_ys -> c_List_Olast(T_a,V_ys) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_last__appendR) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4152 (all V_x_2 all T_c (hBOOL(hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_bot__empty__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4153 (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]. 9.04/9.08 4154 (all T_d all V_f_2 all T_c (c_FunDef_Ois__measure(T_c,V_f_2) -> c_FunDef_Ois__measure(tc_prod(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,tc_Nat_Onat,tc_prod(T_d,T_c)),V_f_2),c_Product__Type_Osnd(T_d,T_c))))) # label(fact_measure__snd) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4155 (all V_Pa_2 all T_c all T_d (class_Enum_Oenum(T_d) & class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_Enum_Oenum__class_Oenum__all(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,T_d),V_Pa_2)),c_Product__Type_OPair(T_c,T_d))))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(tc_prod(T_c,T_d)),V_Pa_2))))) # label(fact_enum__all__prod__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4156 (all V_n all T_a (class_Int_Onumber__ring(T_a) -> c_Groups_Oone__class_Oone(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)))) # label(fact_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4157 (all V_w_2 all T_c (class_Int_Onumber__ring(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Int_Oring__1__class_OInts(T_c))))) # label(fact_Ints__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4158 (all V_A_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_A_2))) -> (V_b_2 != V_a_2 -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2))))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4159 (all V_a_2 all V_r_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,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_d,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4160 (all V_A_2 all V_f_2 all T_d all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_member(T_d),hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,V_f_2),V_x_2)),V_A_2)))) # label(fact_inv__into__into) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4161 (all V_y all V_ys all V_x all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys)) = 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)) # label(fact_list__update__length) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4162 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) <-> c_List_Oremdups(T_c,V_xs_2) = V_xs_2)) # label(fact_remdups__id__iff__distinct) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4163 (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]. 9.04/9.08 4164 (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]. 9.04/9.08 4165 (all V_p_2 all V_f_2 all T_c all T_e all T_d hAPP(hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_p_2)),hAPP(c_Product__Type_Osnd(T_d,T_e),V_p_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2),V_p_2)) # label(fact_prod__case__beta) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4166 (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]. 9.04/9.08 4167 (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_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_Y_2)) -> (V_Y_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_x_2),V_y_2)),V_r_2))))))))) # label(fact_quotient__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4168 (all V_A_2 all V_f_2 all T_c hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(c_Int_Onat,hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_c),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2))) # label(fact_transfer__nat__int__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.08 4169 (all V_b_2 all V_a_2 all V_fb_2 all V_fa_2 all T_d all T_c c_Nitpick_Opair__box_Opair__box__size(T_c,T_d,V_fa_2,V_fb_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,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]. 9.04/9.09 4170 (all V_ys all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Orev(T_a),V_ys)),hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_rev__append) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4171 (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]. 9.04/9.09 4172 (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)) <-> V_i_2 != hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_code__numeral__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4173 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> 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__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2))))) # label(fact_setsum__abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4174 (all V_y all V_x 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)) = 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))) # label(fact_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4175 (all V_c_2 all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_b_2,V_a_2)),hAPP(hAPP(c_Power_Opower__class_Opower(T_d),V_c_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_S_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),V_c_2)),V_S_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),V_c_2)),V_S_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_d),V_c_2),hAPP(c_Finite__Set_Ocard(T_c),V_S_2)))))) # label(fact_setprod__gen__delta) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4176 (all T_c all V_R_2 all T_d all V_Pa_2 ((all B_x (hBOOL(hAPP(V_Pa_2,B_x)) -> c_Wellfounded_Owf(T_d,hAPP(V_R_2,B_x)))) -> c_Wellfounded_Owf(tc_prod(T_c,T_d),c_Recdef_Osame__fst(T_c,T_d,V_Pa_2,V_R_2)))) # label(fact_wf__same__fst) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4177 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4178 (all V_A_2 all T_c V_A_2 = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))) # label(fact_Collect__mem__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4179 (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]. 9.04/9.09 4180 (all V_ys_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = V_xs_2 & V_ys_2 = c_List_Olist_ONil(T_c))) # label(fact_Nil__is__append__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4181 (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]. 9.04/9.09 4182 (all V_w_2 all V_b_2 all V_a_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))))) # label(fact_less__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4183 (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]. 9.04/9.09 4184 (all V_n all V_i c_Int_Onat__aux(V_i,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_i)),V_n)) # label(fact_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4185 (all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,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]. 9.04/9.09 4186 (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]. 9.04/9.09 4187 (all V_B_2 all V_A_2 all T_c (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)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_B_2)) -> B_x != B_xa)))))) # label(fact_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4188 (all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,T_d)),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,T_d),tc_HOL_Obool)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_finite__fun__UNIVD2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4189 (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]. 9.04/9.09 4190 (all V_i_2 all V_x_2 all T_c c_List_Oreplicate(T_c,V_i_2,V_x_2) = hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),hAPP(c_COMBK(T_c,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]. 9.04/9.09 4191 (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]. 9.04/9.09 4192 (all T_c c_Relation_Otrans(tc_fun(T_c,tc_HOL_Obool),c_Wellfounded_Ofinite__psubset(T_c))) # label(fact_trans__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4193 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_c),V_B_2))))) # label(fact_PowI) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4194 (all V_a_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)) = V_a_2)))) # label(fact_sup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4195 (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]. 9.04/9.09 4196 (all V_g_2 all V_f_2 all T_d all V_A_2 all V_a_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,V_A_2),V_a_2))) # label(fact_override__on__apply__notin) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4197 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_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,V_x_2),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4198 (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),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4199 (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]. 9.04/9.09 4200 (all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Relation_OId(T_c))) <-> V_b_2 = V_a_2)) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4201 (all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_l_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_l_2))) # label(fact_ivl__disj__un_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4202 (all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> c_List_OtakeWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),c_List_Olast(T_c,V_xs_2))),V_xs_2) = c_List_Obutlast(T_c,V_xs_2)))) # label(fact_takeWhile__not__last) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4203 (all V_x all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Oreplicate(T_a,V_n,V_x)) = V_n) # label(fact_length__replicate) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4204 (all V_x_2 all V_f_2 all V_A_2 all T_d all T_c c_Hilbert__Choice_OEps(T_c,hAPP(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(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_x_2))) = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_x_2)) # label(fact_inv__into__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4205 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)))))) # label(fact_listrel__reflcl__if__listrel1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4206 (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)) -> V_y = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4207 (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]. 9.04/9.09 4208 (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]. 9.04/9.09 4209 (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]. 9.04/9.09 4210 (all V_a_2 all T_c c_HOL_OThe(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2)) = V_a_2) # label(fact_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4211 (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_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> 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_d,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_folding__image__simple__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4212 (all V_r_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),c_List_Olexn(T_c,V_r_2)) = c_List_Olex(T_c,V_r_2)) # label(fact_lex__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4213 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_c))),c_List_Olists(T_c,V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_c))),c_List_Olists(T_c,V_B_2))) = c_List_Olistsp(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),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_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2)))) # label(fact_lists__Int__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4214 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))))) # label(fact_less__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4215 (all V_p_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_p_2),V_r_2)) -> c_Wellfounded_Owf(T_c,V_p_2)))) # label(fact_wf__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4216 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2))) <-> (exists B_f (c_Fun_Oinj__on(T_c,T_d,B_f,V_A_2) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,B_f),V_A_2)),V_B_2)))))))) # label(fact_inj__on__iff__card__le) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4217 (all V_z_2 all V_y_2 all V_A_2 all V_b_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c),V_b_2,V_A_2),V_y_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c),V_z_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_A_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_z_2),V_y_2))))))) # label(fact_fold__graph__insert__swap) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4218 (all V_A_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_A_2)) # label(fact_empty__Diff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4219 (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]. 9.04/9.09 4220 (all V_x_2 all V_n_2 all T_c (c_List_Oreplicate(T_c,V_n_2,V_x_2) = c_List_Olist_ONil(T_c) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_replicate__empty) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4221 (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]. 9.04/9.09 4222 (all V_m all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = 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)))) # label(fact_plus__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4223 (all V_xs_2 all V_f_2 all T_d all T_c c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(T_d,tc_List_Olist(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_d),c_List_Olist_OCons(T_c)),V_f_2)),c_List_Olist_ONil(T_c))),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) # label(fact_concat__map__singleton) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4224 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal)),hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))))))))) # label(fact_min__max_Oinf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4225 (all V_x_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oremdups(T_c,V_x_2) <-> V_x_2 = c_List_Olist_ONil(T_c))) # label(fact_remdups__eq__nil__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4226 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4227 (all V_x_2 all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2))) # label(fact_fst__apfst) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4228 (all V_k (V_k != hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> 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]. 9.04/9.09 4229 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> (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_c_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_trancl__rtrancl__trancl) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4230 (all V_s_2 all V_r_2 all V_I_2 all T_f all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)),V_I_2),V_r_2)),V_s_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBC(T_f,tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_f),c_Relation_Orel__comp(T_c,T_e,T_d)),V_r_2)),V_s_2))) # label(fact_rel__comp__UNION__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4231 (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]. 9.04/9.09 4232 (all V_A_2 all V_x_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)))))) # label(fact_Sup__upper) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4233 (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]. 9.04/9.09 4234 (all V_ys_2 all V_g_2 all T_e all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_e,T_c),V_g_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_e)),V_ys_2))) # label(fact_map__eq__imp__length__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4235 (all V_Pa_2 all V_B_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(V_Pa_2,B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_B_2)) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_ball__Un) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4236 (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 (hAPP(V_f_2,B_i) = V_k_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)))))))) # label(fact_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4237 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_r_2),V_r_2)),V_r_2)))) # label(fact_trans__O__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4238 (all T_c (class_Rings_Oring__1(T_c) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Groups_Oone__class_Oone(T_c)),c_Int_Oring__1__class_OInts(T_c))))) # label(fact_Ints__1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4239 (all V_d_2 all T_c (class_Lazy__Sequence_Osmall__lazy(T_c) -> c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_List_Olist(T_c),V_d_2) = c_Lazy__Sequence_Oappend(tc_List_Olist(T_c),hAPP(c_Lazy__Sequence_Osingle(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),hAPP(hAPP(hAPP(c_If(tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_c))),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_c,tc_List_Olist(T_c)),tc_List_Olist(T_c),c_Lazy__Sequence_Oproduct(T_c,tc_List_Olist(T_c),c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_c,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_c),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_c,tc_List_Olist(T_c),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_c))),T_c),hAPP(c_COMBB(tc_List_Olist(T_c),tc_Lazy__Sequence_Olazy__sequence(tc_List_Olist(T_c)),tc_List_Olist(T_c)),c_Lazy__Sequence_Osingle(tc_List_Olist(T_c)))),c_List_Olist_OCons(T_c))))),c_Lazy__Sequence_Oempty(tc_List_Olist(T_c)))))) # label(fact_small__lazy__list_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4240 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 9.04/9.09 4241 (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_a != V_b -> 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]. 9.04/9.09 4242 (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]. 9.04/9.09 4243 (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]. 9.04/9.09 4244 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> 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]. 9.04/9.09 4245 (all V_t_2 all V_D_2 all V_d_2 all T_c (class_Rings_Ocomm__ring(T_c) & class_Rings_Odvd(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),V_D_2)) -> (all B_x all B_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),B_x),V_t_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),B_k),V_D_2))),V_t_2)))))))) # label(fact_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4246 (all V_f_2 all V_A_2 all V_B_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) & class_Groups_Oordered__ab__semigroup__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2)))))))) # label(fact_setsum__mono3) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4247 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (V_x_2 = c_Groups_Ozero__class_Ozero(T_c) & c_Groups_Ozero__class_Ozero(T_c) = V_y_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),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_c),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_c)))) # label(fact_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4248 (all V_zs_2 all V_ys_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_zs_2) <-> V_ys_2 = V_zs_2)) # label(fact_same__append__eq) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4249 (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_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4250 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2))))) # label(fact_atLeastatMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4251 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)))) # label(fact_ivl__disj__un_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4252 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_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_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4253 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> V_x = 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)))) # label(fact_sup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4254 (all T_c (class_Orderings_Olinorder(T_c) -> c_Orderings_Oord_Omax(T_c,hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_c))) = c_Orderings_Oord__class_Omin(T_c))) # label(fact_dual__max) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4255 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)),hAPP(c_Set_OPow(T_c),V_B_2))),hAPP(c_Set_OPow(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Un__Pow__subset) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4256 (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)) -> 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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))))) # label(fact_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4257 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_INT__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4258 (all V_xs all T_a all V_y all V_x ((V_y != V_x -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Oremove1(T_a,V_x,V_xs)) = c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),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]. 9.04/9.09 4259 (all V_b_2 all V_a_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,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_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2))))) # label(fact_equiv__class__nondisjoint) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4260 (all V_z_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2)))))) # label(fact_fun__left__comm_Ofold__graph__fold) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4261 (all V_f_2 all V_x_2 all T_d all T_e all T_c hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_e,T_c,T_d),hAPP(c_Product__Type_OPair(T_e,T_c),V_x_2)),V_f_2) = hAPP(V_f_2,V_x_2)) # label(fact_Pair__scomp) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4262 (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_Int_Oint),V_x_2),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))))))) # label(fact_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4263 (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]. 9.04/9.09 4264 (all V_n hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n)) # label(fact_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4265 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_UnI1) # label(axiom) # label(non_clause). [assumption]. 9.04/9.09 4266 (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]. 9.04/9.09 4267 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)))))) # label(fact_Min__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4268 (all V_B_2 all V_A_2 all T_c all T_d hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OAll(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fimplies),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_d)),c_member(T_c))),V_B_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_INTER__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4269 (all V_Pa_2 all V_xs_2 all T_c 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_c)),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),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_c)),V_xs_2))) -> -hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_n_2)))) -> c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2)))) # label(fact_takeWhile__eq__take__P__nth) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4270 (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_Omin(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__max_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4271 (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]. 9.15/9.09 4272 (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,V_a,V_a))) # label(fact_mod__self) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4273 (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),V_xs) = hAPP(hAPP(c_List_Otake(T_a),V_n),c_List_Obutlast(T_a,V_xs)))) # label(fact_take__butlast) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4274 (all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_ComplE) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4275 (all T_d all T_c 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_c,T_d,V_r_2),V_b_2),V_a_2)))) # label(fact_conversep_Ointros) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4276 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_x)),V_r_2)))))) # label(fact_sym__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4277 (all V_g_2 all V_f_2 all T_d all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_g_2,V_a_2) = hAPP(c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,V_A_2),V_a_2))) # label(fact_override__on__apply__in) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4278 (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),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4279 (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]. 9.15/9.09 4280 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ocard(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),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_c),V_A_2)))) # label(fact_card__Pow) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4281 (all V_l_2 all V_k_2 (V_l_2 = V_k_2 <-> c_Int_OBit0(V_l_2) = c_Int_OBit0(V_k_2))) # label(fact_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4282 (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]. 9.15/9.09 4283 (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_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4284 (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]. 9.15/9.09 4285 (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]. 9.15/9.09 4286 (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]. 9.15/9.09 4287 (all V_xs_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremove1(T_c,V_x_2,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremove1(T_c,V_x_2,V_xs_2))))) # label(fact_length__remove1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4288 (all V_b all V_a ((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(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),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(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]. 9.15/9.09 4289 (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),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_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4290 (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]. 9.15/9.09 4291 (all V_w_2 all V_b_2 all V_a_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))))) # label(fact_le__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4292 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_B_2))))))) # label(fact_setsum__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4293 (all V_g_2 all T_c all T_d hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),V_g_2) = V_g_2) # label(fact_id__o) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4294 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Rings_Odivision__ring(T_c) -> (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2) = V_a_2 <-> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))) # label(fact_nonzero__divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4295 (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]. 9.15/9.09 4296 (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]. 9.15/9.09 4297 (all V_r_2 all T_c (c_Predicate_Osymp(T_c,V_r_2) <-> c_Relation_Osym(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))))) # label(fact_symp__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4298 (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]. 9.15/9.09 4299 (all V_x_2 all V_a_2 all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_x_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_a_2)),V_r_2))))) # label(fact_wf__asym) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4300 (all V_g_2 all V_f_2 ((all B_x hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)) -> V_g_2 = V_f_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4301 (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]. 9.15/9.09 4302 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_List_Oset(T_d),V_xs_2))) # label(fact_set__map) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4303 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_d),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_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))))))) # label(fact_folding__image__simple_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4304 (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_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4305 (all V_c_2 all V_R_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_R_2)) -> (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_c_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_R_2)))))) # label(fact_r__r__into__trancl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4306 (all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I16_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4307 (all V_i_2 all V_B_2 all V_f_2 all V_s_2 all T_c all T_d (class_Groups_Oordered__ab__group__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_s_2)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_i))))) -> (hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_s_2) = V_B_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,V_i_2)),V_B_2)))))))) # label(fact_setsum__nonneg__leq__bound) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4308 (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) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & 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(tc_Int_Oint),V_i_2),V_k_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2)) | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4309 (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]. 9.15/9.09 4310 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4311 (all V_xs_2 all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_xs_2))),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel1I1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.09 4312 (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]. 9.15/9.09 4313 (all V_x_2 all V_Pa_2 all V_xs_2 all T_c ((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_c)),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i))))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,V_x_2))))) # label(fact_all__nth__imp__all__set) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4314 (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]. 9.15/9.10 4315 (all V_xs_2 all V_a_2 all T_c (class_Groups_Omonoid__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),c_List_Ofoldr(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_xs_2,c_Groups_Ozero__class_Ozero(T_c))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_xs_2))) # label(fact_foldl__foldr1__lemma) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4316 (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_d all T_c (c_Finite__Set_Ofolding__image(T_c,T_d,V_f_2,V_z_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),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]. 9.15/9.10 4317 (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_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_d),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)) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_d),V_x_2),V_A_2)))))) # label(fact_folding__image__simple_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4318 (all V_g_2 all V_f_2 all T_f all T_e all V_R_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_R_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_e,T_f)),hAPP(hAPP(c_Product__Type_OPair(T_e,T_f),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2))),hAPP(c_Set_Oimage(tc_prod(T_c,T_d),tc_prod(T_e,T_f),c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,V_f_2,V_g_2)),V_R_2))))) # label(fact_map__pair__imageI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4319 (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]. 9.15/9.10 4320 (all V_n_2 all V_m_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_m_2)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_m_2),V_n_2))))) # label(fact_lessThan__strict__subset__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4321 (all V_g_2 all V_f_2 all V_B_2 all V_A_2 all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> (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_g_2,B_x) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),V_g_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),V_f_2),V_A_2)))))) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4322 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_a_2 -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2) = V_b_2 <-> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_b_2),V_a_2) = V_c_2))))) # label(fact_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4323 (all V_f_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c),B_xa),V_A_2)) -> (B_xa != B_x -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,B_xa)))))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4324 (all V_c 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),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__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4325 (all V_n_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,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)))) # label(fact_lessThan__Suc__eq__insert__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4326 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)) = V_x)) # label(fact_inf__top__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4327 (all T_e all V_f_2 all V_Qa_2 all V_p_2 all T_c all T_d all V_Pa_2 (hBOOL(hAPP(hAPP(V_Pa_2,hAPP(c_Product__Type_Osnd(T_d,T_c),V_p_2)),hAPP(c_Product__Type_Ofst(T_d,T_c),V_p_2))) -> ((all B_x all B_y (hBOOL(hAPP(hAPP(V_Pa_2,B_y),B_x)) -> hBOOL(hAPP(V_Qa_2,hAPP(hAPP(V_f_2,B_x),B_y))))) -> hBOOL(hAPP(V_Qa_2,hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,T_e),V_f_2),V_p_2)))))) # label(fact_exE__realizer) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4328 (all V_B_2 all V_g_2 all T_e all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),tc_fun(tc_fun(T_e,tc_HOL_Obool),T_c),T_d),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,T_c),tc_fun(T_e,T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,T_c),tc_fun(tc_fun(T_e,T_c),tc_fun(T_e,T_c)),T_d),c_COMBB(T_c,T_c,T_e)),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Otimes__class_Otimes(T_c)),V_f_2))),V_g_2))),V_B_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_c),V_g_2),V_B_2)))) # label(fact_setsum__product) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4329 (all V_n hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n) # label(fact_max__0R) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4330 (all V_x_2 all V_n_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oreplicate(T_c,V_n_2,V_x_2) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_empty__replicate) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4331 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> V_a_2 = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_a_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)))))) # label(fact_min__max_Oinf__Sup__absorb) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4332 (all V_l c_Int_OBit1(V_l) != c_Int_OPls) # label(fact_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4333 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2)))))) # label(fact_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4334 (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]. 9.15/9.10 4335 (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]. 9.15/9.10 4336 (all V_xs_2 all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2)) = hAPP(c_List_OremoveAll(T_d,hAPP(V_f_2,V_x_2)),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)))) # label(fact_map__removeAll__inj__on) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4337 (all V_A_2 all T_c hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),c_List_Olist_ONil(T_c)))) # label(fact_listsp_Oequations_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4338 (all V_r_2 all V_a_2 all T_c hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2)))) # label(fact_rtrancl_Ortrancl__refl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4339 (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]. 9.15/9.10 4340 (all V_lst_2 all V_k_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_COMBK(T_c,T_d),V_k_2)),V_lst_2) = c_List_Oreplicate(T_c,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_lst_2),V_k_2)) # label(fact_map__replicate__const) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4341 (all V_c_2 all V_r_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) -> (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_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),c_Transitive__Closure_Otrancl(T_c,V_r_2)))))) # label(fact_Transitive__Closure_Otrancl__into__trancl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4342 (all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2))) -> B_x = c_List_Olist_ONil(T_c))) <-> c_List_Olist_ONil(tc_List_Olist(T_c)) = c_List_Otranspose(T_c,V_xs_2))) # label(fact_transpose__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4343 (all V_B_2 all V_A_2 all V_C_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,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_d),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_OINFI(T_d,tc_fun(T_c,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]. 9.15/9.10 4344 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) = c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) & (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) # label(fact_dropWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4345 (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) = 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))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2)))) # label(fact_simp__from__to) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4346 (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]. 9.15/9.10 4347 (all V_n all 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) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4348 (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]. 9.15/9.10 4349 (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]. 9.15/9.10 4350 (all V_x_2 all T_c c_List_OremoveAll(T_c,V_x_2) = c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(c_fequal,V_x_2)))) # label(fact_removeAll__filter__not__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4351 (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]. 9.15/9.10 4352 (all V_n all V_m all V_k 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)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4353 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_y_2 = hAPP(V_f_2,V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Fun_Othe__inv__into(T_c,T_d,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]. 9.15/9.10 4354 (all V_v_2 all V_A_2 all V_x_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_v_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),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_c,T_d,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]. 9.15/9.10 4355 (all V_z_2 all V_y_2 all V_x_2 all V_r_2 all T_c (c_Relation_Osingle__valued(T_c,T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_z_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_z_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))))) # label(fact_single__valued__confluent) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4356 (all V_l_2 (c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),V_l_2)) <-> c_List_Onat__list(V_l_2))) # label(fact_nat__list__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4357 (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]. 9.15/9.10 4358 (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]. 9.15/9.10 4359 (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]. 9.15/9.10 4360 (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]. 9.15/9.10 4361 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_y_2))) # label(fact_rtrancl__listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4362 (all V_A_2 all T_c hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,T_c),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)) # label(fact_card__eq__setsum) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4363 (all V_n_2 all V_m_2 (V_m_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__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4364 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,V_r_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_xa != B_x -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_xa)),V_r_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_xa),B_x)),V_r_2))))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4365 (all V_Pa_2 all T_c all T_d (class_Enum_Oenum(T_d) & class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_Enum_Oenum__class_Oenum__ex(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,T_d),V_Pa_2)),c_Product__Type_OPair(T_c,T_d))))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(tc_prod(T_c,T_d)),V_Pa_2))))) # label(fact_enum__ex__prod__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4366 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),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_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice__class_OSup__fin(T_c,V_N_2))))))) # label(fact_Sup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4367 (all V_f_2 all V_g_2 all T_e all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_e,V_f_2),V_A_2)),V_g_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Fun_Ocomp(T_e,tc_fun(T_c,tc_HOL_Obool),T_d,V_g_2),V_f_2))) # label(fact_UN__o) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4368 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_k_2)))) # label(fact_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4369 (all T_d all T_c 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_c,tc_fun(T_d,tc_HOL_Obool))),V_A_2),V_B_2),V_x_2),V_y_2)))) # label(fact_sup2CI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4370 (all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(T_c,V_A_2,V_r_2) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)),V_r_2))))) # label(fact_refl__on__comp__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4371 (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]. 9.15/9.10 4372 (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]. 9.15/9.10 4373 (all V_r_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_ORange(T_d,T_c),V_r_2))) -> -(all B_x -hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_x),V_b_2)),V_r_2))))) # label(fact_RangeE) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4374 (all T_e all T_d all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_fun(T_d,tc_fun(T_e,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]. 9.15/9.10 4375 (all V_B_2 all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),hAPP(hAPP(c_COMBS(T_c,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_c),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_B_2))) # label(fact_Sigma__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4376 (all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) <-> hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_x_2)))) # label(fact_List_Omember__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4377 (all V_B_2 all V_A_2 all V_g_2 all V_f_2 all T_d all T_c all T_e (class_Rings_Osemiring__0(T_e) -> (c_Fun_Oinj__on(tc_prod(T_c,T_d),T_e,hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_e),tc_fun(T_d,T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_e),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,T_e)),T_c),c_COMBB(T_e,T_e,T_d)),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_e,T_e),T_c),c_Groups_Otimes__class_Otimes(T_e)),V_f_2))),V_g_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_e,T_e),c_Fun_Oid(T_e)),hAPP(c_Set_OCollect(T_e),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_e),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_e),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_e,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(T_c,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_e),c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool))),T_e),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_c),c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_e),hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,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_d),c_fconj))),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_c,tc_fun(T_d,T_e)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_d,T_e)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),T_e),c_COMBB(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool)),T_e),c_COMBB(T_e,tc_HOL_Obool,T_d)),c_fequal))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_e),tc_fun(T_d,T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_e),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,T_e)),T_c),c_COMBB(T_e,T_e,T_d)),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_e,T_e),T_c),c_Groups_Otimes__class_Otimes(T_e)),V_f_2))),V_g_2)))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(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_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),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(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_e),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_g_2),V_B_2))))) # label(fact_setsum__mult__setsum__if__inj) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4378 (all V_x_2 all V_y_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_x_2)) -> (V_x_2 = V_y_2 <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4379 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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_2),V_B_2) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4380 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4381 (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]. 9.15/9.10 4382 (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]. 9.15/9.10 4383 (all V_code__numeral_2 all V_f2_2 all V_f1_2 all T_c hAPP(V_f2_2,V_code__numeral_2) = c_Code__Numeral_Ocode__numeral_Ocode__numeral__case(T_c,V_f1_2,V_f2_2,c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2))) # label(fact_code__numeral_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4384 (all V_Pa_2 all T_c hAPP(c_Set_OCollect(T_c),V_Pa_2) = V_Pa_2) # label(fact_Collect__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4385 (all V_S_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),V_S_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_Relation_ORange(T_d,T_c))) # label(fact_Range__Union) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4386 (all V_B_H_2 all V_B_2 all V_g_2 all T_e all T_f all V_A_H_2 all V_A_2 all V_f_2 all T_c all T_d (V_A_H_2 = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) -> (V_B_H_2 = hAPP(c_Set_Oimage(T_f,T_e,V_g_2),V_B_2) -> hAPP(hAPP(c_Product__Type_OSigma(T_c,T_e),V_A_H_2),hAPP(c_COMBK(tc_fun(T_e,tc_HOL_Obool),T_c),V_B_H_2)) = hAPP(c_Set_Oimage(tc_prod(T_d,T_f),tc_prod(T_c,T_e),c_Product__Type_Omap__pair(T_d,T_c,T_f,T_e,V_f_2,V_g_2)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_f),V_A_2),hAPP(c_COMBK(tc_fun(T_f,tc_HOL_Obool),T_d),V_B_2)))))) # label(fact_map__pair__surj__on) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4387 (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]. 9.15/9.10 4388 (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(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4389 (all V_r_2 all T_c c_Relation_Osym(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_sym__Int__converse) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4390 (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_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_I16_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4391 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)))) = 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))) # label(fact_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4392 (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_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z))) # label(fact_mult__minus1__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4393 (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]. 9.15/9.10 4394 (all V_z_2 all V_h_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Oab__semigroup__mult(T_d) -> (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(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_g_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_h_2),V_z_2),V_A_2))))) # label(fact_fold__image__cong) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4395 (all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_a_2 = c_List_Olist_ONil(T_c) | (exists B_xs exists B_x (V_a_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) & c_List_Olinorder__class_Osorted(T_c,B_xs) & (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),hAPP(c_List_Oset(T_c),B_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_x),B_xa)))))) <-> c_List_Olinorder__class_Osorted(T_c,V_a_2)))) # label(fact_sorted_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4396 (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_x = V_a)) # label(fact_fst__eqD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4397 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4398 (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),V_b),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))))) # label(fact_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4399 (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]. 9.15/9.10 4400 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) = hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_sup__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4401 (all T_a c_Enum_Osublists(T_a,c_List_Olist_ONil(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)))) # label(fact_sublists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4402 (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_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))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4403 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))))) # label(fact_neg__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4404 (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),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)))) # label(fact_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4405 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),c_Orderings_Obot__class_Obot(T_c)),V_A_2) = c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)))) # label(fact_Sup__fold__sup) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4406 (all T_c all T_d c_Product__Type_Oapsnd(T_d,T_d,T_c,c_Fun_Oid(T_d)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_apsnd__id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4407 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_x_2),V_A_2))))) # label(fact_Min_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4408 (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]. 9.15/9.10 4409 (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]. 9.15/9.10 4410 (all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_a_2))) # label(fact_atLeastAtMost__singleton) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4411 (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]. 9.15/9.10 4412 (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]. 9.15/9.10 4413 (all V_z_2 -(all B_x all B_y 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))))) != V_z_2)) # label(fact_eq__Abs__Integ) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4414 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> c_Equiv__Relations_Oequiv(tc_List_Olist(T_c),c_List_Olists(T_c,V_A_2),c_List_Olistrel(T_c,V_r_2)))) # label(fact_equiv__listrel) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4415 (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]. 9.15/9.10 4416 (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]. 9.15/9.10 4417 (all V_xq_2 all V_x_2 all V_f_2 all T_c all T_d c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_d,V_x_2,V_xq_2)) = c_Lazy__Sequence_Olazy__sequence_OInsert(T_c,hAPP(V_f_2,V_x_2),c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,V_xq_2))) # label(fact__06) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4418 (all V_Pa_2 all T_c c_Enum_Oenum__the(T_c,V_Pa_2) = c_HOL_OThe(T_c,V_Pa_2)) # label(fact_enum__the__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4419 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a))) # label(fact_diff__self) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4420 (all V_A_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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_d,T_c,V_f_2),V_A_2))),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)))))) # label(fact_surj__Compl__image__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4421 (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__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2))))) # label(fact_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4422 (all V_xss_2 all T_c hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xss_2)),c_List_Oset(T_c))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremdups(T_c,c_List_Oconcat(T_c,V_xss_2)))) # label(fact_length__remdups__concat) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4423 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,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_C_2)) # label(fact_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4424 (all V_A_2 all V_f_2 all T_c all T_d (c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = V_A_2 <-> hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4425 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,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_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)))))) # label(fact_psubset__card__mono) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4426 (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_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]. 9.15/9.10 4427 (all V_y_2 all V_x_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_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))))),V_y_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,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]. 9.15/9.10 4428 (all V_B_2 all V_A_2 all T_c (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_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card__Un__Int) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4429 (all V_f_2 all V_xs_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),hAPP(c_List_Oset(T_d),V_xs_2)),V_f_2) = hAPP(c_List_Ofoldl(T_c,T_d,hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(T_c)),V_xs_2))) # label(fact_SUPR__set__fold) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4430 (all V_f_2 all V_S_2 all V_T_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_d))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_S_2)))))) # label(fact_setprod__mono__one__left) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4431 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(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(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)))))))) # label(fact_Sup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4432 (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]. 9.15/9.10 4433 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Orderings_Otop__class_Otop(T_c) = c_Complete__Lattice_OInf__class_OInf(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Inf__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4434 (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_Nat__Transfer_Otsub(V_x,V_y)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4435 (all V_x_2 all V_A_2 all V_y_2 all T_c (hBOOL(hAPP(V_A_2,V_x_2)) | V_y_2 = V_x_2 <-> hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),V_A_2),V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4436 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4437 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> 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_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__int_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4438 (all V_b_2 all V_a_2 (V_b_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))) = c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_b_2))) # label(fact_dvd_OatLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4439 (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]. 9.15/9.10 4440 (all V_A_2 all T_c all V_b_2 all V_a_2 (V_a_2 = V_b_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2))))) # label(fact_eq__mem__trans) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4441 (all V_xs_2 all V_Pa_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) -> -hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))))))) # label(fact_nth__length__takeWhile) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4442 (all V_xs all V_n all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_drop__tl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4443 (all V_xs all V_x all T_a all V_n ((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)) & (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_drop__Cons_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4444 (all V_R_2 all T_c hAPP(c_Relation_ODomain(T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_R_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4445 (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]. 9.15/9.10 4446 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),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)))) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2))))) # label(fact_finite__cartesian__productD2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4447 (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]. 9.15/9.10 4448 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_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)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) <-> hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_c,T_c,V_r_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_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4449 (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]. 9.15/9.10 4450 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)))))))) # label(fact_min__max_OInf__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4451 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OField(T_c),V_r_2)),hAPP(c_Relation_OField(T_c),V_s_2))))) # label(fact_mono__Field) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4452 (all V_xs_2 all V_f_2 all T_d all T_c (V_xs_2 = c_List_Olist_ONil(T_d) <-> c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_Nil__is__map__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4453 (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)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (-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))))) # label(fact_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4454 (all V_x_2 all V_xs_2 all T_c 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_c)),V_xs_2))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_n_2),V_x_2)))))) # label(fact_set__update__memI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.10 4455 (all V_Y_2 all V_X_2 all V_f_2 all T_d all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_c,tc_fun(T_d,tc_HOL_Obool),V_r_2,V_f_2) -> (hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_Y_2),V_f_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_X_2),V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),V_A_2)) -> (hAPP(V_f_2,B_x) = hAPP(V_f_2,B_y) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2))))))) -> V_X_2 = V_Y_2))))))) # label(fact_UN__equiv__class__inject) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4456 (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]. 9.15/9.11 4457 (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),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_diff__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4458 (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_x_2 = V_y_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]. 9.15/9.11 4459 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_length__0__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4460 (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_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_H_2),V_b_H_2))),c_Wellfounded_Olex__prod(T_c,T_d,V_r_2,V_s_2))) <-> V_a_H_2 = V_a_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),V_b_2),V_b_H_2)),V_s_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_H_2)),V_r_2)))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4461 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_x_2 != V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_x_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4462 (all V_r_2 all V_A_2 all V_X_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> -(all B_x (V_X_2 = hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_quotientE) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4463 (all V_xs_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),V_xs_2))),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_c,T_c),c_Groups_Oabs__class_Oabs(T_c)),V_xs_2)))))) # label(fact_listsum__abs) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4464 (all V_n_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),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_c),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]. 9.15/9.11 4465 (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_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))))) <-> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Nat_OSuc,V_n_2))) # label(fact_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4466 (all V_f_2 all T_d all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> c_Wellfounded_Owf(T_d,hAPP(c_Set_Oimage(tc_prod(T_c,T_c),tc_prod(T_d,T_d),c_Product__Type_Omap__pair(T_c,T_d,T_c,T_d,V_f_2,V_f_2)),V_r_2))))) # label(fact_wf__map__pair__image) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4467 (all V_n_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,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_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4468 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_c (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_c),V_i_2),c_SetInterval_Oord_OatLeastLessThan(T_c,V_less__eq_2,V_less_2,V_l_2,V_u_2))))) # label(fact_ord_OatLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4469 (all V_com2_2 all V_com1_2 all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 c_Com_Ocom_OSemi(V_com1_2,V_com2_2) != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2)) # label(fact_com_Osimps_I45_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4470 (all V_Pa_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_Pa_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_Pa_2,B_x)))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4471 (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]. 9.15/9.11 4472 c_Hoare__Mirabelle_Ohoare__derivs(t_b,v_G,hAPP(c_Set_Oimage(t_a,tc_Hoare__Mirabelle_Otriple(t_b),hAPP(hAPP(c_COMBS(t_a,tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b)),hAPP(hAPP(c_COMBS(t_a,tc_Com_Ocom,tc_fun(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b))),hAPP(hAPP(c_COMBB(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_fun(tc_Com_Ocom,tc_fun(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b))),t_a),c_Hoare__Mirabelle_Otriple_Otriple(t_b)),v_P_H)),v_c0)),v_Q_H)),v_F)) -> c_Hoare__Mirabelle_Ohoare__derivs(t_b,v_G,hAPP(c_Set_Oimage(t_a,tc_Hoare__Mirabelle_Otriple(t_b),hAPP(hAPP(c_COMBS(t_a,tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b)),hAPP(hAPP(c_COMBS(t_a,tc_Com_Ocom,tc_fun(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b))),hAPP(hAPP(c_COMBB(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_fun(tc_Com_Ocom,tc_fun(tc_fun(t_b,tc_fun(tc_Com_Ostate,tc_HOL_Obool)),tc_Hoare__Mirabelle_Otriple(t_b))),t_a),c_Hoare__Mirabelle_Otriple_Otriple(t_b)),v_P)),v_c0)),v_Q)),v_F)) # label(conj_3) # label(hypothesis) # label(non_clause). [assumption]. 9.15/9.11 4473 (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]. 9.15/9.11 4474 (all V_k ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_k -> (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)))) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> 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) != 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)))) & (V_k = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(c_Code__Numeral_Oint__of,V_k) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_int__of__code) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4475 (all V_m_2 all 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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4476 (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]. 9.15/9.11 4477 (all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image__simple(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = V_z_2)) # label(fact_folding__image__simple_Oempty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4478 (all T_c all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_c) -> c_Random_Oselect__weight(T_c,hAPP(hAPP(c_List_Omap(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,T_c)),hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,T_c),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),V_xs_2)) = c_Random_Oselect(T_c,V_xs_2))) # label(fact_select__weigth__select) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4479 (all V_Pa_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(V_Pa_2,V_A_2)) -> ((all B_a all B_A (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_A)) -> (hBOOL(hAPP(hAPP(c_member(T_c),B_a),B_A)) -> (hBOOL(hAPP(V_Pa_2,B_A)) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),B_A),hAPP(hAPP(c_Set_Oinsert(T_c),B_a),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) -> hBOOL(hAPP(V_Pa_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))) # label(fact_finite__empty__induct) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4480 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(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))) # label(fact_take__1__Cons) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4481 (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]. 9.15/9.11 4482 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) <-> V_a_2 = V_b_2))) # label(fact_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4483 (all V_b_2 all V_a_2 c_Divides_Odivmod__int__rel(V_a_2,V_b_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)))))))) # label(fact_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4484 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))))) # label(fact_pos__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4485 (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]. 9.15/9.11 4486 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = 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)))) # label(fact_compl__sup) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4487 (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_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__sup__aci_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4488 (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]. 9.15/9.11 4489 (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]. 9.15/9.11 4490 (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]. 9.15/9.11 4491 (all V_r_2 all V_S_2 all T_e all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_S_2),V_r_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_S_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_e),c_Relation_Oconverse(T_d,T_c)),V_r_2))) # label(fact_converse__UNION) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4492 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2))))) # label(fact_Max_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4493 (all V_B_2 all V_A_2 all T_d all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,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)) = 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_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_Compl__UN) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4494 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_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_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_B_2))))))))) # label(fact_min__max_OSup__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4495 (all V_a_2 all V_A_H_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_H_2,c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2)),V_a_2)))) # label(fact_inv__into__inv__into__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4496 (all V_Pa_2 all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2))) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d)),V_Pa_2))) # label(fact_Domain__Collect__split) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4497 (all V_z 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_OMin)),V_z) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z))) # label(fact_mult__minus1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4498 (all V_f_2 all T_d all T_c V_f_2 = hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_f_2),c_Fun_Oid(T_c))) # label(fact_o__id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4499 (all V_ys_2 all V_xs_2 all T_c (V_xs_2 = V_ys_2 <-> hAPP(c_List_Orev(T_c),V_ys_2) = hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4500 (all V_a_2 all V_f_2 all V_A_2 all T_c all T_d (class_Fields_Ofield(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Groups_Ozero__class_Ozero(T_d) != hAPP(V_f_2,V_a_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) # label(fact_setprod__diff1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4501 (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]. 9.15/9.11 4502 (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]. 9.15/9.11 4503 (all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) & class_Groups_Ocomm__monoid__mult(T_c) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,tc_Nat_Onat),V_f_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,T_d),c_Nat_Osemiring__1__class_Oof__nat(T_c)),V_f_2)),V_A_2))) # label(fact_of__nat__setprod) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4504 (all V_c all V_b all V_v all T_b (class_Int_Onumber(T_b) & class_Rings_Oring(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]. 9.15/9.11 4505 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBS(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_c),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_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,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_c),c_fconj))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal))),c_Orderings_Oord__class_Omax(T_c)))))),hAPP(hAPP(c_COMBC(T_c,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_c),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_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(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)))))))) # label(fact_min__max_Osup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4506 (all V_r_2 all T_c c_Relation_Otrans(T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2))) # label(fact_trans__rtrancl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4507 (all V_v_H_2 all V_v_2 ((-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_2 = V_v_H_2) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2))) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = 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_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)))) <-> 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_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4508 (all V_xs_2 all T_d all T_c (class_Groups_Omonoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_COMBK(T_c,T_d),c_Groups_Ozero__class_Ozero(T_c))),V_xs_2)) = c_Groups_Ozero__class_Ozero(T_c))) # label(fact_listsum__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4509 (all V_k V_k = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OPls)) # label(fact_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4510 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) <-> V_A_2 = V_B_2))) # label(fact_inj__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4511 (all T_e all T_d all T_c c_Product__Type_Ointernal__split(T_c,T_d,T_e) = c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e)) # label(fact_internal__split__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4512 (all V_xy all V_i all V_ys all V_xs all T_b all T_a 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) = 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)))) # label(fact_update__zip) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4513 (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]. 9.15/9.11 4514 (all V_xs_2 all V_x_2 all T_c ((exists B_ys exists B_zs hAPP(hAPP(c_List_Oappend(T_c),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_zs)) = V_xs_2) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__decomp) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4515 (all T_c (class_Power_Opower(T_c) -> c_Power_Opower_Opower(T_c,c_Groups_Oone__class_Oone(T_c),c_Groups_Otimes__class_Otimes(T_c)) = c_Power_Opower__class_Opower(T_c))) # label(fact_power__power__power) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4516 (all V_n_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_c)),hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),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_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),V_A_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),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_c)),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_n_2))))))) # label(fact_finite__lists__length__le) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4517 (all V_n all V_m 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)) = 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)) # label(fact_int__power) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4518 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omax(T_c)),V_A_2)))) # label(fact_min__max_OSup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4519 (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]. 9.15/9.11 4520 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_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(c_Fun_Ocomp(T_d,T_d,T_d,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]. 9.15/9.11 4521 (all V_xs all V_n all T_a 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) = hAPP(c_List_Orotate(T_a,V_n),V_xs)) # label(fact_rotate__conv__mod) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4522 (all V_com_2 all V_fun_2 all V_loc_2 hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Com_Ocom_Ocom__size(V_com_2)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Com_Ocom_Ocom__size(c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2))) # label(fact_com_Osize_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4523 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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)),V_B_2),V_C_2)) -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2))))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4524 (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]. 9.15/9.11 4525 (all V_xs_2 all V_n_2 all T_c hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)) = hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),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_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),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_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_n_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__n__lists) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4526 (all T_c all V_r_2 all T_d ((all B_i c_Wellfounded_OwfP(T_d,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_d,tc_HOL_Obool)),c_Predicate_ODomainP(T_d,T_d,hAPP(V_r_2,B_i))),c_Predicate_ORangeP(T_d,T_d,hAPP(V_r_2,B_j))) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) -> c_Wellfounded_OwfP(T_d,hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_fun(T_d,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))))) # label(fact_wfP__SUP) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4527 (all V_f_2 all T_d all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2)),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_B_2))))) # label(fact_vimage__mono) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4528 (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,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),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,V_l) = c_SMT_Oz3mod(V_k,V_l)))) # label(fact_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4529 (all V_c_2 all V_b_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (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)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),B_x)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_c_2)),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_c_2),V_A_2))))))) # label(fact_min__max_Oinf__le__fold__inf) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4530 (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]. 9.15/9.11 4531 (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]. 9.15/9.11 4532 (all V_ys_2 all V_g_2 all T_f all V_xs_2 all V_f_2 all T_e all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Omap(T_e,T_c),V_f_2),V_xs_2)),hAPP(hAPP(c_List_Omap(T_f,T_d),V_g_2),V_ys_2)) = hAPP(hAPP(c_List_Omap(tc_prod(T_e,T_f),tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_f,T_d),tc_fun(T_f,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_f,T_d),tc_fun(T_f,tc_prod(T_c,T_d))),T_e),c_COMBB(T_d,tc_prod(T_c,T_d),T_f)),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_f_2))),V_g_2))),hAPP(hAPP(c_List_Ozip(T_e,T_f),V_xs_2),V_ys_2))) # label(fact_zip__map__map) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4533 (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]. 9.15/9.11 4534 (all V_c_2 all V_b_2 all V_a_2 all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (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_c_2)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_c_2)),V_r_2)))))) # label(fact_transD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4535 (all V_s_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,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_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,V_r_2)),V_s_2))))) # label(fact_trancl__Int__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4536 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_u_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_ivl__disj__un_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4537 (all V_C_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))))) # label(fact_subset__psubset__trans) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4538 (all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_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_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)) = V_f_2)) # label(fact_inv__inv__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4539 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_less__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4540 (all V_A_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),V_A_2)))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_List_Olistrel(T_c,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olists(T_c,V_A_2)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_List_Olist(T_c)),c_List_Olists(T_c,V_A_2))))))) # label(fact_listrel__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4541 (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_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4542 (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]. 9.15/9.11 4543 (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(c_List_Orev(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_y)) = 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))) # label(fact_rev__update) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4544 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_a_2)))) # label(fact_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4545 (all V_h_2 all V_g_2 all T_h all V_f_2 all T_e all T_f all T_g all V_F_2 all V_fa_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_fa_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_g,T_f,T_e,V_f_2),hAPP(c_Fun_Ocomp(T_h,T_g,T_e,V_g_2),V_h_2)) = hAPP(c_Fun_Ocomp(T_h,T_f,T_e,hAPP(c_Fun_Ocomp(T_g,T_f,T_h,V_f_2),V_g_2)),V_h_2))) # label(fact_folding_Ocommute__comps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4546 (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_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_Y_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_X_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_Y_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) -> V_X_2 = V_Y_2))))))) # label(fact_quotient__eqI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4547 (all V_R_2 all V_Qa_2 all V_Pa_2 all T_c all T_e all T_d hAPP(hAPP(hAPP(c_COMBS(T_d,T_e,T_c),V_Pa_2),V_Qa_2),V_R_2) = hAPP(hAPP(V_Pa_2,V_R_2),hAPP(V_Qa_2,V_R_2))) # label(help_c__COMBS__1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4548 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring(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__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4549 (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]. 9.15/9.11 4550 (all V_xs_2 all T_c 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))),c_List_Otranspose(T_c,V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),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_xs_2))) # label(fact_transpose__max__length) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4551 (all V_r_2 all T_c hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool))) # label(fact_quotient__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4552 (all V_m all V_l c_List_Oreturn__list(hAPP(hAPP(c_List_Oappend(tc_Int_Oint),c_List_Oembed__list(V_l)),c_List_Oembed__list(V_m))) = hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),V_l),V_m)) # label(fact_transfer__nat__int__list__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4553 (all V_a_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one__idem(T_c,V_f_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_a_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,V_f_2),V_a_2),V_A_2)))) # label(fact_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4554 (all V_f2_2 all V_f1_2 all T_d all T_c c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_c,T_d,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_d)) = V_f1_2) # label(fact_lazy__sequence_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4555 (all V_m all V_n 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) = V_m) # label(fact_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4556 (all V_n_2 all V_m_2 (V_n_2 != 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)))) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4557 (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]. 9.15/9.11 4558 (all V_n all V_m (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) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_div__less) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4559 (all V_A_2 all T_d all V_I_2 all V_k_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_INT__absorb) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4560 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Un__absorb2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4561 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> 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]. 9.15/9.11 4562 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d hAPP(hAPP(V_f1_2,V_a_2),V_b_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2))) # label(fact_prod_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4563 (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]. 9.15/9.11 4564 (all V_xs_2 all V_x_2 all V_Qa_2 all T_c c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Qa_2),c_List_Oremove1(T_c,V_x_2,V_xs_2))) # label(fact_filter__remove1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4565 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)) = hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_x_2,V_y_2)))) # label(fact_image__uminus__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4566 (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]. 9.15/9.11 4567 (all V_A_2 all T_c all V_y_2 all V_x_2 (V_x_2 = V_y_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))))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4568 (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]. 9.15/9.11 4569 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_neg__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4570 (all V_A_2 all T_c (class_Finite__Set_Ofinite(T_c) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_finite) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4571 (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]. 9.15/9.11 4572 (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]. 9.15/9.11 4573 (all V_c_2 all T_d all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_image__constant) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4574 (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]. 9.15/9.11 4575 (all V_xs_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),V_xs_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_c),c_COMBI(T_c)),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_distinct__listsum__conv__Setsum) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4576 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Predicate_OPowp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_x_2),hAPP(c_Set_OPow(T_c),V_A_2))))) # label(fact_Powp__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4577 (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_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),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]. 9.15/9.11 4578 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2))))))) # label(fact_min__max_OInf__le__Sup) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4579 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2) = c_Groups_Ozero__class_Ozero(T_d)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2) = hAPP(V_b_2,V_a_2))))) # label(fact_setsum__delta_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4580 (all T_c c_Wellfounded_Ofinite__psubset(T_c) = hAPP(c_Set_OCollect(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBS(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_fconj)),c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool))))),c_Finite__Set_Ofinite(T_c))))) # label(fact_finite__psubset__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4581 (all V_A_2 all T_c c_Fun_Oinj__on(T_c,T_c,c_COMBI(T_c),V_A_2)) # label(fact_inj__on__id2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4582 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_prod(T_c,T_d)),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_card__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4583 (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]. 9.15/9.11 4584 (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]. 9.15/9.11 4585 (all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofoldl(tc_List_Olist(T_c),T_c,hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c))),c_List_Olist_ONil(T_c)),V_xs_2) = c_List_Olinorder__class_Osort__key(T_c,T_c,c_COMBI(T_c),V_xs_2))) # label(fact_sort__foldl__insort) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4586 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)))))) # label(fact_le__minus__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4587 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_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_list__all2__lengthD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4588 (all V_z_2 all V_w_2 all T_c (class_Int_Oring__char__0(T_c) -> (hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_w_2) <-> V_w_2 = V_z_2))) # label(fact_of__int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4589 (all V_p_2 all V_g_2 all T_g all T_f all V_f_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_f,T_g,tc_prod(T_d,T_e)),V_g_2),V_p_2)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_f,T_g,T_c),hAPP(hAPP(c_COMBB(tc_fun(T_g,tc_prod(T_d,T_e)),tc_fun(T_g,T_c),T_f),hAPP(c_COMBB(tc_prod(T_d,T_e),T_c,T_g),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2))),V_g_2)),V_p_2)) # label(fact_split__twice) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4590 (all V_n all T_a c_Nitpick_Onumber__of__frac(T_a,V_n) = 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)))) # label(fact_Nitpick_Onumber__of__frac__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4591 (all V_B_2 all V_A_2 all V_R_2 all T_d all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,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(c_Relation_OImage(T_d,T_c,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_B_2))))) # label(fact_Image__Int__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4592 (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]. 9.15/9.11 4593 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4594 (all V_r_2 all T_c ((all B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Wellfounded_Oacc(T_c,V_r_2)))) <-> c_Wellfounded_Owf(T_c,V_r_2))) # label(fact_wf__acc__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4595 (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]. 9.15/9.11 4596 (all V_fs_2 all V_f_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_c,tc_Nat_Onat)),V_f_2),V_fs_2)))) <-> hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_List_Omeasures(T_c,V_fs_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))))) # label(fact_in__measures_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4597 (all V_xs_2 all V_x_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_xs_2)))))) # label(fact_sorted__insort__key) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4598 (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]. 9.15/9.11 4599 (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]. 9.15/9.11 4600 (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_x = V_y)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4601 (all V_B_2 all V_A_2 all V_R_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_R_2)) -> hAPP(c_Relation_OImage(T_d,T_c,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,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(c_Relation_OImage(T_d,T_c,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_B_2)))) # label(fact_Image__Int__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4602 (all V_l_2 all V_k_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_f_2),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_k_2,V_l_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(tc_Int_Oint,T_c),V_f_2),c_List_Oupto(V_k_2,V_l_2))))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4603 (all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(V_f_2,B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(V_F_2,V_A_2)),V_A_2))))))) # label(fact_folding__one_Oclosed) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4604 (all V_A_2 all T_c hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),c_List_Olist_ONil(T_c)))) # label(fact_listsp_ONil) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4605 (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]. 9.15/9.11 4606 (all V_g_2 all V_f_2 all T_c all T_d all T_e hAPP(c_List_Omap(T_c,T_d),hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_f_2),V_g_2)) = hAPP(c_Fun_Ocomp(tc_List_Olist(T_e),tc_List_Olist(T_d),tc_List_Olist(T_c),hAPP(c_List_Omap(T_e,T_d),V_f_2)),hAPP(c_List_Omap(T_c,T_e),V_g_2))) # label(fact_map__comp__map) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4607 (all T_d all T_c c_Product__Type_Oapfst(T_c,T_c,T_d,c_Fun_Oid(T_c)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_apfst__id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.11 4608 (all V_A_2 all V_B_2 all V_f_2 all T_e all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_HOL_Obool),T_c),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_c),tc_fun(tc_fun(T_e,tc_HOL_Obool),T_c),T_d),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_c)),V_f_2)),V_B_2)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(T_d,T_e),T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_f_2)),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))))) # label(fact_setsum__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4609 (all V_list_2 all V_a_2 all V_fa_2 all T_c c_List_Olist_Olist__size(T_c,V_fa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),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_c,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]. 9.15/9.12 4610 (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]. 9.15/9.12 4611 (all T_c c_Lazy__Sequence_Oanamorph(tc_Lazy__Sequence_Olazy__sequence(T_c),T_c,c_Lazy__Sequence_Oyield(T_c)) = c_Lazy__Sequence_Oyieldn(T_c)) # label(fact_yieldn__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4612 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d 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_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,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_d),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2))) # label(fact_INT__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4613 (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]. 9.15/9.12 4614 (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]. 9.15/9.12 4615 (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]. 9.15/9.12 4616 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4617 (all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),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))) # label(fact_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4618 (all V_r_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_Transitive__Closure_Otrancl(T_c,V_r_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_c),hAPP(c_Relation_OField(T_c),V_r_2)),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_Relation_OField(T_c),V_r_2)))))) # label(fact_trancl__subset__Field2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4619 (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(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]. 9.15/9.12 4620 (all T_c hAPP(c_Relation_ODomain(T_c,T_c),c_Relation_OId(T_c)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4621 (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(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)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n))))) # label(fact_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4622 (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]. 9.15/9.12 4623 (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]. 9.15/9.12 4624 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> V_m != V_n)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4625 (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]. 9.15/9.12 4626 (all V_r_2 all T_d all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_c,T_d),V_r_2) <-> V_r_2 = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)))) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4627 (all V_z_2 all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> hAPP(hAPP(V_f_2,V_y_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_y_2),V_z_2)))) # label(fact_fun__left__comm_Ofun__left__comm) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4628 (all V_M_2 all T_c all V_A_2 all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_M_2 = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),hAPP(c_COMBK(T_d,T_c),V_M_2))))) # label(fact_SUP__const) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4629 (all V_f2_2 all V_f1_2 all T_d all T_c hAPP(c_List_Olist_Olist__case(T_c,T_d,V_f1_2,V_f2_2),c_List_Olist_ONil(T_d)) = V_f1_2) # label(fact_list_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4630 (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]. 9.15/9.12 4631 (all V_r_2 all V_yx_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),V_yx_2),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) -> -(all B_x all B_y (V_yx_2 = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_y),B_x) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_x),B_y)),V_r_2)))))) # label(fact_converseE) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4632 (all V_Pa_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,T_d),V_Pa_2)),c_Product__Type_OPair(T_c,T_d)))) = c_HOL_OThe(tc_prod(T_c,T_d),V_Pa_2)) # label(fact_split__paired__The) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4633 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (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_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),c_Groups_Oone__class_Oone(T_c)))))) # label(fact_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4634 (all V_b_2 all V_a_2 all T_d all T_c all V_f_2 (hBOOL(hAPP(V_f_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))) -> hBOOL(hAPP(hAPP(c_Product__Type_Ocurry(T_c,T_d,tc_HOL_Obool,V_f_2),V_a_2),V_b_2)))) # label(fact_curryI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4635 (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]. 9.15/9.12 4636 (all V_b_2 all V_a_2 all V_f_2 all T_c (V_b_2 = V_a_2 <-> hBOOL(hAPP(c_Finite__Set_Ofold1Set(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)))),V_b_2)))) # label(fact_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4637 (all V_ys_2 all V_x_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c))) = V_ys_2 <-> c_List_Olist_ONil(T_c) != V_ys_2 & V_xs_2 = c_List_Obutlast(T_c,V_ys_2) & V_x_2 = c_List_Olast(T_c,V_ys_2))) # label(fact_snoc__eq__iff__butlast) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4638 (all V_a_2 all V_r_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),B_x)))) <-> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),V_a_2)))) # label(fact_accp_Oequations) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4639 (all V_r_2 all V_x_2 all V_y_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),V_r_2)) <-> c_Wellfounded_Owf(T_c,V_r_2) & -hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))))) # label(fact_wf__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4640 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Odrop(T_a),V_n),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)),V_n)) # label(fact_length__drop) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4641 (all V_A_2 all V_z_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_Finite__Set_Ofold(T_d,T_c,V_f_2),V_z_2),V_A_2) = c_HOL_OThe(T_c,c_Finite__Set_Ofold__graph(T_d,T_c,V_f_2,V_z_2,V_A_2))) # label(fact_fold__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4642 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_a_2,V_b_2) -> (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_c_2,V_d_2) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_a_2),V_c_2),hAPP(hAPP(c_List_Oappend(T_d),V_b_2),V_d_2))))) # label(fact_list__all2__appendI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4643 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2)))) # label(fact_Compl__greaterThan) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4644 (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]. 9.15/9.12 4645 (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]. 9.15/9.12 4646 (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]. 9.15/9.12 4647 (all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I17_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4648 (all V_z 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)),V_z)),V_z)) # label(fact_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4649 (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),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),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)))) # label(fact_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4650 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_d_2)) -> V_a_2 = V_c_2))))) # label(fact_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4651 (all V_n all V_i ((V_i != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> 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_Code__Numeral_Onat__of__aux(V_i,V_n)) & (V_i = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> V_n = c_Code__Numeral_Onat__of__aux(V_i,V_n)))) # label(fact_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4652 (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_c all T_d (class_Orderings_Oord(T_c) & class_Groups_Ocomm__monoid__add(T_d) -> (V_a_2 = V_c_2 -> (V_d_2 = V_b_2 -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),B_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),B_x),V_d_2)) -> hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2))))))) # label(fact_setsum__ivl__cong) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4653 (all V_g_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),V_B_2)) -> (c_Fun_Oinj__on(T_d,T_c,V_g_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_d,T_c,V_g_2),V_B_2)),V_A_2)) -> (exists B_h c_Fun_Obij__betw(T_c,T_d,B_h,V_A_2,V_B_2))))))) # label(fact_Cantor__Bernstein) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4654 (all V_w c_Int_OPls = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OPls),V_w)) # label(fact_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4655 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_c_2),V_d_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_d_2)))))) # label(fact_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4656 (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]. 9.15/9.12 4657 (all V_B_2 all T_c V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_B_2)) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4658 (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),V_b) = 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)))) # label(fact_minus__mult__minus) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4659 (all V_i_1 all V_x all T_a c_Lazy__Sequence_Ohb__single(T_a,V_x) = hAPP(c_New__DSequence_Oneg__single(T_a,V_x),V_i_1)) # label(fact_New__DSequence_Oneg__single__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4660 (all V_g_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_g_2 = hAPP(c_Fun_Ocomp(T_d,T_e,T_c,hAPP(c_Fun_Ocomp(T_c,T_e,T_d,V_g_2),c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2))),V_f_2))) # label(fact_o__inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4661 (all 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)))) <-> c_Nat__Transfer_Onat__set(V_S_2))) # label(fact_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4662 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2)) = V_A_2)) # label(fact_Union__quotient) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4663 (all V_w_2 all T_c (class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_w_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Nat_Osemiring__1__class_ONats(T_c)))))) # label(fact_Nats__number__of) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4664 (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]. 9.15/9.12 4665 (all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) <-> c_Fun_Oid(T_c) = hAPP(c_Fun_Ocomp(T_d,T_c,T_c,V_f_2),c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_f_2)))) # label(fact_surj__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4666 (all V_xs_2 all V_r_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Wellfounded_Oacc(T_c,V_r_2))) -> (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_Wellfounded_Oacc(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2)))) -> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),c_Wellfounded_Oacc(tc_List_Olist(T_c),c_List_Olistrel1(T_c,V_r_2))))))) # label(fact_Cons__acc__listrel1I) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4667 (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_y),V_x)) | V_x = V_y | 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]. 9.15/9.12 4668 (all V_lazy__sequence_2 all V_a_2 all V_f2_2 all V_f1_2 all T_d all T_c hAPP(hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_c,T_d,V_f1_2,V_f2_2,V_lazy__sequence_2)) = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_c,T_d,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_d,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4669 (all V_xs_2 all V_x_2 all T_c ((exists B_ys exists B_zs (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),B_zs))) & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_zs)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__decomp__last) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4670 (all V_a_2 all V_A_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)),V_a_2),V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)))))) # label(fact_fold1__eq__fold) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4671 (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]. 9.15/9.12 4672 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),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_finite__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4673 (all V_n_2 all V_a_2 all T_c (class_Rings_Omult__zero(T_c) & class_Rings_Ozero__neq__one(T_c) & class_Rings_Ono__zero__divisors(T_c) & class_Power_Opower(T_c) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 & c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4674 (all V_A_2 all V_r_2 all T_d all V_b_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Relation_OImage(T_d,T_c,V_r_2),V_A_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),B_x),V_b_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)))))) # label(fact_ImageE) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4675 (all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Oring__1(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,T_c,T_d),c_Int_Oring__1__class_Oof__int(T_c)),V_f_2)),V_A_2) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,tc_Int_Oint),V_f_2),V_A_2)))) # label(fact_of__int__setsum) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4676 (all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Oone__class_Oone(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)))) # label(fact_setprod__infinite) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4677 (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]. 9.15/9.12 4678 (all V_B_2 all V_C_2 all T_d all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(V_B_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_INT__greatest) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4679 (all V_k_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_k_2)) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4680 (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]. 9.15/9.12 4681 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (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(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))) # label(fact_Inf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4682 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),c_List_Olinorder__class_Osorted__list__of__set(T_c,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)))))) = c_List_Olinorder__class_Osorted__list__of__set(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_sorted__list__of__set__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4683 (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]. 9.15/9.12 4684 (all V_xs all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs))) # label(fact_listsum__rev) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4685 (all V_Pa_2 all V_Qa_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> ((all B_x (hBOOL(hAPP(V_Qa_2,B_x)) -> hBOOL(hAPP(V_Pa_2,B_x)))) -> ((exists B_x1 hBOOL(hAPP(V_Qa_2,B_x1))) -> (exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Quotient_ORespects(T_c,V_R_2))))))))) # label(fact_bex__reg__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4686 (all V_ys_2 all V_xs_2 all T_c hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_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))) # label(fact_set__append) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4687 (all V_R_2 all T_c c_Predicate_Opred__comp(T_c,T_c,T_c,c_fequal,c_Predicate_Opred__comp(T_c,T_c,T_c,V_R_2,c_fequal)) = V_R_2) # label(fact_eq__comp__r) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4688 (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]. 9.15/9.12 4689 (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]. 9.15/9.12 4690 (all V_x_2 all V_f_2 all T_c all T_d all V_k_2 ((V_k_2 = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),T_d),c_List_Olist_ONil(T_c)),V_x_2) = hAPP(hAPP(c_Lazy__Sequence_Oanamorph(T_d,T_c,V_f_2),V_k_2),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_c),T_d),tc_prod(T_c,T_d),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),T_d),c_List_Olist_ONil(T_c)),V_x_2),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_prod(tc_List_Olist(T_c),T_d)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d)),tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(tc_List_Olist(T_c),T_d),tc_prod(tc_List_Olist(T_c),T_d)),tc_fun(tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d)),tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d))),T_c),c_COMBB(tc_prod(tc_List_Olist(T_c),T_d),tc_prod(tc_List_Olist(T_c),T_d),T_d)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d))),tc_fun(tc_prod(tc_List_Olist(T_c),T_d),tc_prod(tc_List_Olist(T_c),T_d)),T_c),c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),T_d,tc_prod(tc_List_Olist(T_c),T_d))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d))),T_c),hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(T_d,tc_prod(tc_List_Olist(T_c),T_d)),tc_List_Olist(T_c)),c_Product__Type_OPair(tc_List_Olist(T_c),T_d))),c_List_Olist_OCons(T_c))))),hAPP(c_Lazy__Sequence_Oanamorph(T_d,T_c,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_d,T_c,V_f_2),V_k_2),V_x_2)))) # label(fact_anamorph_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4691 (all V_ys_2 all V_xs_2 all T_c (V_ys_2 = c_List_Olist_ONil(T_c) & V_xs_2 = c_List_Olist_ONil(T_c) <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = c_List_Olist_ONil(T_c))) # label(fact_append__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4692 (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]. 9.15/9.12 4693 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) = hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4694 (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_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_d)) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4695 (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]. 9.15/9.12 4696 (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]. 9.15/9.12 4697 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_b_2),V_c_2)) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_b_2),V_a_2)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_c_2),V_a_2)))))))) # label(fact_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4698 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4699 (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))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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))))) # label(fact_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4700 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4701 (all V_c_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2))))) # label(fact_psubsetD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4702 (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]. 9.15/9.12 4703 (all V_R_2 all V_a2_2 all V_a1_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_xa)),V_R_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_a2_2)))))) & V_a2_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a2_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_a1_2),V_a2_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))))) # label(fact_max__ext_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4704 (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]. 9.15/9.12 4705 (all V_x_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) -> c_List_Olinorder__class_Osorted(T_c,c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2))))) # label(fact_sorted__insort__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4706 (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_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),V_a))) # label(fact_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4707 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_c),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),V_B_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_fun(tc_prod(T_c,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_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),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_fun(tc_prod(T_c,T_d),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_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_d),c_Set_Oinsert(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)) # label(fact_Sigma__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4708 (all V_y all V_x (-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_y),V_x)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_y)) -> -(-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_y)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_y),V_x))))) # label(fact_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4709 (all V_n_2 all V_A_2 all V_f_2 all T_c (hAPP(c_Nat_OSuc,V_n_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2) -> (exists B_x (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_member(T_c),B_x),V_A_2)))))) # label(fact_setsum__SucD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4710 (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]. 9.15/9.12 4711 (all V_a_2 all V_b_2 all V_A_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_b_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_fold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4712 (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]. 9.15/9.12 4713 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & (all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_y)) -> B_y = B_x)) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_list__ex1__iff__raw) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4714 (all V_As_2 all V_A_2 all T_c c_List_Oset__Cons(T_c,V_A_2,c_List_Olistset(T_c,V_As_2)) = c_List_Olistset(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_As_2))) # label(fact_listset_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4715 (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)) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Omod(T_a,V_b,V_a)))) # label(fact_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4716 (all V_A_2 all V_a_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) <-> -hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_a_2)),hAPP(c_Set_Oimage(T_c,T_d,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_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) & c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2))) # label(fact_inj__on__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4717 (all V_i_2 all V_f_2 all V_s_2 all T_c all T_d (class_Groups_Oordered__ab__group__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_s_2)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_i))))) -> (hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_s_2) = c_Groups_Ozero__class_Ozero(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),V_s_2)) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,V_i_2))))))) # label(fact_setsum__nonneg__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4718 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Oset(T_c),c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_set__insort__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4719 (all V_j_2 all V_i_2 all V_Pa_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_Pa_2,B_x)))) <-> c_List_Oall__interval__nat(V_Pa_2,V_i_2,V_j_2))) # label(fact_all__interval__nat__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4720 (all T_c all T_d all V_y_2 all V_x_2 all V_f_2 (hAPP(V_f_2,V_x_2) = V_y_2 -> V_f_2 = c_Fun_Ofun__upd(T_d,T_c,V_f_2,V_x_2,V_y_2))) # label(fact_fun__upd__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4721 (all V_A_2 all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Un__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4722 (all V_s_2 all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) -> (c_Relation_Osym(T_c,V_s_2) -> c_Relation_Osym(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_sym__Int) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4723 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (V_y_2 = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2))))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4724 (all V_n_2 all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2),hAPP(hAPP(c_List_Otake(T_d),V_n_2),V_ys_2)))) # label(fact_list__all2__takeI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4725 (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]. 9.15/9.12 4726 (all T_1 all T_2 (class_Lazy__Sequence_Osmall__lazy(T_1) & class_Lazy__Sequence_Osmall__lazy(T_2) -> class_Lazy__Sequence_Osmall__lazy(tc_prod(T_2,T_1)))) # label(arity_prod__Lazy__Sequence_Osmall__lazy) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4727 (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)) -> V_xs = hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))) # label(fact_take__all) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4728 (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]. 9.15/9.12 4729 (all V_xs_2 all V_x_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_Enum_Osublists(T_c,V_xs_2)),hAPP(hAPP(c_COMBS(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_List_Olist(tc_List_Olist(T_c)),tc_fun(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),tc_List_Olist(tc_List_Olist(T_c))),c_List_Oappend(tc_List_Olist(T_c))),hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(c_List_Olist_OCons(T_c),V_x_2)))),c_COMBI(tc_List_Olist(tc_List_Olist(T_c))))) = c_Enum_Osublists(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_sublists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4730 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) -> -(all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2)) -> (all B_us all B_vs (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_vs)) -> hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_vs)) != V_ys_2)))))) # label(fact_listrel1E) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4731 (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]. 9.15/9.12 4732 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> c_Relation_Otrans(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c))))) # label(fact_trans__reflclI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4733 (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]. 9.15/9.12 4734 (all V_x_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Wellfounded_Oacc(T_c,V_r_2))) <-> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)),V_x_2)))) # label(fact_accp__acc__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4735 (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]. 9.15/9.12 4736 (all V_x_2 all V_g_2 all T_f all V_f_2 all T_d all T_c all T_e hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_f),V_x_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_e,T_f),V_x_2))) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),hAPP(c_Product__Type_Oapsnd(T_f,T_d,T_e,V_g_2),V_x_2))) # label(fact_apfst__apsnd) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4737 (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]. 9.15/9.12 4738 (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]. 9.15/9.12 4739 (all V_ys all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_ys)) # label(fact_append__Cons) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4740 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2) = hAPP(c_Product__Type_Ofst(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Opartition(T_c,V_Pa_2,V_xs_2))) # label(fact_partition__filter1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4741 (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]. 9.15/9.12 4742 (all T_c hAPP(c_Relation_OField(T_c),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Field__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4743 (all V_xs_2 all V_Pa_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist_ONil(T_c) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_filter__empty__conv) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4744 (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]. 9.15/9.12 4745 (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]. 9.15/9.12 4746 (all T_e all T_d all V_c_2 all V_b_2 all V_B_2 all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),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_c,tc_fun(T_d,tc_fun(T_e,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]. 9.15/9.12 4747 (all V_a_2 all V_b_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2))))) # label(fact_neg__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4748 (all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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_c,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_c,tc_HOL_Obool)),tc_fun(T_c,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_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),tc_Nat_Onat),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(T_c,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]. 9.15/9.12 4749 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4750 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> V_xs_2 = c_List_Oinsert(T_c,V_x_2,V_xs_2))) # label(fact_in__set__insert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4751 (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]. 9.15/9.12 4752 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) & hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = V_B_2 <-> c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2))) # label(fact_bij__betw__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4753 (all V_a_2 all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d c_List_Ofoldr(T_d,T_c,V_f_2,hAPP(hAPP(c_List_Oappend(T_d),V_xs_2),V_ys_2),V_a_2) = c_List_Ofoldr(T_d,T_c,V_f_2,V_xs_2,c_List_Ofoldr(T_d,T_c,V_f_2,V_ys_2,V_a_2))) # label(fact_foldr__append) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4754 (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_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)) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n),V_x)) & (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)))) # label(fact_list__update__append) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4755 (all V_A_2 all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) -> V_A_2 = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,c_Hilbert__Choice_Oinv__into(T_d,T_c,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)),V_f_2)),V_A_2)))) # label(fact_image__surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 9.15/9.12 4756 (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]. 9.15/9.12 4757 (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]. 9.15/9.13 4758 (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]. 9.15/9.13 4759 (all V_Pa_2 all V_Qa_2 (-hBOOL(V_Qa_2) | hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)))) # label(help_c__fdisj__2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4760 (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]. 9.15/9.13 4761 (all V_a_2 all V_f_2 all T_c V_a_2 = hAPP(c_Finite__Set_Ofold1(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))))) # label(fact_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4762 (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]. 9.15/9.13 4763 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) & hBOOL(hAPP(V_Pa_2,V_x_2)) <-> c_List_Olist__all(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) # label(fact_list__all__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4764 (all V_i_1 all V_x all T_a hAPP(c_Lazy__Sequence_Osingle(T_a),V_x) = hAPP(c_New__DSequence_Opos__single(T_a,V_x),V_i_1)) # label(fact_New__DSequence_Opos__single__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4765 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_f all T_e all T_d hAPP(hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_f),hAPP(V_f_2,V_x_2))),hAPP(c_Product__Type_Osnd(T_e,T_f),hAPP(V_f_2,V_x_2))) = hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_d,T_e,T_f,T_c),V_f_2),V_g_2),V_x_2)) # label(fact_scomp__unfold) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4766 (all V_A_2 all V_Pa_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,hAPP(c_Set_OCollect(tc_prod(T_d,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,tc_HOL_Obool),V_Pa_2))),V_A_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_A_2)))),hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Pa_2))))) # label(fact_Image__Collect__split) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4767 (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]. 9.15/9.13 4768 (all V_g_2 all V_f_2 all V_K_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) & class_Groups_Oordered__ab__semigroup__add(T_d) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_K_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(V_f_2,B_i)),hAPP(V_g_2,B_i))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_K_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_K_2)))))) # label(fact_setsum__mono) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4769 (all V_y_2 all T_c (class_Rings_Olinordered__idom(T_c) & class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),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]. 9.15/9.13 4770 (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]. 9.15/9.13 4771 (all V_z_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_z_2)) & hAPP(c_Int_Onat,V_z_2) = V_m_2 <-> V_z_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))) # label(fact_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4772 (all V_x_2 -(all B_y (hAPP(c_Code__Numeral_Oof__nat,B_y) = V_x_2 -> -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]. 9.15/9.13 4773 (all V_Pa_2 (-hBOOL(V_Pa_2) | -hBOOL(hAPP(c_fNot,V_Pa_2)))) # label(help_c__fNot__1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4774 (all V_xs all T_a all V_n (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_xs = hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4775 (all V_v_2 all V_f_2 all T_d all V_xs_2 all V_y_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_List_Omap(T_c,T_d),c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_y_2,V_v_2)),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))) # label(fact_map__fun__upd) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4776 (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]. 9.15/9.13 4777 (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]. 9.15/9.13 4778 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (V_y_2 = c_Groups_Ozero__class_Ozero(T_c) & c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_y_2),V_y_2))),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4779 (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))) -> 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)) = 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_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]. 9.15/9.13 4780 (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]. 9.15/9.13 4781 (all V_A_2 all T_c c_List_Oset__Cons(T_c,V_A_2,hAPP(hAPP(c_Set_Oinsert(tc_List_Olist(T_c)),c_List_Olist_ONil(T_c)),c_Orderings_Obot__class_Obot(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_c,tc_List_Olist(T_c),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c))),V_A_2)) # label(fact_set__Cons__sing__Nil) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4782 (all V_n_2 all V_m_2 (V_n_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_m_2 | c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2 & 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) = c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4783 (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_m_2 | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4784 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2) <-> V_y_2 = V_x_2))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4785 (all V_f_H_2 all T_e all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (c_Fun_Oinj__on(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_f_H_2),V_f_2),V_A_2) <-> c_Fun_Oinj__on(T_d,T_e,V_f_H_2,hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))))) # label(fact_comp__inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4786 (all V_r_2 all T_c hAPP(c_Relation_ORange(T_c,T_c),V_r_2) = hAPP(c_Relation_ORange(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2))) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4787 (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]. 9.15/9.13 4788 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2) <-> V_b_2 = V_a_2 & V_c_2 = V_b_2))) # label(fact_atLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4789 (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]. 9.15/9.13 4790 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Oall__interval__int(V_Pa_2,V_i_2,V_j_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_Pa_2,B_x)))))) # label(fact_all__interval__int__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4791 (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]. 9.15/9.13 4792 (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]. 9.15/9.13 4793 (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]. 9.15/9.13 4794 (all V_A_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),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))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4795 (all V_Pa_2 all T_c c_List_Olist_ONil(T_c) = c_List_OdropWhile(T_c,V_Pa_2,c_List_Olist_ONil(T_c))) # label(fact_dropWhile_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4796 (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]. 9.15/9.13 4797 (all V_B_2 all V_A_2 all T_c (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_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_card_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4798 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(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)),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)),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(T_c,tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_c),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_c)),V_xs_2)))))) # label(fact_set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4799 (all V_j all V_i all V_m c_List_Oupt(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m),V_j) = hAPP(hAPP(c_List_Odrop(tc_Nat_Onat),V_m),c_List_Oupt(V_i,V_j))) # label(fact_drop__upt) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4800 (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__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4801 (all V_m_2 all V_n_2 all V_i_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_i_2),V_n_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_i_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_i_2),V_n_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_n_2),V_m_2)))) # label(fact_ivl__diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4802 (all T_d all T_c all V_Pa_2 ((all B_a all B_b hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b)))) <-> (all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4803 (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]. 9.15/9.13 4804 (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),V_q)),V_r) = 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__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]. 9.15/9.13 4805 (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]. 9.15/9.13 4806 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__bot(T_c) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2) = c_Orderings_Obot__class_Obot(T_c) <-> c_Orderings_Obot__class_Obot(T_c) = V_x_2 & c_Orderings_Obot__class_Obot(T_c) = V_y_2))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4807 (all V_R_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Image__empty) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4808 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (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(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_x_2),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Inf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4809 (all V_a_2 all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),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]. 9.15/9.13 4810 (all V_j_2 all V_i_2 all V_Pa_2 (-c_List_Oall__interval__nat(hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat,c_fNot),V_Pa_2),V_i_2,V_j_2) <-> c_List_Olist__ex(tc_Nat_Onat,V_Pa_2,c_List_Oupt(V_i_2,V_j_2)))) # label(fact_list__ex__iff__not__all__inverval__nat) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4811 (all V_M_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_List_Olist(T_c)),V_M_2)) -> (exists B_n all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),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_c)),B_x)),B_n)))))) # label(fact_finite__maxlen) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4812 (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_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)))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4813 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OgreaterThanAtMost(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)),V_j_2))) # label(fact_greaterThanAtMost__upto) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4814 (all V_r_2 all T_d all V_a_2 all T_c ((exists B_y hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),B_y)),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4815 (all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Fun_Othe__inv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,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]. 9.15/9.13 4816 (all V_g_2 all V_f_2 all T_c all T_d (class_Enum_Oenum(T_c) & class_Orderings_Oorder(T_d) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2)) & hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBS(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_g_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,T_d)),V_f_2),V_g_2))))) # label(fact_order__fun_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4817 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,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_c),V_B_2)),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))) -> V_A_2 = V_B_2)))) # label(fact_card__seteq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4818 (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]. 9.15/9.13 4819 (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]. 9.15/9.13 4820 (all V_A_2 all V_y_2 all V_x_2 all T_c (V_x_2 = V_y_2 & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Relation_OId__on(T_c,V_A_2))))) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4821 (all V_s_2 all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> (c_Relation_Otrans(T_c,V_s_2) -> c_Relation_Otrans(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),V_s_2))))) # label(fact_trans__Int) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4822 (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]. 9.15/9.13 4823 (all V_y_2 all V_x_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)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2)) # label(fact_nat) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4824 (all V_com_H_2 all V_fun_H_2 all V_loc_H_2 all V_com_2 all V_fun_2 all V_loc_2 (V_com_2 = V_com_H_2 & V_fun_H_2 = V_fun_2 & V_loc_2 = V_loc_H_2 <-> c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) = c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2))) # label(fact_com_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4825 (all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Olinordered__semidom(T_d) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_x))))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)))))) # label(fact_setprod__pos) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4826 (all V_n_2 all V_Pa_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(V_Pa_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__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_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) = V_n_2 -> hBOOL(hAPP(hAPP(V_Pa_2,B_i),B_j))))))) # label(fact_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4827 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (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_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_B_2)))))) # label(fact_setprod__Un__Int) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4828 (all V_B_2 all V_a_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) # label(fact_insert__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4829 (all V_xs_2 all V_n_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__dropD) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4830 (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]. 9.15/9.13 4831 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),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]. 9.15/9.13 4832 (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]. 9.15/9.13 4833 (all V_r_2 all T_c (c_Wellfounded_Oacyclic(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),c_Predicate_Oconversep(T_c,T_c,V_r_2)))) <-> c_Wellfounded_Oacyclic(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))))) # label(fact_acyclicP__converse) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4834 (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(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4835 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Set_OPow(T_c),V_B_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_Pow__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4836 (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_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)) = 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)))))) # label(fact_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4837 (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]. 9.15/9.13 4838 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)))))) # label(fact_pos__divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4839 (all V_x all V_y all T_a (class_Divides_Oring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_y),V_x)) -> hAPP(c_Groups_Ouminus__class_Ouminus(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_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y))))) # label(fact_dvd__div__neg) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4840 (all V_A_2 all T_c (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)),V_A_2))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_subset__Compl__self__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4841 (all V_r_2 all T_c all T_d (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)) = V_r_2 <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ORange(T_d,T_c),V_r_2))) # label(fact_Range__empty__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4842 (all V_Pa_2 all V_b_2 all T_d all V_a_2 all T_c (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_a_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_b_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_c)),V_a_2))) -> hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_a_2),B_n)),hAPP(hAPP(c_List_Onth(T_d),V_b_2),B_n))))) -> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_a_2,V_b_2)))) # label(fact_list__all2__all__nthI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4843 (all V_B_2 all V_X_2 all V_f_2 all T_d all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (c_Equiv__Relations_Ocongruent(T_c,tc_fun(T_d,tc_HOL_Obool),V_r_2,V_f_2) -> (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_member(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_f_2,B_x)),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_c,tc_fun(T_d,tc_HOL_Obool)),V_X_2),V_f_2)),V_B_2))))))) # label(fact_UN__equiv__class__type) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4844 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_y_2) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_x_2) <-> V_x_2 = V_y_2))) # label(fact_atMost__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4845 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olinorder__class_OMin(T_c,V_B_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)))))) # label(fact_Min_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4846 (all V_Pa_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_Pa_2,V_n_2),V_m_2))) -> ((V_n_2 = V_m_2 -> hBOOL(hAPP(hAPP(V_Pa_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_Pa_2,V_n_2),V_m_2))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_n_2),V_m_2)))))) # label(fact_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4847 (all V_r_2 all T_c c_Relation_Orefl__on(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))) # label(fact_refl__reflcl) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4848 (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(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs)) # label(fact_drop__Suc) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4849 (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]. 9.15/9.13 4850 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Inf__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4851 (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]. 9.15/9.13 4852 (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)),V_a_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)))))) # label(fact_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4853 (all V_ys_2 all V_y_2 all V_xs_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_c,T_d)),T_c,c_List_Olist_ONil(tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d)))),T_c),c_COMBB(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),T_c),c_List_Olist_OCons(tc_prod(T_c,T_d))),hAPP(hAPP(c_COMBC(T_c,T_d,tc_prod(T_c,T_d)),c_Product__Type_OPair(T_c,T_d)),V_y_2)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),c_List_Ozip(T_c,T_d)),V_ys_2))),V_xs_2)) # label(fact_zip__Cons) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4854 (all V_y_2 all V_x_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,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_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_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),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_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)),c_fequal),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))))) # label(fact_reflcl__set__eq) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4855 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_c_2,V_d_2)))) # label(fact_Int__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4856 (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]. 9.15/9.13 4857 (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_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))))))) # label(fact_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4858 (all V_prod_2 all V_i_2 all V_h_2 all T_h all T_g all V_g_2 all V_f_2 all T_d all T_f all T_c all T_e hAPP(c_Product__Type_Omap__pair(T_g,T_c,T_h,T_d,hAPP(c_Fun_Ocomp(T_e,T_c,T_g,V_f_2),V_h_2),hAPP(c_Fun_Ocomp(T_f,T_d,T_h,V_g_2),V_i_2)),V_prod_2) = hAPP(c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2),hAPP(c_Product__Type_Omap__pair(T_g,T_e,T_h,T_f,V_h_2,V_i_2),V_prod_2))) # label(fact_map__pair_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4859 (all V_xs_2 all T_c (class_Groups_Omonoid__add(T_c) -> c_List_Ofoldr(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_xs_2,c_Groups_Ozero__class_Ozero(T_c)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_xs_2))) # label(fact_foldl__foldr1) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4860 (all V_a_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)) = V_a_2)))) # label(fact_inf__Sup__absorb) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4861 (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]. 9.15/9.13 4862 (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]. 9.15/9.13 4863 (all V_i_2 all V_xs_2 all T_c (c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_Nat_Onat),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),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_c))),V_xs_2))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),hAPP(c_List_Ofilter(tc_List_Olist(T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c)),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),c_List_Otranspose(T_c,V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),V_i_2))))) # label(fact_transpose__column__length) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4864 (all V_n (V_n != 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)))) # label(fact_gr0I) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4865 (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_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) = V_x))) # label(fact_nth__list__update) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4866 (all V_l_2 all V_Pa_2 all T_c (class_Rings_Osemiring__0(T_c) & class_Rings_Odvd(T_c) -> ((exists B_x hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_l_2),B_x)))) <-> (exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_l_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),B_x),c_Groups_Ozero__class_Ozero(T_c))))))))) # label(fact_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4867 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Groups_Oordered__ab__semigroup__add__imp__le(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4868 (all V_B_2 all V_C_2 all T_d all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,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_d),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_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_UN__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4869 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),V_A_2)) = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OSup__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4870 (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]. 9.15/9.13 4871 (all V_xs_2 all T_c c_Random_Oselect__weight(T_c,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_c,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_c),tc_Code__Numeral_Ocode__numeral),c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_c)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_Code__Numeral_Ocode__numeral),c_Product__Type_OPair(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random_Opick(T_c,V_xs_2)))) # label(fact_select__weight__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4872 (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]. 9.15/9.13 4873 (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]. 9.15/9.13 4874 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_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_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(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__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2))))))))) # label(fact_Inf__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4875 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (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(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4876 (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))) -> 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))) = 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)))) & (-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)))))) # label(fact_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4877 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olistrel1(T_c,V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) & V_y_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_x_2),V_y_2)),V_r_2)) & V_ys_2 = V_xs_2)) # label(fact_Cons__listrel1__Cons) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4878 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_g_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Oplus__class_Oplus(T_c)),V_f_2)),V_g_2)),V_A_2))) # label(fact_setsum__addf) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4879 (all V_xs_2 all V_Pa_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4880 (all V_xs_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_d,T_c,V_f_2,hAPP(c_List_Oset(T_d),V_xs_2)) & hBOOL(hAPP(c_List_Odistinct(T_d),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))))) # label(fact_distinct__map) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4881 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> 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)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)))) # label(fact_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4882 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_a)) # label(fact_power__one__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4883 (all V_v_2 all V_u_2 all V_y_2 all V_x_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),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))))) = 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))))))) # label(fact_mult) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4884 (all V_n all V_i all V_m 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) = 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))) # label(fact_min__diff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4885 (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]. 9.15/9.13 4886 (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]. 9.15/9.13 4887 (all V_m_2 all V_n_2 (hAPP(c_Code__Numeral_Oof__nat,V_m_2) = hAPP(c_Code__Numeral_Oof__nat,V_n_2) <-> V_m_2 = V_n_2)) # label(fact_Code__Numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4888 (all V_xss_2 all V_xs_2 all V_x_2 all T_c hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_Olist__case(tc_List_Olist(T_c),T_c,c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),T_c),c_COMBK(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)),c_List_Olist_ONil(T_c))))),V_xss_2)))),c_List_Otranspose(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),V_xs_2),c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_c)),T_c,c_List_Olist_ONil(tc_List_Olist(T_c)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_OCons(tc_List_Olist(T_c))),c_List_Olist_ONil(tc_List_Olist(T_c)))))),V_xss_2))))) = c_List_Otranspose(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_xss_2))) # label(fact_transpose_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4889 (all T_d all T_c 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_c,T_d,tc_HOL_Obool),V_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2))))) # label(fact_prod__caseI) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4890 (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_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2) = V_k_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) = V_j_2))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4891 (all V_A_2 all V_f_2 all T_c hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(c_Int_Onat,hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_c),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2))) # label(fact_transfer__nat__int__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4892 (all V_fa_2 all T_c c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_List_Olist_Olist__size(T_c,V_fa_2,c_List_Olist_ONil(T_c))) # label(fact_list_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4893 (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) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 & c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n_2)) # label(fact_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4894 (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]. 9.15/9.13 4895 (all V_B_2 all V_A_2 all T_c hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_A_2)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2))))) # label(fact_Union__Int__subset) # label(axiom) # label(non_clause). [assumption]. 9.15/9.13 4896 (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]. 9.15/9.14 4897 (all V_zs_2 all V_ys_2 all V_xs_2 all T_c (V_ys_2 = hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),V_zs_2) & hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),V_zs_2) = V_xs_2 <-> V_zs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_append__eq__conv__conj) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4898 (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]. 9.15/9.14 4899 (all V_com2_H_2 all V_com1_H_2 all V_fun_H_2 c_Com_Ocom_OSKIP != c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2)) # label(fact_com_Osimps_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4900 (all V_s_2 all V_r_2 all T_e all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Relation_Orel__comp(T_d,T_e,T_c),V_r_2),V_s_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(c_Relation_Oconverse(T_e,T_c),V_s_2)),hAPP(c_Relation_Oconverse(T_d,T_e),V_r_2))) # label(fact_converse__rel__comp) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4901 (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]. 9.15/9.14 4902 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),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)))) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) = c_Groups_Oone__class_Oone(T_c)))) # label(fact_eq__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4903 (all V_g_2 all V_x_2 all V_y_2 all T_e all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_y_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_y_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),V_g_2)))) # label(fact_folding_Ocommute__left__comp) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4904 (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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(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_n_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 9.15/9.14 4905 (all V_n_2 all V_Pa_2 all V_Qa_2 ((all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_Qa_2,B_x)),hAPP(V_Pa_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_Pa_2)),V_Qa_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_Pa_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_Qa_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))))) # label(fact_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 9.37/9.38 4906 (all V_A_2 all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))) # label(fact_Compl__eq) # label(axiom) # label(non_clause). [assumption]. 9.37/9.38 4907 (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]. 9.37/9.38 9.37/9.38 ============================== end of process non-clausal formulas === 9.37/9.38 9.37/9.38 ============================== PROCESS INITIAL CLAUSES =============== 9.37/9.38 9.37/9.38 ============================== PREDICATE ELIMINATION ================= 9.37/9.38 4908 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 9.37/9.38 4909 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(B,f1(C,B,A))),hAPP(B,f2(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),f1(C,B,A)),f2(C,B,A))) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Big__Operators_Olinorder__class_OMax(A,hAPP(c_Set_Oimage(A,A,B),C)) = hAPP(B,c_Big__Operators_Olinorder__class_OMax(A,C)) # label(fact_Max_Ohom__commute) # label(axiom). [clausify(1)]. 9.37/9.38 4910 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) # label(fact_min__max_Oinf_Oleft__idem) # label(axiom). [clausify(17)]. 9.37/9.38 4911 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B) # label(fact_min__max_OInf__fin_Oeq__fold__idem_H) # label(axiom). [clausify(21)]. 9.37/9.38 4912 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_ONil) # label(axiom). [clausify(67)]. 9.37/9.38 4913 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)) # label(fact_ivl__disj__un_I5_J) # label(axiom). [clausify(90)]. 9.37/9.38 4914 -class_Orderings_Olinorder(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) # label(axiom). [clausify(95)]. 9.37/9.38 4915 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_OdropWhile(A,C,B)) # label(fact_sorted__dropWhile) # label(axiom). [clausify(111)]. 9.37/9.38 4916 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),C)),hAPP(hAPP(c_List_Onth(A),B),D))) # label(fact_sorted__nth__mono) # label(axiom). [clausify(148)]. 9.37/9.38 4917 -class_Orderings_Olinorder(A) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = B # label(fact_min__max_OSup__fin_Osingleton) # label(axiom). [clausify(179)]. 9.37/9.38 4918 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),C)) | hBOOL(hAPP(c_List_Odistinct(B),c_List_Olinorder__class_Oinsort__insert__key(B,A,D,E,C))) # label(fact_distinct__insort__insert) # label(axiom). [clausify(253)]. 9.37/9.38 4919 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C),D) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(A),B)),c_SetInterval_Oord__class_OatLeastAtMost(A,C,D)) # label(fact_Int__atLeastAtMostR2) # label(axiom). [clausify(265)]. 9.37/9.38 4920 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,hAPP(c_List_Otl(A),B)) # label(fact_sorted__tl) # label(axiom). [clausify(270)]. 9.37/9.38 4921 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),B) # label(fact_min__max_Oinf_Ocommute) # label(axiom). [clausify(281)]. 9.37/9.38 4922 -class_Orderings_Olinorder(A) | hBOOL(hAPP(B,C)) | hAPP(c_List_Ofilter(D,B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,A,E),C),F)) = hAPP(c_List_Ofilter(D,B),F) # label(fact_filter__insort__triv) # label(axiom). [clausify(284)]. 9.37/9.38 4923 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_OtakeWhile(A,C,B)) # label(fact_sorted__takeWhile) # label(axiom). [clausify(307)]. 9.37/9.38 4924 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),B) = B # label(fact_min__max_Oinf__absorb2) # label(axiom). [clausify(326)]. 9.37/9.38 4925 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C),D) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,D)),hAPP(c_SetInterval_Oord__class_OatLeast(A),C)) # label(fact_Int__atLeastAtMostL2) # label(axiom). [clausify(332)]. 9.37/9.38 4926 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__infE) # label(axiom). [clausify(353)]. 9.37/9.38 4927 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__infE) # label(axiom). [clausify(353)]. 9.37/9.38 4928 -class_Orderings_Olinorder(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(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_Max__insert) # label(axiom). [clausify(367)]. 9.37/9.38 4929 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Complete__Lattice_OInf__class_OInf(A,C)),D)) # label(fact_Inf__less__iff) # label(axiom). [clausify(387)]. 9.37/9.38 4930 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f43(B,C,A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Complete__Lattice_OInf__class_OInf(A,C)),B)) # label(fact_Inf__less__iff) # label(axiom). [clausify(387)]. 9.37/9.38 4931 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f43(B,C,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Complete__Lattice_OInf__class_OInf(A,C)),B)) # label(fact_Inf__less__iff) # label(axiom). [clausify(387)]. 9.37/9.38 4932 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | C = B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(388)]. 9.37/9.38 4933 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | C != B # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(388)]. 9.37/9.38 4934 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B)) # label(fact_min__max_Ofold__inf__insert) # label(axiom). [clausify(398)]. 9.37/9.38 4935 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Osup__inf__distrib1) # label(axiom). [clausify(400)]. 9.37/9.38 4936 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OlessThan(A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)) # label(fact_ivl__disj__un_I8_J) # label(axiom). [clausify(406)]. 9.37/9.38 4937 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B)) # label(fact_min__max_Ofold__sup__insert) # label(axiom). [clausify(408)]. 9.37/9.38 4938 -class_Orderings_Olinorder(A) | c_List_Oremove1(A,B,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),C)) = C # label(fact_remove1__insort) # label(axiom). [clausify(432)]. 9.37/9.38 4939 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatMost(A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(A),B)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)) # label(fact_ivl__disj__un_I9_J) # label(axiom). [clausify(434)]. 9.37/9.38 4940 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C)) # label(fact_min__max_Oless__infI1) # label(axiom). [clausify(439)]. 9.37/9.38 4941 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)) # label(fact_min__max_Osup__left__commute) # label(axiom). [clausify(497)]. 9.37/9.38 4942 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),B))) # label(fact_le__maxI2) # label(axiom). [clausify(549)]. 9.37/9.38 4943 -class_Orderings_Olinorder(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) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olinorder__class_OMax(A,B)),c_Big__Operators_Olinorder__class_OMax(A,C)) # label(fact_Max_Ounion__disjoint) # label(axiom). [clausify(552)]. 9.37/9.38 4944 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),B) # label(fact_min__max_Osup_Ocommute) # label(axiom). [clausify(581)]. 9.37/9.38 4945 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(584)]. 9.37/9.38 4946 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),B)),C)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(584)]. 9.37/9.38 4947 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(584)]. 9.37/9.38 4948 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C)),hAPP(c_SetInterval_Oord__class_OatMost(A),D)) # label(fact_Int__atLeastAtMostL1) # label(axiom). [clausify(590)]. 9.37/9.38 4949 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),D),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) # label(fact_min__max_Ofold__inf__le__inf) # label(axiom). [clausify(625)]. 9.37/9.38 4950 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),B),D)),C)) # label(fact_min__max_Ole__infI1) # label(axiom). [clausify(637)]. 9.37/9.38 4951 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = C # label(fact_Max_Oinsert__remove) # label(axiom). [clausify(645)]. 9.37/9.38 4952 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olinorder__class_OMax(A,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)))))) # label(fact_Max_Oinsert__remove) # label(axiom). [clausify(645)]. 9.37/9.38 4953 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)) # label(fact_Compl__lessThan) # label(axiom). [clausify(654)]. 9.37/9.38 4954 -class_Orderings_Olinorder(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_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olinorder__class_OMin(A,B)),c_Big__Operators_Olinorder__class_OMin(A,C)) # label(fact_Min__Un) # label(axiom). [clausify(655)]. 9.37/9.38 4955 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)) = B # label(fact_min__max_Oinf__sup__absorb) # label(axiom). [clausify(667)]. 9.37/9.38 4956 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(c_List_Orev(A),hAPP(hAPP(c_List_Omap(B,A),C),D))) | c_List_OtakeWhile(B,hAPP(hAPP(c_COMBB(A,tc_HOL_Obool,B),hAPP(c_Orderings_Oord__class_Oless(A),E)),C),D) = hAPP(c_List_Ofilter(B,hAPP(hAPP(c_COMBB(A,tc_HOL_Obool,B),hAPP(c_Orderings_Oord__class_Oless(A),E)),C)),D) # label(fact_filter__equals__takeWhile__sorted__rev) # label(axiom). [clausify(680)]. 9.37/9.38 4957 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) # label(fact_min__max_OSup__fin_Oinsert) # label(axiom). [clausify(685)]. 9.37/9.38 4958 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(A),C)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,D)) # label(fact_Int__atLeastAtMostR1) # label(axiom). [clausify(687)]. 9.37/9.38 4959 -class_Lattices_Osemilattice__sup(A) | -class_Orderings_Olinorder(A) | c_Lattices_Osemilattice__sup__class_Osup(A) = c_Orderings_Oord__class_Omax(A) # label(fact_sup__max) # label(axiom). [clausify(688)]. 9.37/9.38 4960 -class_Orderings_Olinorder(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_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__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_COMBB(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool),A),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_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),A),c_COMBS(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(A,tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),A),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_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),A),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,tc_fun(A,A)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_fun(A,A)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),A),c_COMBB(tc_fun(A,A),tc_fun(A,tc_HOL_Obool),A)),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))),c_Orderings_Oord__class_Omin(A)))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,A)),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))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)) # label(fact_min__max_Oinf__Sup2__distrib) # label(axiom). [clausify(693)]. 9.37/9.38 4961 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B) = hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B) # label(fact_min__max_OInf__fin_OF__eq) # label(axiom). [clausify(696)]. 9.37/9.38 4962 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)),c_SetInterval_Oord__class_OatLeastAtMost(A,C,D)) # label(fact_ivl__disj__un_I19_J) # label(axiom). [clausify(731)]. 9.37/9.38 4963 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B) # label(fact_min__max_OSup__fin_Oeq__fold__idem_H) # label(axiom). [clausify(734)]. 9.37/9.38 4964 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),D),B)),C)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(768)]. 9.37/9.38 4965 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),B),D)),C)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(768)]. 9.37/9.38 4966 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),B)),C)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(768)]. 9.37/9.38 4967 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(c_SetInterval_Oord__class_OlessThan(A),C))) # label(fact_lessThan__subset__iff) # label(axiom). [clausify(785)]. 9.37/9.38 4968 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(c_SetInterval_Oord__class_OlessThan(A),C))) # label(fact_lessThan__subset__iff) # label(axiom). [clausify(785)]. 9.37/9.38 4969 -class_Orderings_Olinorder(A) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C))) = hAPP(c_List_Ofoldl(A,A,c_Orderings_Oord__class_Omax(A),B),C) # label(fact_min__max_OSup__fin__set__fold) # label(axiom). [clausify(786)]. 9.37/9.38 4970 -class_Orderings_Olinorder(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) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olinorder__class_OMin(A,B)),c_Big__Operators_Olinorder__class_OMin(A,C)) # label(fact_Min_Ounion__disjoint) # label(axiom). [clausify(797)]. 9.37/9.38 4971 -class_Orderings_Olinorder(A) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = B # label(fact_Min_Osingleton) # label(axiom). [clausify(816)]. 9.37/9.38 4972 -class_Orderings_Olinorder(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) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)) # label(fact_min__max_OInf__fin_Ounion__disjoint) # label(axiom). [clausify(823)]. 9.37/9.38 4973 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(834)]. 9.37/9.38 4974 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f84(C,B,A))),hAPP(B,f85(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f84(C,B,A)),f85(C,B,A))) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Big__Operators_Olinorder__class_OMin(A,hAPP(c_Set_Oimage(A,A,B),C)) = hAPP(B,c_Big__Operators_Olinorder__class_OMin(A,C)) # label(fact_Min_Ohom__commute) # label(axiom). [clausify(845)]. 9.37/9.38 4975 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) # label(fact_min__max_OSup__fin_Osubset__idem) # label(axiom). [clausify(872)]. 9.37/9.38 4976 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) # label(fact_min__max_Osup__assoc) # label(axiom). [clausify(878)]. 9.37/9.38 4977 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Big__Operators_Olinorder__class_OMax(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olinorder__class_OMax(A,C)),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_Max_Osubset__idem) # label(axiom). [clausify(907)]. 9.37/9.38 4978 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(B,A),C),D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(D,f99(E,D,C,B,A))),E)) # label(fact_INF__less__iff) # label(axiom). [clausify(930)]. 9.37/9.38 4979 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(B,A),C),D)),E)) | hBOOL(hAPP(hAPP(c_member(B),f99(E,D,C,B,A)),C)) # label(fact_INF__less__iff) # label(axiom). [clausify(930)]. 9.37/9.38 4980 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(B,A),C),D)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(D,F)),E)) | -hBOOL(hAPP(hAPP(c_member(B),F),C)) # label(fact_INF__less__iff) # label(axiom). [clausify(930)]. 9.37/9.38 4981 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) # label(fact_min__max_OInf__fin_Osubset__idem) # label(axiom). [clausify(956)]. 9.37/9.38 4982 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,C) | hBOOL(hAPP(hAPP(c_member(A),f102(C,B,A)),hAPP(c_List_Oset(A),B))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4983 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,C) | hBOOL(hAPP(hAPP(c_member(A),f103(C,B,A)),hAPP(c_List_Oset(A),C))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4984 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f102(C,B,A)),f103(C,B,A))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4985 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4986 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),C),B)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4987 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -hBOOL(hAPP(hAPP(c_member(A),D),hAPP(c_List_Oset(A),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),C),E)) # label(fact_sorted__append) # label(axiom). [clausify(974)]. 9.37/9.38 4988 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) # label(fact_min__max_Osup_Oassoc) # label(axiom). [clausify(994)]. 9.37/9.38 4989 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),hAPP(c_List_Ofilter(B,hAPP(hAPP(c_COMBC(B,A,tc_HOL_Obool),hAPP(hAPP(c_COMBB(A,tc_fun(A,tc_HOL_Obool),B),c_fequal),C)),hAPP(D,E))),E))) # label(fact_sorted__map__same) # label(axiom). [clausify(1011)]. 9.37/9.38 4990 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),C),B)) # label(fact_sorted__insort) # label(axiom). [clausify(1023)]. 9.37/9.38 4991 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),C),B)) # label(fact_sorted__insort) # label(axiom). [clausify(1023)]. 9.37/9.38 4992 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),c_List_Olinorder__class_Osort__key(B,A,C,D))) | hBOOL(hAPP(c_List_Odistinct(B),D)) # label(fact_distinct__sort) # label(axiom). [clausify(1060)]. 9.37/9.38 4993 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_List_Odistinct(B),c_List_Olinorder__class_Osort__key(B,A,C,D))) | -hBOOL(hAPP(c_List_Odistinct(B),D)) # label(fact_distinct__sort) # label(axiom). [clausify(1060)]. 9.37/9.38 4994 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Complete__Lattice_OSup__class_OSup(A,C))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1084)]. 9.37/9.38 4995 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f111(B,C,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Complete__Lattice_OSup__class_OSup(A,B))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1084)]. 9.37/9.38 4996 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),f111(C,B,A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Complete__Lattice_OSup__class_OSup(A,C))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1084)]. 9.37/9.38 4997 -class_Orderings_Olinorder(A) | c_Orderings_Oord_Omin(A,hAPP(c_COMBC(A,A,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(A))) = c_Orderings_Oord__class_Omax(A) # label(fact_dual__min) # label(axiom). [clausify(1086)]. 9.37/9.38 4998 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__le) # label(axiom). [clausify(1104)]. 9.37/9.38 4999 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__le) # label(axiom). [clausify(1104)]. 9.37/9.38 5000 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),E))) # label(fact_min__max_Osup__mono) # label(axiom). [clausify(1117)]. 9.37/9.38 5001 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) # label(fact_min__max_Osup__left__idem) # label(axiom). [clausify(1149)]. 9.37/9.38 5002 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(B,f119(C,B,A))),hAPP(B,f120(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),f119(C,B,A)),f120(C,B,A))) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(c_Set_Oimage(A,A,B),C)) = hAPP(B,c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)) # label(fact_min__max_OSup__fin_Ohom__commute) # label(axiom). [clausify(1163)]. 9.37/9.38 5003 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | -hBOOL(hAPP(E,F)) | hAPP(c_List_Ofilter(B,E),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),F),D)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),F),hAPP(c_List_Ofilter(B,E),D)) # label(fact_filter__insort) # label(axiom). [clausify(1180)]. 9.37/9.38 5004 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)) # label(fact_min__max_Oinf__sup__distrib1) # label(axiom). [clausify(1188)]. 9.37/9.38 5005 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),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)))))) # label(fact_min__max_OSup__fin_Oinsert__remove) # label(axiom). [clausify(1220)]. 9.37/9.38 5006 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = C # label(fact_min__max_OSup__fin_Oinsert__remove) # label(axiom). [clausify(1220)]. 9.37/9.38 5007 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),E)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) # label(fact_Int__atLeastLessThan) # label(axiom). [clausify(1229)]. 9.37/9.38 5008 -class_Orderings_Olinorder(A) | hAPP(c_List_Oset(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E)) = hAPP(hAPP(c_Set_Oinsert(B),D),hAPP(c_List_Oset(B),E)) # label(fact_set__insort) # label(axiom). [clausify(1243)]. 9.37/9.38 5009 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) # label(fact_min__max_OInf__fin_Oinsert) # label(axiom). [clausify(1271)]. 9.37/9.38 5010 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(c_List_Odistinct(A),B)) | -c_List_Olinorder__class_Osorted(A,C) | -hBOOL(hAPP(c_List_Odistinct(A),C)) | hAPP(c_List_Oset(A),C) != hAPP(c_List_Oset(A),B) | C = B # label(fact_sorted__distinct__set__unique) # label(axiom). [clausify(1276)]. 9.37/9.38 5011 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) # label(fact_min__max_Ole__infI) # label(axiom). [clausify(1278)]. 9.37/9.38 5012 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f138(B,C,A)),hAPP(c_List_Oset(A),B))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(1310)]. 9.37/9.38 5013 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f138(B,C,A))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(1310)]. 9.37/9.38 5014 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(1310)]. 9.37/9.38 5015 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),B)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),D),C)) # label(fact_sorted__Cons) # label(axiom). [clausify(1310)]. 9.37/9.38 5016 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) # label(fact_min__max_Osup_Oleft__idem) # label(axiom). [clausify(1337)]. 9.37/9.38 5017 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,C,D)) # label(fact_ivl__disj__un_I16_J) # label(axiom). [clausify(1349)]. 9.37/9.38 5018 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__cases) # label(axiom). [clausify(1378)]. 9.37/9.38 5019 -class_Orderings_Olinorder(A) | -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_Orderings_Oord__class_Omax(A),c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olinorder__class_OMax(A,B)),c_Big__Operators_Olinorder__class_OMax(A,C)) # label(fact_Max_Ounion__inter) # label(axiom). [clausify(1383)]. 9.37/9.39 5020 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f147(C,D,B,A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C))) # label(fact_min__max_Ofold__sup__le__sup) # label(axiom). [clausify(1408)]. 9.37/9.39 5021 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f147(C,D,B,A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C))) # label(fact_min__max_Ofold__sup__le__sup) # label(axiom). [clausify(1408)]. 9.37/9.39 5022 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(B),f150(C,D,E,B,A)),hAPP(c_List_Oset(B),E))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,D),C),E) = hAPP(hAPP(c_List_Olist_OCons(B),C),E) # label(fact_insort__is__Cons) # label(axiom). [clausify(1421)]. 9.37/9.39 5023 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,C)),hAPP(B,f150(C,B,D,E,A)))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),D) = hAPP(hAPP(c_List_Olist_OCons(E),C),D) # label(fact_insort__is__Cons) # label(axiom). [clausify(1421)]. 9.37/9.39 5024 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(c_List_Oset(A),c_List_Olinorder__class_Osorted__list__of__set(A,B)) = B # label(fact_sorted__list__of__set) # label(axiom). [clausify(1423)]. 9.37/9.39 5025 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted(A,c_List_Olinorder__class_Osorted__list__of__set(A,B)) # label(fact_sorted__list__of__set) # label(axiom). [clausify(1423)]. 9.37/9.39 5026 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osorted__list__of__set(A,B))) # label(fact_sorted__list__of__set) # label(axiom). [clausify(1423)]. 9.37/9.39 5027 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C))) # label(fact_min__max_Oless__supI2) # label(axiom). [clausify(1447)]. 9.37/9.39 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(A,f1(B,A,tc_Nat_Onat))),hAPP(A,f2(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),f1(B,A,tc_Nat_Onat)),f2(B,A,tc_Nat_Onat))) | -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_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B)). [resolve(4908,a,4909,a)]. 9.37/9.39 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B). [resolve(4908,a,4910,a)]. 9.37/9.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Nat_Onat)),B),A). [resolve(4908,a,4911,a)]. 9.37/9.39 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olist_ONil(tc_Nat_Onat)). [resolve(4908,a,4912,a)]. 9.37/9.39 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)). [resolve(4908,a,4913,a)]. 9.46/9.42 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_OdropWhile(tc_Nat_Onat,B,A)). [resolve(4908,a,4915,a)]. 9.46/9.42 Derived: -c_List_Olinorder__class_Osorted(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),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),B)),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),C))). [resolve(4908,a,4916,a)]. 9.46/9.42 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) = A. [resolve(4908,a,4917,a)]. 9.46/9.42 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Nat_Onat,C,D,B))). [resolve(4908,a,4918,a)]. 9.46/9.42 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)). [resolve(4908,a,4919,a)]. 9.46/9.42 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Otl(tc_Nat_Onat),A)). [resolve(4908,a,4920,a)]. 9.46/9.42 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),A). [resolve(4908,a,4921,a)]. 9.46/9.42 Derived: hBOOL(hAPP(A,B)) | hAPP(c_List_Ofilter(C,A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Nat_Onat,D),B),E)) = hAPP(c_List_Ofilter(C,A),E). [resolve(4908,a,4922,a)]. 9.46/9.42 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_OtakeWhile(tc_Nat_Onat,B,A)). [resolve(4908,a,4923,a)]. 9.46/9.42 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),A) = A. [resolve(4908,a,4924,a)]. 9.46/9.42 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,C)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B)). [resolve(4908,a,4925,a)]. 9.46/9.42 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(4908,a,4926,a)]. 9.46/9.42 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(4908,a,4927,a)]. 9.46/9.42 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(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)). [resolve(4908,a,4928,a)]. 9.46/9.42 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Complete__Lattice_OInf__class_OInf(tc_Nat_Onat,B)),C)). [resolve(4908,a,4929,b)]. 9.46/9.42 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f43(A,B,tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Complete__Lattice_OInf__class_OInf(tc_Nat_Onat,B)),A)). [resolve(4908,a,4930,b)]. 9.46/9.44 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f43(A,B,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Complete__Lattice_OInf__class_OInf(tc_Nat_Onat,B)),A)). [resolve(4908,a,4931,b)]. 9.46/9.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),B),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),B),A)). [resolve(4908,a,4934,a)]. 9.46/9.44 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)). [resolve(4908,a,4935,a)]. 9.46/9.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)). [resolve(4908,a,4936,a)]. 9.46/9.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A)). [resolve(4908,a,4937,a)]. 9.46/9.44 Derived: c_List_Oremove1(tc_Nat_Onat,A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),A),B)) = B. [resolve(4908,a,4938,a)]. 9.46/9.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)). [resolve(4908,a,4939,a)]. 9.46/9.44 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_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B)). [resolve(4908,a,4940,a)]. 9.46/9.44 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)). [resolve(4908,a,4941,a)]. 9.46/9.44 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),A))). [resolve(4908,a,4942,a)]. 9.46/9.44 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) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B)). [resolve(4908,a,4943,a)]. 9.46/9.44 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),A). [resolve(4908,a,4944,a)]. 9.46/9.44 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_Orderings_Oord__class_Omin(tc_Nat_Onat),C),A)),B)). [resolve(4908,a,4946,a)]. 9.46/9.46 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B)). [resolve(4908,a,4947,a)]. 9.46/9.46 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),C)). [resolve(4908,a,4948,a)]. 9.46/9.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),C),A)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))). [resolve(4908,a,4949,a)]. 9.46/9.46 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_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B)). [resolve(4908,a,4950,a)]. 9.46/9.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = B. [resolve(4908,a,4951,a)]. 9.46/9.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(4908,a,4952,a)]. 9.46/9.46 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)). [resolve(4908,a,4953,a)]. 9.46/9.46 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_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,B)). [resolve(4908,a,4954,a)]. 9.46/9.46 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)) = A. [resolve(4908,a,4955,a)]. 9.46/9.46 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C))) | c_List_OtakeWhile(A,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D)),B),C) = hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D)),B)),C). [resolve(4908,a,4956,a)]. 9.46/9.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(4908,a,4957,a)]. 9.46/9.46 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,C)). [resolve(4908,a,4958,a)]. 9.46/9.46 Derived: -class_Lattices_Osemilattice__sup(tc_Nat_Onat) | c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat) = c_Orderings_Oord__class_Omax(tc_Nat_Onat). [resolve(4908,a,4959,b)]. 9.46/9.46 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_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__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_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),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_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_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),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_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),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_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_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Nat_Onat),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_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat)),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))),c_Orderings_Oord__class_Omin(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_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_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat)),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)),A)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),B)). [resolve(4908,a,4960,a)]. 9.56/9.49 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A). [resolve(4908,a,4961,a)]. 9.56/9.49 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)). [resolve(4908,a,4962,a)]. 9.56/9.49 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A). [resolve(4908,a,4963,a)]. 9.56/9.49 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_Orderings_Oord__class_Omin(tc_Nat_Onat),C),A)),B)). [resolve(4908,a,4964,a)]. 9.56/9.49 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),A)),B)). [resolve(4908,a,4966,a)]. 9.56/9.49 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_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))). [resolve(4908,a,4967,a)]. 9.56/9.49 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_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))). [resolve(4908,a,4968,a)]. 9.56/9.49 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_List_Oset(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B))) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B). [resolve(4908,a,4969,a)]. 9.56/9.49 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) = A. [resolve(4908,a,4971,a)]. 9.56/9.49 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) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)). [resolve(4908,a,4972,a)]. 9.56/9.49 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),B),A)). [resolve(4908,a,4973,a)]. 9.56/9.49 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f84(B,A,tc_Nat_Onat))),hAPP(A,f85(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f84(B,A,tc_Nat_Onat)),f85(B,A,tc_Nat_Onat))) | -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_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,B)). [resolve(4908,a,4974,a)]. 9.57/9.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),B)),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(4908,a,4975,a)]. 9.57/9.50 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)). [resolve(4908,a,4976,a)]. 9.57/9.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B)),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)). [resolve(4908,a,4977,a)]. 9.57/9.50 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Nat_Onat),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(C,f99(D,C,B,A,tc_Nat_Onat))),D)). [resolve(4908,a,4978,a)]. 9.57/9.50 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Nat_Onat),B),C)),D)) | hBOOL(hAPP(hAPP(c_member(A),f99(D,C,B,A,tc_Nat_Onat)),B)). [resolve(4908,a,4979,a)]. 9.57/9.50 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Nat_Onat),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(C,E)),D)) | -hBOOL(hAPP(hAPP(c_member(A),E),B)). [resolve(4908,a,4980,a)]. 9.57/9.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)). [resolve(4908,a,4981,a)]. 9.57/9.50 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f102(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)). [resolve(4908,a,4982,a)]. 9.57/9.50 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f103(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),B))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)). [resolve(4908,a,4983,a)]. 9.57/9.50 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f102(B,A,tc_Nat_Onat)),f103(B,A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)). [resolve(4908,a,4984,a)]. 9.58/9.53 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)). [resolve(4908,a,4985,a)]. 9.58/9.53 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),B),A)). [resolve(4908,a,4986,a)]. 9.58/9.53 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),hAPP(c_List_Oset(tc_Nat_Onat),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),B),D)). [resolve(4908,a,4987,a)]. 9.58/9.53 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),A),c_fequal),B)),hAPP(C,D))),D))). [resolve(4908,a,4989,a)]. 9.58/9.53 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),B),A)). [resolve(4908,a,4990,a)]. 9.58/9.53 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),B),A)). [resolve(4908,a,4991,a)]. 9.58/9.53 Derived: -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,C))) | hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(4908,a,4992,a)]. 9.58/9.53 Derived: hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,C))) | -hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(4908,a,4993,a)]. 9.58/9.53 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,B))). [resolve(4908,a,4994,b)]. 9.58/9.53 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f111(A,B,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,A))). [resolve(4908,a,4995,b)]. 9.58/9.53 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),f111(B,A,tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,B))). [resolve(4908,a,4996,b)]. 9.58/9.53 Derived: c_Orderings_Oord_Omin(tc_Nat_Onat,hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat))) = c_Orderings_Oord__class_Omax(tc_Nat_Onat). [resolve(4908,a,4997,a)]. 9.58/9.53 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),B),A)). [resolve(4908,a,4999,a)]. 9.58/9.53 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),D))). [resolve(4908,a,5000,a)]. 9.58/9.53 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B). [resolve(4908,a,5001,a)]. 9.58/9.53 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(A,f119(B,A,tc_Nat_Onat))),hAPP(A,f120(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),f119(B,A,tc_Nat_Onat)),f120(B,A,tc_Nat_Onat))) | -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_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,A),B)) = hAPP(A,c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),B)). [resolve(4908,a,5002,a)]. 9.58/9.55 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | -hBOOL(hAPP(D,E)) | hAPP(c_List_Ofilter(A,D),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),E),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),E),hAPP(c_List_Ofilter(A,D),C)). [resolve(4908,a,5003,a)]. 9.58/9.55 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)). [resolve(4908,a,5004,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(4908,a,5005,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = B. [resolve(4908,a,5006,a)]. 9.58/9.55 Derived: hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),D)). [resolve(4908,a,5007,a)]. 9.58/9.55 Derived: hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D)) = hAPP(hAPP(c_Set_Oinsert(A),C),hAPP(c_List_Oset(A),D)). [resolve(4908,a,5008,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)). [resolve(4908,a,5009,a)]. 9.58/9.55 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),A)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),B)) | hAPP(c_List_Oset(tc_Nat_Onat),B) != hAPP(c_List_Oset(tc_Nat_Onat),A) | B = A. [resolve(4908,a,5010,a)]. 9.58/9.55 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),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))). [resolve(4908,a,5011,a)]. 9.58/9.57 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f138(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(4908,a,5012,a)]. 9.58/9.57 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f138(A,B,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(4908,a,5013,a)]. 9.58/9.57 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(4908,a,5014,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),A)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),C),B)). [resolve(4908,a,5015,a)]. 9.58/9.57 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,B,C)). [resolve(4908,a,5017,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),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(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B)). [resolve(4908,a,5019,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f147(B,C,A,tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B))). [resolve(4908,a,5020,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f147(B,C,A,tc_Nat_Onat)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B))). [resolve(4908,a,5021,a)]. 9.58/9.57 Derived: hBOOL(hAPP(hAPP(c_member(A),f150(B,C,D,A,tc_Nat_Onat)),hAPP(c_List_Oset(A),D))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,C),B),D) = hAPP(hAPP(c_List_Olist_OCons(A),B),D). [resolve(4908,a,5022,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,B)),hAPP(A,f150(B,A,C,D,tc_Nat_Onat)))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),C) = hAPP(hAPP(c_List_Olist_OCons(D),B),C). [resolve(4908,a,5023,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(c_List_Oset(tc_Nat_Onat),c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,A)) = A. [resolve(4908,a,5024,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,A)). [resolve(4908,a,5025,a)]. 9.58/9.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,A))). [resolve(4908,a,5026,a)]. 9.58/9.60 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B))). [resolve(4908,a,5027,a)]. 9.58/9.60 5028 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),hAPP(c_List_Ofilter(B,E),D))) # label(fact_sorted__filter) # label(axiom). [clausify(1497)]. 9.58/9.60 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(c_List_Ofilter(A,D),C))). [resolve(5028,a,4908,a)]. 9.58/9.60 5029 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(1498)]. 9.58/9.60 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_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5029,a,4908,a)]. 9.58/9.60 5030 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | hBOOL(hAPP(hAPP(c_member(A),f160(C,B,A)),B)) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(1498)]. 9.58/9.60 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_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f160(B,A,tc_Nat_Onat)),A)). [resolve(5030,a,4908,a)]. 9.58/9.60 5031 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),f160(C,B,A))) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(1498)]. 9.58/9.60 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_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),f160(B,A,tc_Nat_Onat))). [resolve(5031,a,4908,a)]. 9.58/9.60 5032 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D))),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)))) # label(fact_min__max_Odistrib__inf__le) # label(axiom). [clausify(1506)]. 9.58/9.60 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)))). [resolve(5032,a,4908,a)]. 9.58/9.60 5033 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),B),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Big__Operators_Olinorder__class_OMax(A,B)),c_Big__Operators_Olinorder__class_OMax(A,C))) # label(fact_Max__mono) # label(axiom). [clausify(1529)]. 9.67/9.61 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B))). [resolve(5033,a,4908,a)]. 9.67/9.61 5034 -class_Orderings_Olinorder(A) | hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),c_List_Olinorder__class_Osort__key(B,A,C,D)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),D) # label(fact_length__sort) # label(axiom). [clausify(1534)]. 9.67/9.61 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,C)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C). [resolve(5034,a,4908,a)]. 9.67/9.61 5035 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),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)))))) # label(fact_min__max_OInf__fin_Oremove) # label(axiom). [clausify(1552)]. 9.67/9.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5035,a,4908,a)]. 9.67/9.61 5036 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B) = C # label(fact_min__max_OInf__fin_Oremove) # label(axiom). [clausify(1552)]. 9.67/9.61 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = B. [resolve(5036,a,4908,a)]. 9.67/9.61 5037 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | c_List_Olinorder__class_Oinsort__insert__key(A,A,c_COMBI(A),B,C) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),C) # label(fact_insort__insert__insort) # label(axiom). [clausify(1565)]. 9.67/9.61 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),A,B) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),A),B). [resolve(5037,a,4908,a)]. 9.67/9.63 5038 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | C != E | D != B | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1570)]. 9.67/9.63 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)) | B != D | C != A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B). [resolve(5038,a,4908,a)]. 9.67/9.63 5039 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | C = E | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1570)]. 9.67/9.63 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)) | B = D | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B). [resolve(5039,a,4908,a)]. 9.67/9.63 5040 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | D = B | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1570)]. 9.67/9.63 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)) | C = A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B). [resolve(5040,a,4908,a)]. 9.67/9.63 5041 -class_Orderings_Olinorder(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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),D)) # label(fact_fold1__below__iff) # label(axiom). [clausify(1597)]. 9.67/9.63 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),B),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(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),C)). [resolve(5041,a,4908,a)]. 9.67/9.63 5042 -class_Orderings_Olinorder(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),f166(C,B,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) # label(fact_fold1__below__iff) # label(axiom). [clausify(1597)]. 9.67/9.63 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),f166(B,A,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),B)). [resolve(5042,a,4908,a)]. 9.67/9.63 5043 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless__eq(A),f166(C,B,A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) # label(fact_fold1__below__iff) # label(axiom). [clausify(1597)]. 9.67/9.64 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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f166(B,A,tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),B)). [resolve(5043,a,4908,a)]. 9.67/9.64 5044 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olinorder__class_OMax(A,B) = hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omax(A)),B) # label(fact_Max_OF__eq) # label(axiom). [clausify(1599)]. 9.67/9.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),A). [resolve(5044,a,4908,a)]. 9.67/9.64 5045 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olinorder__class_OMin(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olinorder__class_OMin(A,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)))))) # label(fact_Min_Oremove) # label(axiom). [clausify(1601)]. 9.67/9.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5045,a,4908,a)]. 9.67/9.64 5046 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olinorder__class_OMin(A,B) = C # label(fact_Min_Oremove) # label(axiom). [clausify(1601)]. 9.67/9.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = B. [resolve(5046,a,4908,a)]. 9.67/9.64 5047 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted__list__of__set(A,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))))) = c_List_Oremove1(A,C,c_List_Olinorder__class_Osorted__list__of__set(A,B)) # label(fact_sorted__list__of__set__remove) # label(axiom). [clausify(1605)]. 9.67/9.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) = c_List_Oremove1(tc_Nat_Onat,B,c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,A)). [resolve(5047,a,4908,a)]. 9.67/9.68 5048 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),D),B))) # label(fact_min__max_Osup__le__fold__sup) # label(axiom). [clausify(1614)]. 9.67/9.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),C),A))). [resolve(5048,a,4908,a)]. 9.67/9.68 5049 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),C),D)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),D)) # label(fact_insort__left__comm) # label(axiom). [clausify(1645)]. 9.67/9.68 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),B),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),A),C)). [resolve(5049,a,4908,a)]. 9.67/9.68 5050 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_min__max_Oinf_Oidem) # label(axiom). [clausify(1650)]. 9.67/9.68 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),A) = A. [resolve(5050,a,4908,a)]. 9.67/9.68 5051 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B) # label(fact_Max_Oeq__fold_H) # label(axiom). [clausify(1684)]. 9.67/9.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A). [resolve(5051,a,4908,a)]. 9.67/9.68 5052 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OatLeastAtMost(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),E)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,D)),c_SetInterval_Oord__class_OatLeastAtMost(A,C,E)) # label(fact_Int__atLeastAtMost) # label(axiom). [clausify(1688)]. 9.67/9.68 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,C)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,D)). [resolve(5052,a,4908,a)]. 9.67/9.68 5053 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1716)]. 9.67/9.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5053,a,4908,a)]. 9.67/9.68 5054 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1716)]. 9.77/9.72 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5054,a,4908,a)]. 9.77/9.72 5055 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1716)]. 9.77/9.72 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5055,a,4908,a)]. 9.77/9.72 5056 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),D),B)),C)) # label(fact_min__max_Ole__infI2) # label(axiom). [clausify(1718)]. 9.77/9.72 5057 -class_Orderings_Olinorder(A) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C))) = hAPP(c_List_Ofoldl(A,A,c_Orderings_Oord__class_Omin(A),B),C) # label(fact_Min__fin__set__fold) # label(axiom). [clausify(1726)]. 9.77/9.72 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(c_List_Oset(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B))) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B). [resolve(5057,a,4908,a)]. 9.77/9.72 5058 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),f181(B,A)),f182(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f181(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f182(B,A)),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(A),c_Big__Operators_Olinorder__class_OMin(A,B)),B)) # label(fact_Min_Oclosed) # label(axiom). [clausify(1730)]. 9.77/9.72 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_Orderings_Oord__class_Omin(tc_Nat_Onat),f181(A,tc_Nat_Onat)),f182(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f181(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f182(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),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),A)). [resolve(5058,a,4908,a)]. 9.77/9.72 5059 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) != hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C) | B = C # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(1781)]. 9.77/9.72 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B) | A = B. [resolve(5059,a,4908,a)]. 9.77/9.72 5060 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) = hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C) | B != C # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(1781)]. 9.77/9.72 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B) | A != B. [resolve(5060,a,4908,a)]. 9.77/9.72 5061 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_linorder__le__cases) # label(axiom). [clausify(1796)]. 9.77/9.72 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),A)). [resolve(5061,a,4908,a)]. 9.77/9.72 5062 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olinorder__class_OMin(A,B) = hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B) # label(fact_Min_OF__eq) # label(axiom). [clausify(1814)]. 9.77/9.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A). [resolve(5062,a,4908,a)]. 9.77/9.75 5063 -class_Orderings_Olinorder(A) | -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_Orderings_Oord__class_Omin(A),c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olinorder__class_OMin(A,B)),c_Big__Operators_Olinorder__class_OMin(A,C)) # label(fact_Min_Ounion__inter) # label(axiom). [clausify(1819)]. 9.77/9.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),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(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,B)). [resolve(5063,a,4908,a)]. 9.77/9.75 5064 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omax(A),f200(B,A)),f201(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f200(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f201(B,A)),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(A),c_Big__Operators_Olinorder__class_OMax(A,B)),B)) # label(fact_Max_Oclosed) # label(axiom). [clausify(1828)]. 9.77/9.77 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_Orderings_Oord__class_Omax(tc_Nat_Onat),f200(A,tc_Nat_Onat)),f201(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f200(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f201(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),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),A)). [resolve(5064,a,4908,a)]. 9.77/9.77 5065 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(c_List_Orev(A),B)) | c_List_Olist_ONil(A) = B | c_List_Ofoldr(A,A,c_Orderings_Oord__class_Omax(A),B,C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_List_Onth(A),B),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),C) # label(fact_foldr__max__sorted) # label(axiom). [clausify(1873)]. 9.77/9.77 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),A)) | c_List_Olist_ONil(tc_Nat_Onat) = A | c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5065,a,4908,a)]. 9.77/9.77 5066 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(c_List_Orev(A),B)) | c_List_Olist_ONil(A) != B | c_List_Ofoldr(A,A,c_Orderings_Oord__class_Omax(A),B,C) = C # label(fact_foldr__max__sorted) # label(axiom). [clausify(1873)]. 9.77/9.77 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(tc_Nat_Onat),A)) | c_List_Olist_ONil(tc_Nat_Onat) != A | c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A,B) = B. [resolve(5066,a,4908,a)]. 9.77/9.77 5067 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = C # label(fact_Min_Oinsert__remove) # label(axiom). [clausify(1876)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = B. [resolve(5067,a,4908,a)]. 9.77/9.77 5068 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olinorder__class_OMin(A,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)))))) # label(fact_Min_Oinsert__remove) # label(axiom). [clausify(1876)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5068,a,4908,a)]. 9.77/9.77 5069 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f210(C,B,A))),hAPP(B,f211(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f210(C,B,A)),f211(C,B,A))) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(c_Set_Oimage(A,A,B),C)) = hAPP(B,c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)) # label(fact_min__max_OInf__fin_Ohom__commute) # label(axiom). [clausify(1886)]. 9.77/9.77 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f210(B,A,tc_Nat_Onat))),hAPP(A,f211(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f210(B,A,tc_Nat_Onat)),f211(B,A,tc_Nat_Onat))) | -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_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,A),B)) = hAPP(A,c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)). [resolve(5069,a,4908,a)]. 9.77/9.77 5070 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B) = C # label(fact_min__max_OSup__fin_Oremove) # label(axiom). [clausify(1893)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A) = B. [resolve(5070,a,4908,a)]. 9.77/9.77 5071 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),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)))))) # label(fact_min__max_OSup__fin_Oremove) # label(axiom). [clausify(1893)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5071,a,4908,a)]. 9.77/9.77 5072 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olinorder__class_OMax(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olinorder__class_OMax(A,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)))))) # label(fact_Max_Oremove) # label(axiom). [clausify(1901)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5072,a,4908,a)]. 9.77/9.77 5073 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olinorder__class_OMax(A,B) = C # label(fact_Max_Oremove) # label(axiom). [clausify(1901)]. 9.77/9.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = B. [resolve(5073,a,4908,a)]. 9.86/9.79 5074 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__leE) # label(axiom). [clausify(1911)]. 9.86/9.79 5075 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),c_List_Olist_ONil(A))) # label(fact_sorted__single) # label(axiom). [clausify(1923)]. 9.86/9.79 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),c_List_Olist_ONil(tc_Nat_Onat))). [resolve(5075,a,4908,a)]. 9.86/9.79 5076 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C)) # label(fact_ivl__disj__un_I13_J) # label(axiom). [clausify(1936)]. 9.86/9.79 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B)). [resolve(5076,a,4908,a)]. 9.86/9.79 5077 -class_Orderings_Olinorder(A) | -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_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)) # label(fact_min__max_OInf__fin_Ounion__inter) # label(axiom). [clausify(1937)]. 9.86/9.79 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),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(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)). [resolve(5077,a,4908,a)]. 9.86/9.79 5078 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)) = B # label(fact_min__max_Osup__inf__absorb) # label(axiom). [clausify(1946)]. 9.86/9.79 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)) = A. [resolve(5078,a,4908,a)]. 9.86/9.79 5079 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(1947)]. 9.86/9.79 5080 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C != B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(1947)]. 9.87/9.87 5081 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(1947)]. 9.87/9.87 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),B),A)). [resolve(5081,a,4908,a)]. 9.87/9.87 5082 -class_Orderings_Olinorder(A) | hAPP(B,C) = hAPP(B,D) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),D),F)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),D),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),F)) # label(fact_insort__key__left__comm) # label(axiom). [clausify(2012)]. 9.87/9.87 Derived: hAPP(A,B) = hAPP(A,C) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),C),E)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),E)). [resolve(5082,a,4908,a)]. 9.87/9.87 5083 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OgreaterThanAtMost(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),E)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,D)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,C,E)) # label(fact_Int__greaterThanAtMost) # label(axiom). [clausify(2034)]. 9.87/9.87 Derived: c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,C)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,B,D)). [resolve(5083,a,4908,a)]. 9.87/9.87 5084 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatMost(A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C)) # label(fact_ivl__disj__un_I10_J) # label(axiom). [clausify(2062)]. 9.87/9.87 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)). [resolve(5084,a,4908,a)]. 9.87/9.87 5085 -class_Orderings_Olinorder(A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(A),B)) = hAPP(c_SetInterval_Oord__class_OlessThan(A),B) # label(fact_Compl__atLeast) # label(axiom). [clausify(2094)]. 9.87/9.87 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A). [resolve(5085,a,4908,a)]. 9.87/9.87 5086 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | c_List_Olinorder__class_Oinsort__insert__key(A,A,c_COMBI(A),B,C) = C # label(fact_insort__insert__triv) # label(axiom). [clausify(2123)]. 9.87/9.87 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),A,B) = B. [resolve(5086,a,4908,a)]. 9.87/9.87 5087 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C)) # label(fact_ivl__disj__un_I11_J) # label(axiom). [clausify(2137)]. 9.97/9.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B)). [resolve(5087,a,4908,a)]. 9.97/9.92 5088 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f243(B,C,A)),hAPP(c_List_Oset(A),B))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(2164)]. 9.97/9.92 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f243(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(5088,a,4908,a)]. 9.97/9.92 5089 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f243(B,C,A))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(2164)]. 9.97/9.92 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f243(A,B,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(5089,a,4908,a)]. 9.97/9.92 5090 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(2164)]. 9.97/9.92 5091 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),B)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),D),C)) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(2164)]. 9.97/9.92 5092 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osort__key(B,A,C,c_List_Olist_ONil(B)) = c_List_Olist_ONil(B) # label(fact_sort__key__simps_I1_J) # label(axiom). [clausify(2168)]. 9.97/9.92 Derived: c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5092,a,4908,a)]. 9.97/9.92 5093 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,C,D)) # label(fact_ivl__disj__un_I22_J) # label(axiom). [clausify(2191)]. 9.97/9.92 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)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,B,C)). [resolve(5093,a,4908,a)]. 9.97/9.92 5094 -class_Orderings_Olinorder(A) | c_Finite__Set_Ofun__left__comm__idem(A,A,c_Orderings_Oord__class_Omax(A)) # label(fact_min__max_Ofun__left__comm__idem__sup) # label(axiom). [clausify(2218)]. 9.97/9.92 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)). [resolve(5094,a,4908,a)]. 9.97/9.92 5095 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__inf__iff) # label(axiom). [clausify(2242)]. 9.97/9.92 5096 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__inf__iff) # label(axiom). [clausify(2242)]. 9.97/9.97 5097 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__inf__iff) # label(axiom). [clausify(2242)]. 9.97/9.97 5098 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(A) = B | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_Obutlast(A,B)) # label(fact_sorted__butlast) # label(axiom). [clausify(2248)]. 9.97/9.97 Derived: c_List_Olist_ONil(tc_Nat_Onat) = A | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Obutlast(tc_Nat_Onat,A)). [resolve(5098,a,4908,a)]. 9.97/9.97 5099 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),D)) # label(fact_ivl__disj__un_I17_J) # label(axiom). [clausify(2261)]. 9.97/9.97 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(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C) = 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,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)). [resolve(5099,a,4908,a)]. 9.97/9.97 5100 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),B) # label(fact_min__max_Oinf__commute) # label(axiom). [clausify(2269)]. 9.97/9.97 5101 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),B) = C # label(fact_min__max_Osup__absorb1) # label(axiom). [clausify(2277)]. 9.97/9.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),A) = B. [resolve(5101,a,4908,a)]. 9.97/9.97 5102 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) = B # label(fact_min__max_Oinf__absorb1) # label(axiom). [clausify(2293)]. 9.97/9.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B) = A. [resolve(5102,a,4908,a)]. 9.97/9.97 5103 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatMost(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) # label(fact_ivl__disj__un_I2_J) # label(axiom). [clausify(2294)]. 9.97/9.97 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))). [resolve(5103,a,4908,a)]. 9.97/9.97 5104 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olinorder__class_Osort__key(A,A,c_COMBI(A),B)) # label(fact_sorted__sort) # label(axiom). [clausify(2317)]. 9.97/9.97 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olinorder__class_Osort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),A)). [resolve(5104,a,4908,a)]. 9.97/9.97 5105 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Oinf_Oassoc) # label(axiom). [clausify(2325)]. 10.06/10.03 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)). [resolve(5105,a,4908,a)]. 10.06/10.03 5106 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),B)) # label(fact_min__max_Oinf__le1) # label(axiom). [clausify(2349)]. 10.06/10.03 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),A)). [resolve(5106,a,4908,a)]. 10.06/10.03 5107 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_le__max__iff__disj) # label(axiom). [clausify(2373)]. 10.06/10.03 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C))). [resolve(5107,a,4908,a)]. 10.06/10.03 5108 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C))) # label(fact_le__max__iff__disj) # label(axiom). [clausify(2373)]. 10.06/10.03 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B))). [resolve(5108,a,4908,a)]. 10.06/10.03 5109 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_le__max__iff__disj) # label(axiom). [clausify(2373)]. 10.06/10.03 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),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C))). [resolve(5109,a,4908,a)]. 10.06/10.03 5110 -class_Orderings_Olinorder(A) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C))) = hAPP(c_List_Ofoldl(A,A,c_Orderings_Oord__class_Omax(A),B),C) # label(fact_Max__fin__set__fold) # label(axiom). [clausify(2375)]. 10.06/10.03 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(c_List_Oset(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B))) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B). [resolve(5110,a,4908,a)]. 10.06/10.03 5111 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),c_List_Olist_ONil(B)) = hAPP(hAPP(c_List_Olist_OCons(B),D),c_List_Olist_ONil(B)) # label(fact_insort__key_Osimps_I1_J) # label(axiom). [clausify(2408)]. 10.06/10.03 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),c_List_Olist_ONil(A)) = hAPP(hAPP(c_List_Olist_OCons(A),C),c_List_Olist_ONil(A)). [resolve(5111,a,4908,a)]. 10.06/10.03 5112 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMin(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_Min_Oin__idem) # label(axiom). [clausify(2440)]. 10.06/10.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5112,a,4908,a)]. 10.06/10.03 5113 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) # label(fact_min__max_Oinf__greatest) # label(axiom). [clausify(2446)]. 10.06/10.06 5114 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osort__key(B,A,C,hAPP(hAPP(c_List_Olist_OCons(B),D),E)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),c_List_Olinorder__class_Osort__key(B,A,C,E)) # label(fact_sort__key__simps_I2_J) # label(axiom). [clausify(2447)]. 10.06/10.06 Derived: c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,D)). [resolve(5114,a,4908,a)]. 10.06/10.06 5115 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(2480)]. 10.06/10.06 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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5115,a,4908,a)]. 10.06/10.06 5116 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | hBOOL(hAPP(hAPP(c_member(A),f282(C,B,A)),B)) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(2480)]. 10.06/10.06 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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f282(B,A,tc_Nat_Onat)),A)). [resolve(5116,a,4908,a)]. 10.06/10.06 5117 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f282(C,B,A))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(2480)]. 10.06/10.06 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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f282(B,A,tc_Nat_Onat))). [resolve(5117,a,4908,a)]. 10.06/10.06 5118 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Big__Operators_Olinorder__class_OMax(A,B))) # label(fact_Max__ge) # label(axiom). [clausify(2485)]. 10.06/10.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A))). [resolve(5118,a,4908,a)]. 10.06/10.06 5119 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C))) # label(fact_min__max_Ole__supI2) # label(axiom). [clausify(2486)]. 10.06/10.06 5120 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_leD) # label(axiom). [clausify(2495)]. 10.17/10.11 5121 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted__list__of__set(A,c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))) = c_List_Olist_ONil(A) # label(fact_sorted__list__of__set__empty) # label(axiom). [clausify(2496)]. 10.17/10.11 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = c_List_Olist_ONil(tc_Nat_Onat). [resolve(5121,a,4908,a)]. 10.17/10.11 5122 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B) # label(fact_min__max_OSup__fin_Oeq__fold_H) # label(axiom). [clausify(2501)]. 10.17/10.11 5123 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(A),B)) # label(fact_Compl__atMost) # label(axiom). [clausify(2517)]. 10.17/10.11 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A)). [resolve(5123,a,4908,a)]. 10.17/10.11 5124 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__inf) # label(axiom). [clausify(2537)]. 10.17/10.11 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B) != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5124,a,4908,a)]. 10.17/10.11 5125 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__inf) # label(axiom). [clausify(2537)]. 10.17/10.11 5126 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B) # label(fact_Min_Oeq__fold__idem_H) # label(axiom). [clausify(2564)]. 10.17/10.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olinorder__class_OMin(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_Orderings_Oord__class_Omin(tc_Nat_Onat)),B),A). [resolve(5126,a,4908,a)]. 10.17/10.11 5127 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = 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)))) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),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)))))) # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom). [clausify(2567)]. 10.17/10.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5127,a,4908,a)]. 10.17/10.15 5128 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != 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)))) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = C # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom). [clausify(2567)]. 10.17/10.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = B. [resolve(5128,a,4908,a)]. 10.17/10.15 5129 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_Oremdups(A,B)) # label(fact_sorted__remdups) # label(axiom). [clausify(2596)]. 10.17/10.15 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Oremdups(tc_Nat_Onat,A)). [resolve(5129,a,4908,a)]. 10.17/10.15 5130 -class_Orderings_Olinorder(A) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = B # label(fact_Max_Osingleton) # label(axiom). [clausify(2610)]. 10.17/10.15 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) = A. [resolve(5130,a,4908,a)]. 10.17/10.15 5131 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),B)),C)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(2633)]. 10.17/10.15 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),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),A)),B)). [resolve(5131,a,4908,a)]. 10.17/10.15 5132 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),B)),C)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(2633)]. 10.17/10.15 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_Orderings_Oord__class_Omax(tc_Nat_Onat),C),A)),B)). [resolve(5132,a,4908,a)]. 10.17/10.15 5133 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(2633)]. 10.17/10.15 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_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)),B)). [resolve(5133,a,4908,a)]. 10.17/10.15 5134 -class_Orderings_Olinorder(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_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olinorder__class_OMax(A,B)),c_Big__Operators_Olinorder__class_OMax(A,C)) # label(fact_Max__Un) # label(axiom). [clausify(2634)]. 10.17/10.15 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_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,B)). [resolve(5134,a,4908,a)]. 10.17/10.17 5135 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)))) # label(fact_min__max_Odistrib__sup__le) # label(axiom). [clausify(2635)]. 10.17/10.17 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)))). [resolve(5135,a,4908,a)]. 10.17/10.17 5136 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),hAPP(c_SetInterval_Oord__class_OatLeast(A),C)) # label(fact_ivl__disj__un_I14_J) # label(axiom). [clausify(2638)]. 10.17/10.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = 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,A),B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B)). [resolve(5136,a,4908,a)]. 10.17/10.17 5137 -class_Orderings_Olinorder(A) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) = B # label(fact_min__max_OInf__fin_Osingleton) # label(axiom). [clausify(2641)]. 10.17/10.17 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) = A. [resolve(5137,a,4908,a)]. 10.17/10.17 5138 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),D),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),D)),hAPP(hAPP(c_List_Onth(A),B),C))) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(2662)]. 10.17/10.17 5139 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f296(B,A)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(2662)]. 10.17/10.17 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f296(A,tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))). [resolve(5139,a,4908,a)]. 10.17/10.17 5140 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f297(B,A)),f296(B,A))) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(2662)]. 10.17/10.17 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f297(A,tc_Nat_Onat)),f296(A,tc_Nat_Onat))). [resolve(5140,a,4908,a)]. 10.17/10.17 5141 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f297(B,A))),hAPP(hAPP(c_List_Onth(A),B),f296(B,A)))) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(2662)]. 10.27/10.21 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f297(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f296(A,tc_Nat_Onat)))). [resolve(5141,a,4908,a)]. 10.27/10.21 5142 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) # label(fact_ivl__disj__un_I6_J) # label(axiom). [clausify(2664)]. 10.27/10.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B) = 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,A),B)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))). [resolve(5142,a,4908,a)]. 10.27/10.21 5143 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Otake(A),C),B)) # label(fact_sorted__take) # label(axiom). [clausify(2666)]. 10.27/10.21 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Otake(tc_Nat_Onat),B),A)). [resolve(5143,a,4908,a)]. 10.27/10.21 5144 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,C)),hAPP(B,D))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),hAPP(hAPP(c_List_Olist_OCons(E),D),F)) = hAPP(hAPP(c_List_Olist_OCons(E),C),hAPP(hAPP(c_List_Olist_OCons(E),D),F)) # label(fact_insort__key_Osimps_I2_J) # label(axiom). [clausify(2678)]. 10.27/10.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)). [resolve(5144,a,4908,a)]. 10.27/10.21 5145 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,C)),hAPP(B,D))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),hAPP(hAPP(c_List_Olist_OCons(E),D),F)) = hAPP(hAPP(c_List_Olist_OCons(E),D),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(E,A,B),C),F)) # label(fact_insort__key_Osimps_I2_J) # label(axiom). [clausify(2678)]. 10.27/10.21 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Nat_Onat,A),B),E)). [resolve(5145,a,4908,a)]. 10.27/10.21 5146 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omin(A),f311(B,A)),f312(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f311(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f312(B,A)),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),B)) # label(fact_min__max_OInf__fin_Oclosed) # label(axiom). [clausify(2722)]. 10.27/10.21 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_Orderings_Oord__class_Omin(tc_Nat_Onat),f311(A,tc_Nat_Onat)),f312(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f311(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f312(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),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),A)). [resolve(5146,a,4908,a)]. 10.27/10.27 5147 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),c_List_Olinorder__class_Osort__key(B,A,C,D))) # label(fact_sorted__sort__key) # label(axiom). [clausify(2726)]. 10.27/10.27 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,C))). [resolve(5147,a,4908,a)]. 10.27/10.27 5148 -class_Orderings_Olinorder(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_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)) # label(fact_min__max_OSup__fin_Ounion__idem) # label(axiom). [clausify(2728)]. 10.27/10.27 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_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),B)). [resolve(5148,a,4908,a)]. 10.27/10.27 5149 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) != hAPP(c_SetInterval_Oord__class_OatLeast(A),C) | B = C # label(fact_atLeast__eq__iff) # label(axiom). [clausify(2734)]. 10.27/10.27 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B) | A = B. [resolve(5149,a,4908,a)]. 10.27/10.27 5150 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(c_SetInterval_Oord__class_OatLeast(A),C) | B != C # label(fact_atLeast__eq__iff) # label(axiom). [clausify(2734)]. 10.27/10.27 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B) | A != B. [resolve(5150,a,4908,a)]. 10.27/10.27 5151 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),C)) # label(fact_min__max_Oinf__le2) # label(axiom). [clausify(2740)]. 10.27/10.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),B)). [resolve(5151,a,4908,a)]. 10.27/10.27 5152 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_linorder__linear) # label(axiom). [clausify(2790)]. 10.27/10.27 5153 -class_Orderings_Olinorder(A) | c_Finite__Set_Ofun__left__comm__idem(A,A,c_Orderings_Oord__class_Omin(A)) # label(fact_min__max_Ofun__left__comm__idem__inf) # label(axiom). [clausify(2806)]. 10.27/10.27 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)). [resolve(5153,a,4908,a)]. 10.27/10.27 5154 class_Orderings_Olinorder(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Orderings_Olinorder) # label(axiom). [assumption]. 10.27/10.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(A,f1(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f2(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f1(B,A,tc_Code__Numeral_Ocode__numeral)),f2(B,A,tc_Code__Numeral_Ocode__numeral))) | -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_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_Oimage(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,4909,a)]. 10.27/10.28 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,a,4910,a)]. 10.27/10.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5154,a,4911,a)]. 10.27/10.28 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)). [resolve(5154,a,4912,a)]. 10.27/10.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5154,a,4913,a)]. 10.27/10.28 Derived: A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4914,a)]. 10.27/10.28 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_OdropWhile(tc_Code__Numeral_Ocode__numeral,B,A)). [resolve(5154,a,4915,a)]. 10.27/10.28 Derived: -c_List_Olinorder__class_Osorted(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(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5154,a,4916,a)]. 10.27/10.28 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) = A. [resolve(5154,a,4917,a)]. 10.27/10.28 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Code__Numeral_Ocode__numeral,C,D,B))). [resolve(5154,a,4918,a)]. 10.27/10.28 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5154,a,4919,a)]. 10.27/10.28 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Otl(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,4920,a)]. 10.27/10.30 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5154,a,4921,a)]. 10.27/10.30 Derived: hBOOL(hAPP(A,B)) | hAPP(c_List_Ofilter(C,A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Code__Numeral_Ocode__numeral,D),B),E)) = hAPP(c_List_Ofilter(C,A),E). [resolve(5154,a,4922,a)]. 10.27/10.30 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_OtakeWhile(tc_Code__Numeral_Ocode__numeral,B,A)). [resolve(5154,a,4923,a)]. 10.27/10.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),A) = A. [resolve(5154,a,4924,a)]. 10.27/10.30 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,C)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,4925,a)]. 10.27/10.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4926,a)]. 10.27/10.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5154,a,4927,a)]. 10.27/10.30 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(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,4928,a)]. 10.27/10.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Complete__Lattice_OInf__class_OInf(tc_Code__Numeral_Ocode__numeral,B)),C)). [resolve(5154,a,4929,b)]. 10.27/10.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f43(A,B,tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Complete__Lattice_OInf__class_OInf(tc_Code__Numeral_Ocode__numeral,B)),A)). [resolve(5154,a,4930,b)]. 10.27/10.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f43(A,B,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Complete__Lattice_OInf__class_OInf(tc_Code__Numeral_Ocode__numeral,B)),A)). [resolve(5154,a,4931,b)]. 10.27/10.30 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),A),B)) | B = A. [resolve(5154,a,4932,a)]. 10.27/10.30 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),A),B)) | B != A. [resolve(5154,a,4933,a)]. 10.27/10.30 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B),A)). [resolve(5154,a,4934,a)]. 10.37/10.31 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5154,a,4935,a)]. 10.37/10.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)). [resolve(5154,a,4936,a)]. 10.37/10.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A)). [resolve(5154,a,4937,a)]. 10.37/10.31 Derived: c_List_Oremove1(tc_Code__Numeral_Ocode__numeral,A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),A),B)) = B. [resolve(5154,a,4938,a)]. 10.37/10.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5154,a,4939,a)]. 10.37/10.31 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),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B)). [resolve(5154,a,4940,a)]. 10.37/10.31 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5154,a,4941,a)]. 10.37/10.31 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),A))). [resolve(5154,a,4942,a)]. 10.37/10.31 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) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,4943,a)]. 10.37/10.32 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5154,a,4944,a)]. 10.37/10.32 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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5154,a,4946,a)]. 10.37/10.32 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B)). [resolve(5154,a,4947,a)]. 10.37/10.32 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),C)). [resolve(5154,a,4948,a)]. 10.37/10.32 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),C),A)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5154,a,4949,a)]. 10.37/10.32 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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B)). [resolve(5154,a,4950,a)]. 10.37/10.32 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)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = B. [resolve(5154,a,4951,a)]. 10.37/10.32 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,4952,a)]. 10.37/10.32 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,4953,a)]. 10.37/10.33 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_Big__Operators_Olinorder__class_OMin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,4954,a)]. 10.37/10.33 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)) = A. [resolve(5154,a,4955,a)]. 10.37/10.33 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C))) | c_List_OtakeWhile(A,hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),D)),B),C) = hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),D)),B)),C). [resolve(5154,a,4956,a)]. 10.37/10.33 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,4957,a)]. 10.37/10.33 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,C)). [resolve(5154,a,4958,a)]. 10.37/10.33 Derived: -class_Lattices_Osemilattice__sup(tc_Code__Numeral_Ocode__numeral) | c_Lattices_Osemilattice__sup__class_Osup(tc_Code__Numeral_Ocode__numeral) = c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral). [resolve(5154,a,4959,b)]. 10.37/10.33 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_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__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_COMBB(tc_fun(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),tc_Code__Numeral_Ocode__numeral),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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(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_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBS(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_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_Code__Numeral_Ocode__numeral,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),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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),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_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(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_fun(tc_fun(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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBB(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)),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))),c_Orderings_Oord__class_Omin(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_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_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),hAPP(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_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A)))),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)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,4960,a)]. 10.37/10.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5154,a,4961,a)]. 10.37/10.34 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5154,a,4962,a)]. 10.37/10.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5154,a,4963,a)]. 10.37/10.34 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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5154,a,4964,a)]. 10.37/10.34 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5154,a,4966,a)]. 10.37/10.34 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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B))). [resolve(5154,a,4967,a)]. 10.37/10.34 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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B))). [resolve(5154,a,4968,a)]. 10.37/10.34 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B))) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,a,4969,a)]. 10.37/10.34 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) = A. [resolve(5154,a,4971,a)]. 10.37/10.34 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) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,4972,a)]. 10.37/10.35 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),B),A)). [resolve(5154,a,4973,a)]. 10.37/10.35 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f84(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f85(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f84(B,A,tc_Code__Numeral_Ocode__numeral)),f85(B,A,tc_Code__Numeral_Ocode__numeral))) | -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_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_Oimage(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,4974,a)]. 10.37/10.35 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)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B)),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,4975,a)]. 10.37/10.35 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5154,a,4976,a)]. 10.37/10.35 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)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B)),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,4977,a)]. 10.37/10.35 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Code__Numeral_Ocode__numeral),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(C,f99(D,C,B,A,tc_Code__Numeral_Ocode__numeral))),D)). [resolve(5154,a,4978,a)]. 10.37/10.36 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Code__Numeral_Ocode__numeral),B),C)),D)) | hBOOL(hAPP(hAPP(c_member(A),f99(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5154,a,4979,a)]. 10.37/10.36 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Code__Numeral_Ocode__numeral),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(C,E)),D)) | -hBOOL(hAPP(hAPP(c_member(A),E),B)). [resolve(5154,a,4980,a)]. 10.37/10.36 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)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,4981,a)]. 10.37/10.36 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f102(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4982,a)]. 10.37/10.36 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f103(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4983,a)]. 10.37/10.36 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f102(B,A,tc_Code__Numeral_Ocode__numeral)),f103(B,A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4984,a)]. 10.37/10.36 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,4985,a)]. 10.37/10.36 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,4986,a)]. 10.37/10.36 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5154,a,4987,a)]. 10.37/10.36 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),A),c_fequal),B)),hAPP(C,D))),D))). [resolve(5154,a,4989,a)]. 10.37/10.37 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),B),A)). [resolve(5154,a,4990,a)]. 10.37/10.37 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),B),A)). [resolve(5154,a,4991,a)]. 10.37/10.37 Derived: -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,C))) | hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(5154,a,4992,a)]. 10.37/10.37 Derived: hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,C))) | -hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(5154,a,4993,a)]. 10.37/10.37 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))). [resolve(5154,a,4994,b)]. 10.37/10.37 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f111(A,B,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,A))). [resolve(5154,a,4995,b)]. 10.37/10.37 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),f111(B,A,tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))). [resolve(5154,a,4996,b)]. 10.37/10.37 Derived: c_Orderings_Oord_Omin(tc_Code__Numeral_Ocode__numeral,hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral))) = c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral). [resolve(5154,a,4997,a)]. 10.37/10.37 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),B),A)). [resolve(5154,a,4999,a)]. 10.37/10.37 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5154,a,5000,a)]. 10.37/10.37 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,a,5001,a)]. 10.37/10.37 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(A,f119(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f120(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f119(B,A,tc_Code__Numeral_Ocode__numeral)),f120(B,A,tc_Code__Numeral_Ocode__numeral))) | -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_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(c_Set_Oimage(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),B)) = hAPP(A,c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5002,a)]. 10.37/10.38 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | -hBOOL(hAPP(D,E)) | hAPP(c_List_Ofilter(A,D),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),E),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),E),hAPP(c_List_Ofilter(A,D),C)). [resolve(5154,a,5003,a)]. 10.37/10.38 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5154,a,5004,a)]. 10.37/10.38 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5005,a)]. 10.37/10.38 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)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = B. [resolve(5154,a,5006,a)]. 10.37/10.38 Derived: hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,B),D)). [resolve(5154,a,5007,a)]. 10.37/10.38 Derived: hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D)) = hAPP(hAPP(c_Set_Oinsert(A),C),hAPP(c_List_Oset(A),D)). [resolve(5154,a,5008,a)]. 10.37/10.38 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,5009,a)]. 10.37/10.39 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),A)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),B)) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B) != hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A) | B = A. [resolve(5154,a,5010,a)]. 10.37/10.39 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),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5154,a,5011,a)]. 10.37/10.39 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f138(A,B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5012,a)]. 10.37/10.39 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f138(A,B,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5013,a)]. 10.37/10.39 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5014,a)]. 10.37/10.39 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),A)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),C),B)). [resolve(5154,a,5015,a)]. 10.37/10.39 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5154,a,5017,a)]. 10.37/10.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),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)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(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))),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,5019,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f147(B,C,A,tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5154,a,5020,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f147(B,C,A,tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5154,a,5021,a)]. 10.47/10.40 Derived: hBOOL(hAPP(hAPP(c_member(A),f150(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(A),D))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,C),B),D) = hAPP(hAPP(c_List_Olist_OCons(A),B),D). [resolve(5154,a,5022,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(A,f150(B,A,C,D,tc_Code__Numeral_Ocode__numeral)))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),C) = hAPP(hAPP(c_List_Olist_OCons(D),B),C). [resolve(5154,a,5023,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,A)) = A. [resolve(5154,a,5024,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,5025,a)]. 10.47/10.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,A))). [resolve(5154,a,5026,a)]. 10.47/10.40 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5154,a,5027,a)]. 10.47/10.40 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(A,D),C))). [resolve(5154,a,5028,a)]. 10.47/10.40 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5154,a,5029,a)]. 10.47/10.40 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f160(B,A,tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5154,a,5030,a)]. 10.47/10.41 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),f160(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5154,a,5031,a)]. 10.47/10.41 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)))). [resolve(5154,a,5032,a)]. 10.47/10.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B)) | 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B))). [resolve(5154,a,5033,a)]. 10.47/10.41 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,C)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C). [resolve(5154,a,5034,a)]. 10.47/10.41 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5035,a)]. 10.47/10.41 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A) = B. [resolve(5154,a,5036,a)]. 10.47/10.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),A,B) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),A),B). [resolve(5154,a,5037,a)]. 10.47/10.42 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)) | B != D | C != A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B). [resolve(5154,a,5038,a)]. 10.47/10.42 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)) | B = D | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B). [resolve(5154,a,5039,a)]. 10.47/10.42 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)) | C = A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B). [resolve(5154,a,5040,a)]. 10.47/10.42 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),B),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),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),C)). [resolve(5154,a,5041,a)]. 10.47/10.42 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),f166(B,A,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),B)). [resolve(5154,a,5042,a)]. 10.47/10.42 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_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f166(B,A,tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),B)). [resolve(5154,a,5043,a)]. 10.47/10.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5154,a,5044,a)]. 10.47/10.42 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5045,a)]. 10.47/10.43 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5154,a,5046,a)]. 10.47/10.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))))) = c_List_Oremove1(tc_Code__Numeral_Ocode__numeral,B,c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,5047,a)]. 10.47/10.43 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),C),A))). [resolve(5154,a,5048,a)]. 10.47/10.43 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),B),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),A),C)). [resolve(5154,a,5049,a)]. 10.47/10.43 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5154,a,5050,a)]. 10.47/10.43 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)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5154,a,5051,a)]. 10.47/10.43 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,C)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,B,D)). [resolve(5154,a,5052,a)]. 10.47/10.43 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5154,a,5053,a)]. 10.47/10.43 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,5054,a)]. 10.47/10.44 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,5055,a)]. 10.47/10.44 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B))) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,a,5057,a)]. 10.47/10.44 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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f181(A,tc_Code__Numeral_Ocode__numeral)),f182(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f181(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f182(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),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5154,a,5058,a)]. 10.47/10.44 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B) | A = B. [resolve(5154,a,5059,a)]. 10.47/10.44 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B) | A != B. [resolve(5154,a,5060,a)]. 10.47/10.44 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),B),A)). [resolve(5154,a,5061,a)]. 10.47/10.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5154,a,5062,a)]. 10.47/10.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),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)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(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))),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,5063,a)]. 10.47/10.44 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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f200(A,tc_Code__Numeral_Ocode__numeral)),f201(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f200(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f201(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),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5154,a,5064,a)]. 10.47/10.44 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A | c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5154,a,5065,a)]. 10.47/10.44 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) != A | c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A,B) = B. [resolve(5154,a,5066,a)]. 10.47/10.44 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)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = B. [resolve(5154,a,5067,a)]. 10.47/10.44 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5068,a)]. 10.47/10.44 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f210(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f211(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f210(B,A,tc_Code__Numeral_Ocode__numeral)),f211(B,A,tc_Code__Numeral_Ocode__numeral))) | -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_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(c_Set_Oimage(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),B)) = hAPP(A,c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5069,a)]. 10.47/10.44 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A) = B. [resolve(5154,a,5070,a)]. 10.47/10.45 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5071,a)]. 10.47/10.45 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5072,a)]. 10.47/10.45 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)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5154,a,5073,a)]. 10.47/10.45 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral))). [resolve(5154,a,5075,a)]. 10.47/10.45 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5076,a)]. 10.47/10.45 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),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)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(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))),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5077,a)]. 10.47/10.46 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)) = A. [resolve(5154,a,5078,a)]. 10.47/10.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | B != A. [resolve(5154,a,5080,a)]. 10.47/10.46 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),B),A)). [resolve(5154,a,5081,a)]. 10.47/10.46 Derived: hAPP(A,B) = hAPP(A,C) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),C),E)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),E)). [resolve(5154,a,5082,a)]. 10.47/10.46 Derived: c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,C)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,B,D)). [resolve(5154,a,5083,a)]. 10.47/10.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5154,a,5084,a)]. 10.47/10.46 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A). [resolve(5154,a,5085,a)]. 10.47/10.46 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),A,B) = B. [resolve(5154,a,5086,a)]. 10.47/10.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5087,a)]. 10.47/10.46 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f243(A,B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5088,a)]. 10.47/10.48 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f243(A,B,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5089,a)]. 10.47/10.48 Derived: c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5154,a,5092,a)]. 10.47/10.48 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),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5154,a,5093,a)]. 10.47/10.48 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)). [resolve(5154,a,5094,a)]. 10.47/10.48 Derived: c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Obutlast(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,5098,a)]. 10.47/10.48 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),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,B),C)). [resolve(5154,a,5099,a)]. 10.47/10.48 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),A) = B. [resolve(5154,a,5101,a)]. 10.47/10.48 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B) = A. [resolve(5154,a,5102,a)]. 10.47/10.48 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))). [resolve(5154,a,5103,a)]. 10.47/10.48 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olinorder__class_Osort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,5104,a)]. 10.47/10.48 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5154,a,5105,a)]. 10.47/10.48 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),A)). [resolve(5154,a,5106,a)]. 10.56/10.50 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5154,a,5107,a)]. 10.56/10.50 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5154,a,5108,a)]. 10.56/10.50 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),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5154,a,5109,a)]. 10.56/10.50 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B))) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5154,a,5110,a)]. 10.56/10.50 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),c_List_Olist_ONil(A)) = hAPP(hAPP(c_List_Olist_OCons(A),C),c_List_Olist_ONil(A)). [resolve(5154,a,5111,a)]. 10.56/10.50 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)) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,5112,a)]. 10.56/10.50 Derived: c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,D)). [resolve(5154,a,5114,a)]. 10.56/10.50 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_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5154,a,5115,a)]. 10.56/10.50 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_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f282(B,A,tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5154,a,5116,a)]. 10.56/10.50 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_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f282(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5154,a,5117,a)]. 10.56/10.50 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A))). [resolve(5154,a,5118,a)]. 10.56/10.51 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))) = c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral). [resolve(5154,a,5121,a)]. 10.56/10.51 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5154,a,5123,a)]. 10.56/10.51 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B) != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5154,a,5124,a)]. 10.56/10.51 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olinorder__class_OMin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5154,a,5126,a)]. 10.56/10.51 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5154,a,5127,a)]. 10.56/10.51 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)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = B. [resolve(5154,a,5128,a)]. 10.56/10.51 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Oremdups(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5154,a,5129,a)]. 10.56/10.51 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) = A. [resolve(5154,a,5130,a)]. 10.56/10.51 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),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5154,a,5131,a)]. 10.56/10.51 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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5154,a,5132,a)]. 10.56/10.52 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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)),B)). [resolve(5154,a,5133,a)]. 10.56/10.52 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_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,B)). [resolve(5154,a,5134,a)]. 10.56/10.52 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)))). [resolve(5154,a,5135,a)]. 10.56/10.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5136,a)]. 10.56/10.52 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) = A. [resolve(5154,a,5137,a)]. 10.56/10.52 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f296(A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))). [resolve(5154,a,5139,a)]. 10.56/10.52 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f297(A,tc_Code__Numeral_Ocode__numeral)),f296(A,tc_Code__Numeral_Ocode__numeral))). [resolve(5154,a,5140,a)]. 10.56/10.52 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f297(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f296(A,tc_Code__Numeral_Ocode__numeral)))). [resolve(5154,a,5141,a)]. 10.56/10.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))). [resolve(5154,a,5142,a)]. 10.56/10.52 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Otake(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5154,a,5143,a)]. 10.56/10.55 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)). [resolve(5154,a,5144,a)]. 10.56/10.55 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Code__Numeral_Ocode__numeral,A),B),E)). [resolve(5154,a,5145,a)]. 10.56/10.55 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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f311(A,tc_Code__Numeral_Ocode__numeral)),f312(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f311(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f312(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),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),A)). [resolve(5154,a,5146,a)]. 10.56/10.55 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,C))). [resolve(5154,a,5147,a)]. 10.56/10.55 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_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5154,a,5148,a)]. 10.56/10.55 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B) | A = B. [resolve(5154,a,5149,a)]. 10.56/10.55 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B) | A != B. [resolve(5154,a,5150,a)]. 10.56/10.55 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),B)). [resolve(5154,a,5151,a)]. 10.56/10.55 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)). [resolve(5154,a,5153,a)]. 10.56/10.55 5155 -class_Orderings_Olinorder(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_Orderings_Oord__class_Omax(A),f326(B,A)),f327(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f326(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f327(B,A)),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)),B)) # label(fact_min__max_OSup__fin_Oclosed) # label(axiom). [clausify(2877)]. 10.67/10.63 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_Orderings_Oord__class_Omax(tc_Nat_Onat),f326(A,tc_Nat_Onat)),f327(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f326(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f327(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),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)),A)). [resolve(5155,a,4908,a)]. 10.67/10.63 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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f326(A,tc_Code__Numeral_Ocode__numeral)),f327(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f326(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f327(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),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)),A)). [resolve(5155,a,5154,a)]. 10.67/10.63 5156 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(2899)]. 10.67/10.63 5157 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(2899)]. 10.67/10.63 5158 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_min__max_Osup_Oidem) # label(axiom). [clausify(2928)]. 10.67/10.63 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),A) = A. [resolve(5158,a,4908,a)]. 10.67/10.63 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5158,a,5154,a)]. 10.67/10.63 5159 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_Oremove1(A,C,B)) # label(fact_sorted__remove1) # label(axiom). [clausify(2950)]. 10.67/10.63 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Oremove1(tc_Nat_Onat,B,A)). [resolve(5159,a,4908,a)]. 10.67/10.63 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Oremove1(tc_Code__Numeral_Ocode__numeral,B,A)). [resolve(5159,a,5154,a)]. 10.67/10.63 5160 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Oinf__sup__distrib2) # label(axiom). [clausify(3007)]. 10.67/10.63 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)). [resolve(5160,a,4908,a)]. 10.67/10.63 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5160,a,5154,a)]. 10.67/10.66 5161 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C))) # label(fact_le__maxI1) # label(axiom). [clausify(3020)]. 10.67/10.66 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B))). [resolve(5161,a,4908,a)]. 10.67/10.66 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5161,a,5154,a)]. 10.67/10.66 5162 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(c_List_Orev(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),D)),hAPP(hAPP(c_List_Onth(A),B),C))) # label(fact_sorted__rev__nth__mono) # label(axiom). [clausify(3022)]. 10.67/10.66 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Orev(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),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),C)),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),B))). [resolve(5162,a,4908,a)]. 10.67/10.66 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Orev(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(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5162,a,5154,a)]. 10.67/10.66 5163 -class_Orderings_Olinorder(A) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),C)),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_fold1__antimono) # label(axiom). [clausify(3031)]. 10.67/10.66 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),B)),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))). [resolve(5163,a,4908,a)]. 10.67/10.66 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B)),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))). [resolve(5163,a,5154,a)]. 10.67/10.66 5164 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(A,B,C)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,C,D)) # label(fact_ivl__disj__un_I18_J) # label(axiom). [clausify(3038)]. 10.67/10.67 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),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,B,C)). [resolve(5164,a,4908,a)]. 10.67/10.67 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),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5164,a,5154,a)]. 10.67/10.67 5165 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(B),C),hAPP(c_List_Oset(B),D))) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),E),D)) | hAPP(c_List_Ohd(B),hAPP(c_List_Ofilter(B,hAPP(hAPP(c_COMBB(A,tc_HOL_Obool,B),hAPP(c_fequal,hAPP(E,C))),E)),D)) != C | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,E),C),c_List_Oremove1(B,C,D)) = D # label(fact_insort__key__remove1) # label(axiom). [clausify(3044)]. 10.67/10.67 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),D),C)) | hAPP(c_List_Ohd(A),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,A),hAPP(c_fequal,hAPP(D,B))),D)),C)) != B | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,D),B),c_List_Oremove1(A,B,C)) = C. [resolve(5165,a,4908,a)]. 10.67/10.67 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),D),C)) | hAPP(c_List_Ohd(A),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,A),hAPP(c_fequal,hAPP(D,B))),D)),C)) != B | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,D),B),c_List_Oremove1(A,B,C)) = C. [resolve(5165,a,5154,a)]. 10.67/10.67 5166 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f348(C,B,A)),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMin(A,B) = C # label(fact_Min__eqI) # label(axiom). [clausify(3045)]. 10.67/10.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f348(B,A,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = B. [resolve(5166,a,4908,a)]. 10.67/10.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f348(B,A,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5166,a,5154,a)]. 10.67/10.67 5167 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f348(C,B,A))) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMin(A,B) = C # label(fact_Min__eqI) # label(axiom). [clausify(3045)]. 10.67/10.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f348(B,A,tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = B. [resolve(5167,a,4908,a)]. 10.67/10.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f348(B,A,tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5167,a,5154,a)]. 10.67/10.70 5168 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted__list__of__set(A,hAPP(c_List_Oset(A),B)) = c_List_Olinorder__class_Osort__key(A,A,c_COMBI(A),c_List_Oremdups(A,B)) # label(fact_sorted__list__of__set__sort__remdups) # label(axiom). [clausify(3072)]. 10.67/10.70 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(c_List_Oset(tc_Nat_Onat),A)) = c_List_Olinorder__class_Osort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),c_List_Oremdups(tc_Nat_Onat,A)). [resolve(5168,a,4908,a)]. 10.67/10.70 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A)) = c_List_Olinorder__class_Osort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),c_List_Oremdups(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5168,a,5154,a)]. 10.67/10.70 5169 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(A,C) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),c_List_Oremove1(A,B,C)) = C # label(fact_insort__remove1) # label(axiom). [clausify(3084)]. 10.67/10.70 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),A),c_List_Oremove1(tc_Nat_Onat,A,B)) = B. [resolve(5169,a,4908,a)]. 10.67/10.70 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),A),c_List_Oremove1(tc_Code__Numeral_Ocode__numeral,A,B)) = B. [resolve(5169,a,5154,a)]. 10.67/10.70 5170 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),D)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(3087)]. 10.67/10.70 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_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),C)). [resolve(5170,a,4908,a)]. 10.67/10.70 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),C)). [resolve(5170,a,5154,a)]. 10.67/10.70 5171 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless(A),f354(C,B,A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(3087)]. 10.77/10.74 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_Orderings_Oord__class_Oless(tc_Nat_Onat),f354(B,A,tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),B)). [resolve(5171,a,4908,a)]. 10.77/10.74 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f354(B,A,tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),B)). [resolve(5171,a,5154,a)]. 10.77/10.74 5172 -class_Orderings_Olinorder(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),f354(C,B,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(3087)]. 10.77/10.74 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),f354(B,A,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),B)). [resolve(5172,a,4908,a)]. 10.77/10.74 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),f354(B,A,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),B)). [resolve(5172,a,5154,a)]. 10.77/10.74 5173 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_Oequations_I1_J) # label(axiom). [clausify(3091)]. 10.77/10.74 5174 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(B,C)),hAPP(c_Set_Oimage(D,A,B),hAPP(c_List_Oset(D),E)))) | c_List_Olinorder__class_Oinsort__insert__key(D,A,B,C,E) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,A,B),C),E) # label(fact_insort__insert__insort__key) # label(axiom). [clausify(3129)]. 10.77/10.74 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Nat_Onat,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Nat_Onat,A,B,D) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Nat_Onat,A),B),D). [resolve(5174,a,4908,a)]. 10.77/10.74 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Code__Numeral_Ocode__numeral,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Code__Numeral_Ocode__numeral,A,B,D) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Code__Numeral_Ocode__numeral,A),B),D). [resolve(5174,a,5154,a)]. 10.77/10.74 5175 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),hAPP(B,C)),hAPP(c_Set_Oimage(D,A,B),hAPP(c_List_Oset(D),E)))) | c_List_Olinorder__class_Oinsort__insert__key(D,A,B,C,E) = E # label(fact_insort__insert__key__triv) # label(axiom). [clausify(3133)]. 10.77/10.74 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Nat_Onat,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Nat_Onat,A,B,D) = D. [resolve(5175,a,4908,a)]. 10.77/10.74 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Code__Numeral_Ocode__numeral,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Code__Numeral_Ocode__numeral,A,B,D) = D. [resolve(5175,a,5154,a)]. 10.77/10.78 5176 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E))) | hBOOL(hAPP(c_List_Odistinct(B),E)) # label(fact_distinct__insort) # label(axiom). [clausify(3137)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D))) | hBOOL(hAPP(c_List_Odistinct(A),D)). [resolve(5176,a,4908,a)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D))) | hBOOL(hAPP(c_List_Odistinct(A),D)). [resolve(5176,a,5154,a)]. 10.77/10.78 5177 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E))) | -hBOOL(hAPP(hAPP(c_member(B),D),hAPP(c_List_Oset(B),E))) # label(fact_distinct__insort) # label(axiom). [clausify(3137)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D))) | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5177,a,4908,a)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D))) | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5177,a,5154,a)]. 10.77/10.78 5178 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E))) | -hBOOL(hAPP(c_List_Odistinct(B),E)) | hBOOL(hAPP(hAPP(c_member(B),D),hAPP(c_List_Oset(B),E))) # label(fact_distinct__insort) # label(axiom). [clausify(3137)]. 10.77/10.78 Derived: hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D))) | -hBOOL(hAPP(c_List_Odistinct(A),D)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5178,a,4908,a)]. 10.77/10.78 Derived: hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D))) | -hBOOL(hAPP(c_List_Odistinct(A),D)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5178,a,5154,a)]. 10.77/10.78 5179 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omax(A)),C),B) # label(fact_Max_Oeq__fold__idem_H) # label(axiom). [clausify(3165)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A). [resolve(5179,a,4908,a)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5179,a,5154,a)]. 10.77/10.78 5180 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f368(C,B,A)),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMax(A,B) = C # label(fact_Max__eqI) # label(axiom). [clausify(3180)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f368(B,A,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = B. [resolve(5180,a,4908,a)]. 10.77/10.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f368(B,A,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5180,a,5154,a)]. 10.77/10.78 5181 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f368(C,B,A)),C)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMax(A,B) = C # label(fact_Max__eqI) # label(axiom). [clausify(3180)]. 10.86/10.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f368(B,A,tc_Nat_Onat)),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = B. [resolve(5181,a,4908,a)]. 10.86/10.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f368(B,A,tc_Code__Numeral_Ocode__numeral)),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(5181,a,5154,a)]. 10.86/10.85 5182 -class_Orderings_Olinorder(A) | c_Finite__Set_Ofun__left__comm(A,tc_List_Olist(A),c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A))) # label(fact_fun__left__comm__insort) # label(axiom). [clausify(3250)]. 10.86/10.85 Derived: c_Finite__Set_Ofun__left__comm(tc_Nat_Onat,tc_List_Olist(tc_Nat_Onat),c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat))). [resolve(5182,a,4908,a)]. 10.86/10.85 Derived: c_Finite__Set_Ofun__left__comm(tc_Code__Numeral_Ocode__numeral,tc_List_Olist(tc_Code__Numeral_Ocode__numeral),c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral))). [resolve(5182,a,5154,a)]. 10.86/10.85 5183 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C)),c_SetInterval_Oord__class_OgreaterThanAtMost(A,C,D)) # label(fact_ivl__disj__un_I20_J) # label(axiom). [clausify(3252)]. 10.86/10.85 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)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,B,C)). [resolve(5183,a,4908,a)]. 10.86/10.85 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5183,a,5154,a)]. 10.86/10.85 5184 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f381(B,A)),f382(B,A))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(3296)]. 10.86/10.85 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f381(A,tc_Nat_Onat)),f382(A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5184,a,4908,a)]. 10.86/10.85 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f381(A,tc_Code__Numeral_Ocode__numeral)),f382(A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5184,a,5154,a)]. 10.86/10.85 5185 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f382(B,A)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(3296)]. 10.86/10.85 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f382(A,tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5185,a,4908,a)]. 10.98/10.93 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f382(A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5185,a,5154,a)]. 10.98/10.93 5186 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f381(B,A))),hAPP(hAPP(c_List_Onth(A),B),f382(B,A)))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(3296)]. 10.98/10.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f381(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f382(A,tc_Nat_Onat)))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5186,a,4908,a)]. 10.98/10.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f381(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f382(A,tc_Code__Numeral_Ocode__numeral)))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5186,a,5154,a)]. 10.98/10.93 5187 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)),hAPP(c_SetInterval_Oord__class_OatLeast(A),C)) # label(fact_ivl__disj__un_I12_J) # label(axiom). [clausify(3307)]. 10.98/10.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B)). [resolve(5187,a,4908,a)]. 10.98/10.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5187,a,5154,a)]. 10.98/10.93 5188 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_Min_Oinsert) # label(axiom). [clausify(3397)]. 10.98/10.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),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(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5188,a,4908,a)]. 10.98/10.93 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)) | 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(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5188,a,5154,a)]. 10.98/10.93 5189 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | E = C # label(fact_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(3411)]. 10.98/10.99 5190 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osort__key(B,A,C,hAPP(c_List_Ofilter(B,D),E)) = hAPP(c_List_Ofilter(B,D),c_List_Olinorder__class_Osort__key(B,A,C,E)) # label(fact_filter__sort) # label(axiom). [clausify(3423)]. 10.98/10.99 Derived: c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,hAPP(c_List_Ofilter(A,C),D)) = hAPP(c_List_Ofilter(A,C),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,D)). [resolve(5190,a,4908,a)]. 10.98/10.99 Derived: c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,hAPP(c_List_Ofilter(A,C),D)) = hAPP(c_List_Ofilter(A,C),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,D)). [resolve(5190,a,5154,a)]. 10.98/10.99 5191 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMax(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_Max_Oinsert) # label(axiom). [clausify(3437)]. 10.98/10.99 5192 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_Max_Oidem) # label(axiom). [clausify(3458)]. 10.98/10.99 5193 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Big__Operators_Olinorder__class_OMin(A,B)),C)) # label(fact_Min__le) # label(axiom). [clausify(3488)]. 10.98/10.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),B)). [resolve(5193,a,4908,a)]. 10.98/10.99 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),B)). [resolve(5193,a,5154,a)]. 10.98/10.99 5194 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(3499)]. 10.98/10.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),A)). [resolve(5194,a,4908,a)]. 10.98/10.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5194,a,5154,a)]. 10.98/10.99 5195 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_atLeastLessThan__subset__iff) # label(axiom). [clausify(3499)]. 11.08/11.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),A)). [resolve(5195,a,4908,a)]. 11.08/11.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5195,a,5154,a)]. 11.08/11.02 5196 -class_Orderings_Olinorder(A) | -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_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)) # label(fact_min__max_OSup__fin_Ounion__inter) # label(axiom). [clausify(3510)]. 11.08/11.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),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(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),B)). [resolve(5196,a,4908,a)]. 11.08/11.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),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)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(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))),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5196,a,5154,a)]. 11.08/11.02 5197 -class_Orderings_Olinorder(A) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C))) = hAPP(c_List_Ofoldl(A,A,c_Orderings_Oord__class_Omin(A),B),C) # label(fact_min__max_OInf__fin__set__fold) # label(axiom). [clausify(3559)]. 11.17/11.12 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_List_Oset(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B))) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B). [resolve(5197,a,4908,a)]. 11.17/11.12 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B))) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5197,a,5154,a)]. 11.17/11.12 5198 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),E))) # label(fact_min__max_Oinf__mono) # label(axiom). [clausify(3613)]. 11.17/11.12 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),D))). [resolve(5198,a,4908,a)]. 11.17/11.12 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(5198,a,5154,a)]. 11.17/11.12 5199 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | C != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(3621)]. 11.17/11.12 5200 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(3621)]. 11.17/11.12 5201 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(B) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E) # label(fact_insort__not__Nil) # label(axiom). [clausify(3649)]. 11.17/11.12 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D). [resolve(5201,a,4908,a)]. 11.17/11.12 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D). [resolve(5201,a,5154,a)]. 11.17/11.12 5202 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)) # label(fact_min__max_Osup_Oleft__commute) # label(axiom). [clausify(3658)]. 11.17/11.12 5203 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_leI) # label(axiom). [clausify(3677)]. 11.17/11.12 5204 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) # label(fact_min__max_OInf__fin_Oin__idem) # label(axiom). [clausify(3692)]. 11.17/11.12 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)). [resolve(5204,a,4908,a)]. 11.17/11.14 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)) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5204,a,5154,a)]. 11.17/11.14 5205 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),hAPP(B,C))),C)) # label(fact_sorted__same) # label(axiom). [clausify(3704)]. 11.17/11.14 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(c_List_Ofilter(tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_fequal),hAPP(A,B))),B)). [resolve(5205,a,4908,a)]. 11.17/11.14 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Ofilter(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),c_fequal),hAPP(A,B))),B)). [resolve(5205,a,5154,a)]. 11.17/11.14 5206 -class_Orderings_Olinorder(A) | hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),D),E)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),E)) # label(fact_length__insort) # label(axiom). [clausify(3709)]. 11.17/11.14 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),D)). [resolve(5206,a,4908,a)]. 11.17/11.14 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),D)). [resolve(5206,a,5154,a)]. 11.17/11.14 5207 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) != C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__sup) # label(axiom). [clausify(3714)]. 11.17/11.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5207,a,4908,a)]. 11.17/11.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5207,a,5154,a)]. 11.17/11.14 5208 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__sup) # label(axiom). [clausify(3714)]. 11.17/11.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5208,a,4908,a)]. 11.17/11.14 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5208,a,5154,a)]. 11.17/11.14 5209 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_less__max__iff__disj) # label(axiom). [clausify(3718)]. 11.17/11.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5209,a,4908,a)]. 11.28/11.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5209,a,5154,a)]. 11.28/11.27 5210 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_less__max__iff__disj) # label(axiom). [clausify(3718)]. 11.28/11.27 5211 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_less__max__iff__disj) # label(axiom). [clausify(3718)]. 11.28/11.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5211,a,4908,a)]. 11.28/11.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5211,a,5154,a)]. 11.28/11.27 5212 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_min__max_Ole__supI1) # label(axiom). [clausify(3766)]. 11.28/11.27 5213 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),c_List_Olinorder__class_Oinsort__insert__key(B,A,C,E,D))) # label(fact_sorted__insort__insert__key) # label(axiom). [clausify(3797)]. 11.28/11.27 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Nat_Onat,B,D,C))). [resolve(5213,a,4908,a)]. 11.28/11.27 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Code__Numeral_Ocode__numeral,B,D,C))). [resolve(5213,a,5154,a)]. 11.28/11.27 5214 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hAPP(c_SetInterval_Oord__class_OlessThan(A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(A),B)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)) # label(fact_ivl__disj__un_I7_J) # label(axiom). [clausify(3805)]. 11.28/11.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)). [resolve(5214,a,4908,a)]. 11.28/11.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5214,a,5154,a)]. 11.28/11.27 5215 -class_Lattices_Osemilattice__inf(A) | -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Omin(A) = c_Lattices_Osemilattice__inf__class_Oinf(A) # label(fact_inf__min) # label(axiom). [clausify(3826)]. 11.38/11.35 Derived: -class_Lattices_Osemilattice__inf(tc_Nat_Onat) | c_Orderings_Oord__class_Omin(tc_Nat_Onat) = c_Lattices_Osemilattice__inf__class_Oinf(tc_Nat_Onat). [resolve(5215,b,4908,a)]. 11.38/11.35 Derived: -class_Lattices_Osemilattice__inf(tc_Code__Numeral_Ocode__numeral) | c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral) = c_Lattices_Osemilattice__inf__class_Oinf(tc_Code__Numeral_Ocode__numeral). [resolve(5215,b,5154,a)]. 11.38/11.35 5216 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f429(B,C,A)),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(A,C) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),C)) # label(fact_sorted_OCons) # label(axiom). [clausify(3829)]. 11.38/11.35 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f429(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),B))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)). [resolve(5216,a,4908,a)]. 11.38/11.35 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f429(A,B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5216,a,5154,a)]. 11.38/11.35 5217 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f429(B,C,A))) | -c_List_Olinorder__class_Osorted(A,C) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),C)) # label(fact_sorted_OCons) # label(axiom). [clausify(3829)]. 11.38/11.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f429(A,B,tc_Nat_Onat))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)). [resolve(5217,a,4908,a)]. 11.38/11.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),f429(A,B,tc_Code__Numeral_Ocode__numeral))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5217,a,5154,a)]. 11.38/11.35 5218 -class_Orderings_Olinorder(A) | hAPP(c_List_Oset(B),c_List_Olinorder__class_Osort__key(B,A,C,D)) = hAPP(c_List_Oset(B),D) # label(fact_set__sort) # label(axiom). [clausify(3842)]. 11.38/11.35 Derived: hAPP(c_List_Oset(A),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,C)) = hAPP(c_List_Oset(A),C). [resolve(5218,a,4908,a)]. 11.38/11.35 Derived: hAPP(c_List_Oset(A),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,B,C)) = hAPP(c_List_Oset(A),C). [resolve(5218,a,5154,a)]. 11.38/11.35 5219 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Odrop(A),C),B)) # label(fact_sorted__drop) # label(axiom). [clausify(3850)]. 11.38/11.35 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Odrop(tc_Nat_Onat),B),A)). [resolve(5219,a,4908,a)]. 11.38/11.35 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Odrop(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5219,a,5154,a)]. 11.38/11.35 5220 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B) # label(fact_min__max_OInf__fin_Oeq__fold_H) # label(axiom). [clausify(3901)]. 11.38/11.35 5221 -class_Int_Onumber(A) | -class_Orderings_Olinorder(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(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__number__of__eq__not__less) # label(axiom). [clausify(3918)]. 11.47/11.42 5222 -class_Int_Onumber(A) | -class_Orderings_Olinorder(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(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__number__of__eq__not__less) # label(axiom). [clausify(3918)]. 11.47/11.42 5223 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) # label(fact_min__max_OSup__fin_Oin__idem) # label(axiom). [clausify(3943)]. 11.47/11.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(5223,a,4908,a)]. 11.47/11.42 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)) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5223,a,5154,a)]. 11.47/11.42 5224 -class_Orderings_Olinorder(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),c_Big__Operators_Olinorder__class_OMin(A,B)),B)) # label(fact_Min__in) # label(axiom). [clausify(3951)]. 11.47/11.42 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),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),A)). [resolve(5224,a,4908,a)]. 11.47/11.42 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),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5224,a,5154,a)]. 11.47/11.42 5225 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) = C # label(fact_min__max_Osup__Inf__absorb) # label(axiom). [clausify(3954)]. 11.47/11.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)) = B. [resolve(5225,a,4908,a)]. 11.47/11.42 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(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)) = B. [resolve(5225,a,5154,a)]. 11.47/11.42 5226 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(B),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),E),hAPP(F,C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),E),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,A),D),F))) # label(fact_less__SUP__iff) # label(axiom). [clausify(3996)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D),hAPP(E,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Nat_Onat),C),E))). [resolve(5226,b,4908,a)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),D),hAPP(E,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Code__Numeral_Ocode__numeral),C),E))). [resolve(5226,b,5154,a)]. 11.47/11.43 5227 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(B),f454(C,D,B,E,A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),E),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,A),D),C))) # label(fact_less__SUP__iff) # label(axiom). [clausify(3996)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f454(B,C,A,D,tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Nat_Onat),C),B))). [resolve(5227,b,4908,a)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f454(B,C,A,D,tc_Code__Numeral_Ocode__numeral)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5227,b,5154,a)]. 11.47/11.43 5228 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(C,f454(C,D,E,B,A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(E,A),D),C))) # label(fact_less__SUP__iff) # label(axiom). [clausify(3996)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(B,f454(B,C,D,A,tc_Nat_Onat)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(D,tc_Nat_Onat),C),B))). [resolve(5228,b,4908,a)]. 11.47/11.43 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(B,f454(B,C,D,A,tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(D,tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5228,b,5154,a)]. 11.47/11.43 5229 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B))) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(3998)]. 11.47/11.43 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_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A))). [resolve(5229,a,4908,a)]. 11.47/11.43 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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5229,a,5154,a)]. 11.47/11.47 5230 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B))) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(3998)]. 11.47/11.47 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_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A))). [resolve(5230,a,4908,a)]. 11.47/11.47 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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5230,a,5154,a)]. 11.47/11.47 5231 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) # label(fact_min__max_Ofold1__belowI) # label(axiom). [clausify(4011)]. 11.47/11.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A)),B)). [resolve(5231,a,4908,a)]. 11.47/11.47 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A)),B)). [resolve(5231,a,5154,a)]. 11.47/11.47 5232 -class_Orderings_Olinorder(A) | B != C | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) = hAPP(c_SetInterval_Oord__class_OlessThan(A),C) # label(fact_lessThan__eq__iff) # label(axiom). [clausify(4020)]. 11.47/11.47 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B). [resolve(5232,a,4908,a)]. 11.47/11.47 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5232,a,5154,a)]. 11.47/11.47 5233 -class_Orderings_Olinorder(A) | B = C | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) != hAPP(c_SetInterval_Oord__class_OlessThan(A),C) # label(fact_lessThan__eq__iff) # label(axiom). [clausify(4020)]. 11.47/11.47 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B). [resolve(5233,a,4908,a)]. 11.47/11.47 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5233,a,5154,a)]. 11.47/11.47 5234 -class_Orderings_Olinorder(A) | -c_Fun_Oinj__on(B,A,C,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),hAPP(c_List_Oset(B),D)),hAPP(c_List_Oset(B),E))) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Omap(B,A),C),D))) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),E)) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Omap(B,A),C),E))) | hAPP(c_List_Oset(B),E) != hAPP(c_List_Oset(B),D) | E = D # label(fact_map__sorted__distinct__set__unique) # label(axiom). [clausify(4031)]. 11.47/11.47 Derived: -c_Fun_Oinj__on(A,tc_Nat_Onat,B,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_List_Oset(A),C)),hAPP(c_List_Oset(A),D))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)) | -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D))) | hAPP(c_List_Oset(A),D) != hAPP(c_List_Oset(A),C) | D = C. [resolve(5234,a,4908,a)]. 11.47/11.48 Derived: -c_Fun_Oinj__on(A,tc_Code__Numeral_Ocode__numeral,B,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_List_Oset(A),C)),hAPP(c_List_Oset(A),D))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D)) | -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D))) | hAPP(c_List_Oset(A),D) != hAPP(c_List_Oset(A),C) | D = C. [resolve(5234,a,5154,a)]. 11.47/11.48 5235 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_min__max_Oless__supI1) # label(axiom). [clausify(4040)]. 11.47/11.48 5236 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__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_Orderings_Oord__class_Omax(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) # label(fact_min__max_Osup__Inf1__distrib) # label(axiom). [clausify(4051)]. 11.47/11.48 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_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__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_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)). [resolve(5236,a,4908,a)]. 11.47/11.48 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_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__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_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5236,a,5154,a)]. 11.58/11.51 5237 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),B),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Big__Operators_Olinorder__class_OMin(A,C)),c_Big__Operators_Olinorder__class_OMin(A,B))) # label(fact_Min__antimono) # label(axiom). [clausify(4068)]. 11.58/11.51 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,B)),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A))). [resolve(5237,a,4908,a)]. 11.58/11.51 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),B)) | 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,B)),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A))). [resolve(5237,a,5154,a)]. 11.58/11.51 5238 -class_Orderings_Olinorder(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),c_Big__Operators_Olinorder__class_OMax(A,B)),B)) # label(fact_Max__in) # label(axiom). [clausify(4081)]. 11.58/11.51 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),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),A)). [resolve(5238,a,4908,a)]. 11.59/11.54 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),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5238,a,5154,a)]. 11.59/11.54 5239 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),c_List_Oremove1(B,E,D))) # label(fact_sorted__map__remove1) # label(axiom). [clausify(4088)]. 11.59/11.54 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),c_List_Oremove1(A,D,C))). [resolve(5239,a,4908,a)]. 11.59/11.54 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),c_List_Oremove1(A,D,C))). [resolve(5239,a,5154,a)]. 11.59/11.54 5240 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) = C # label(fact_min__max_Osup__absorb2) # label(axiom). [clausify(4092)]. 11.59/11.54 5241 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__max_Ole__sup__iff) # label(axiom). [clausify(4115)]. 11.59/11.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5241,a,4908,a)]. 11.59/11.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5241,a,5154,a)]. 11.59/11.54 5242 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__sup__iff) # label(axiom). [clausify(4115)]. 11.59/11.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5242,a,4908,a)]. 11.59/11.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5242,a,5154,a)]. 11.59/11.54 5243 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__sup__iff) # label(axiom). [clausify(4115)]. 11.59/11.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C)) | -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),A),C)). [resolve(5243,a,4908,a)]. 11.59/11.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C)) | -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),A),C)). [resolve(5243,a,5154,a)]. 11.59/11.56 5244 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(c_List_Oset(A),f474(B,A)) = B # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(4116)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(c_List_Oset(tc_Nat_Onat),f474(A,tc_Nat_Onat)) = A. [resolve(5244,a,4908,a)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),f474(A,tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5244,a,5154,a)]. 11.59/11.56 5245 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),f474(B,A))) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(4116)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),f474(A,tc_Nat_Onat))). [resolve(5245,a,4908,a)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),f474(A,tc_Code__Numeral_Ocode__numeral))). [resolve(5245,a,5154,a)]. 11.59/11.56 5246 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),C)) | -c_List_Olinorder__class_Osorted(A,C) | hAPP(c_List_Oset(A),C) != B | C = f474(B,A) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(4116)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hAPP(c_List_Oset(tc_Nat_Onat),B) != A | B = f474(A,tc_Nat_Onat). [resolve(5246,a,4908,a)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B) != A | B = f474(A,tc_Code__Numeral_Ocode__numeral). [resolve(5246,a,5154,a)]. 11.59/11.56 5247 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted(A,f474(B,A)) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(4116)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f474(A,tc_Nat_Onat)). [resolve(5247,a,4908,a)]. 11.59/11.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f474(A,tc_Code__Numeral_Ocode__numeral)). [resolve(5247,a,5154,a)]. 11.59/11.56 5248 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) # label(fact_sorted__many__eq) # label(axiom). [clausify(4125)]. 11.59/11.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),C)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),C))). [resolve(5248,a,4908,a)]. 11.59/11.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),C)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5248,a,5154,a)]. 11.59/11.56 5249 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) # label(fact_sorted__many__eq) # label(axiom). [clausify(4125)]. 11.59/11.56 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),C))). [resolve(5249,a,4908,a)]. 11.68/11.66 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5249,a,5154,a)]. 11.68/11.66 5250 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),C)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),D),hAPP(hAPP(c_List_Olist_OCons(A),B),C))) # label(fact_sorted__many__eq) # label(axiom). [clausify(4125)]. 11.68/11.66 5251 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)) # label(fact_min__max_Oinf_Oleft__commute) # label(axiom). [clausify(4138)]. 11.68/11.66 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)). [resolve(5251,a,4908,a)]. 11.68/11.66 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5251,a,5154,a)]. 11.68/11.66 5252 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),D)) # label(fact_ivl__disj__un_I15_J) # label(axiom). [clausify(4177)]. 11.68/11.66 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)). [resolve(5252,a,4908,a)]. 11.68/11.66 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,B),C)). [resolve(5252,a,5154,a)]. 11.68/11.66 5253 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)) # label(fact_ivl__disj__un_I1_J) # label(axiom). [clausify(4201)]. 11.68/11.66 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A)). [resolve(5253,a,4908,a)]. 11.68/11.66 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5253,a,5154,a)]. 11.68/11.69 5254 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__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_Orderings_Oord__class_Omin(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) # label(fact_min__max_Oinf__Sup1__distrib) # label(axiom). [clausify(4224)]. 11.68/11.69 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_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__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_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(5254,a,4908,a)]. 11.68/11.69 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_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__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_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5254,a,5154,a)]. 11.77/11.73 5255 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(A),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)) # label(fact_ivl__disj__un_I3_J) # label(axiom). [clausify(4251)]. 11.77/11.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)). [resolve(5255,a,4908,a)]. 11.77/11.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5255,a,5154,a)]. 11.77/11.73 5256 -class_Orderings_Olinorder(A) | c_Orderings_Oord_Omax(A,hAPP(c_COMBC(A,A,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(A))) = c_Orderings_Oord__class_Omin(A) # label(fact_dual__max) # label(axiom). [clausify(4254)]. 11.77/11.73 Derived: c_Orderings_Oord_Omax(tc_Nat_Onat,hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat))) = c_Orderings_Oord__class_Omin(tc_Nat_Onat). [resolve(5256,a,4908,a)]. 11.77/11.73 Derived: c_Orderings_Oord_Omax(tc_Code__Numeral_Ocode__numeral,hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral))) = c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral). [resolve(5256,a,5154,a)]. 11.77/11.73 5257 -class_Orderings_Olinorder(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(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_Min__insert) # label(axiom). [clausify(4267)]. 11.77/11.73 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(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5257,a,4908,a)]. 11.77/11.73 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(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5257,a,5154,a)]. 11.86/11.79 5258 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Oinf__assoc) # label(axiom). [clausify(4270)]. 11.86/11.79 5259 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) | c_SetInterval_Oord__class_OatLeastAtMost(A,B,D) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)),c_SetInterval_Oord__class_OatLeastAtMost(A,C,D)) # label(fact_ivl__disj__un_I21_J) # label(axiom). [clausify(4310)]. 11.86/11.79 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)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,C) = 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,A),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)). [resolve(5259,a,4908,a)]. 11.86/11.79 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),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Code__Numeral_Ocode__numeral,B,C)). [resolve(5259,a,5154,a)]. 11.86/11.79 5260 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(c_SetInterval_Oord__class_OlessThan(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_lessThan__strict__subset__iff) # label(axiom). [clausify(4320)]. 11.86/11.79 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5260,a,4908,a)]. 11.86/11.79 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5260,a,5154,a)]. 11.86/11.79 5261 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)),hAPP(c_SetInterval_Oord__class_OlessThan(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_lessThan__strict__subset__iff) # label(axiom). [clausify(4320)]. 11.86/11.79 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5261,a,4908,a)]. 11.86/11.79 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5261,a,5154,a)]. 11.89/11.86 5262 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) = C # label(fact_min__max_Oinf__Sup__absorb) # label(axiom). [clausify(4331)]. 11.89/11.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)) = B. [resolve(5262,a,4908,a)]. 11.89/11.86 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(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)) = B. [resolve(5262,a,5154,a)]. 11.89/11.86 5263 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C) # label(fact_min__max_Osup__inf__distrib2) # label(axiom). [clausify(4348)]. 11.89/11.86 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B). [resolve(5263,a,4908,a)]. 11.89/11.86 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5263,a,5154,a)]. 11.89/11.86 5264 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),B)),C)) # label(fact_min__max_Oless__infI2) # label(axiom). [clausify(4359)]. 11.89/11.86 5265 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(4378)]. 11.89/11.86 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(5265,a,4908,a)]. 11.89/11.86 5266 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__antisym__conv3) # label(axiom). [clausify(4378)]. 11.89/11.86 5267 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(A) != B | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.86 Derived: c_List_Olist_ONil(tc_Nat_Onat) != A | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5267,a,4908,a)]. 11.89/11.86 Derived: c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) != A | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5267,a,5154,a)]. 11.89/11.86 5268 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_List_Olist_OCons(A),B),C) != D | -c_List_Olinorder__class_Osorted(A,C) | hBOOL(hAPP(hAPP(c_member(A),f513(D,A,C,B)),hAPP(c_List_Oset(A),C))) | c_List_Olinorder__class_Osorted(A,D) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.86 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f513(C,tc_Nat_Onat,B,A)),hAPP(c_List_Oset(tc_Nat_Onat),B))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,C). [resolve(5268,a,4908,a)]. 11.89/11.87 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f513(C,tc_Code__Numeral_Ocode__numeral,B,A)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,C). [resolve(5268,a,5154,a)]. 11.89/11.87 5269 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_List_Olist_OCons(A),B),C) != D | -c_List_Olinorder__class_Osorted(A,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f513(D,A,C,B))) | c_List_Olinorder__class_Osorted(A,D) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.87 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f513(C,tc_Nat_Onat,B,A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,C). [resolve(5269,a,4908,a)]. 11.89/11.87 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),f513(C,tc_Code__Numeral_Ocode__numeral,B,A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,C). [resolve(5269,a,5154,a)]. 11.89/11.87 5270 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(A) = B | hAPP(hAPP(c_List_Olist_OCons(A),f515(B,A)),f514(B,A)) = B | -c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Nat_Onat) = A | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),f515(A,tc_Nat_Onat)),f514(A,tc_Nat_Onat)) = A | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5270,a,4908,a)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),f515(A,tc_Code__Numeral_Ocode__numeral)),f514(A,tc_Code__Numeral_Ocode__numeral)) = A | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5270,a,5154,a)]. 11.89/11.87 5271 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(A) = B | c_List_Olinorder__class_Osorted(A,f514(B,A)) | -c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Nat_Onat) = A | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f514(A,tc_Nat_Onat)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5271,a,4908,a)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f514(A,tc_Code__Numeral_Ocode__numeral)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5271,a,5154,a)]. 11.89/11.87 5272 -class_Orderings_Olinorder(A) | c_List_Olist_ONil(A) = B | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),f514(B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f515(B,A)),C)) | -c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted_Osimps) # label(axiom). [clausify(4395)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Nat_Onat) = A | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),hAPP(c_List_Oset(tc_Nat_Onat),f514(A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f515(A,tc_Nat_Onat)),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5272,a,4908,a)]. 11.89/11.87 Derived: c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),f514(A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f515(A,tc_Code__Numeral_Ocode__numeral)),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5272,a,5154,a)]. 11.89/11.87 5273 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMin(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B) # label(fact_Min_Oeq__fold_H) # label(axiom). [clausify(4407)]. 11.98/11.95 5274 -class_Orderings_Olinorder(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_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)) # label(fact_min__max_OInf__fin_Ounion__idem) # label(axiom). [clausify(4450)]. 11.98/11.95 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(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)). [resolve(5274,a,4908,a)]. 11.98/11.95 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(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_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5274,a,5154,a)]. 11.98/11.95 5275 -class_Orderings_Olinorder(A) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__neq__iff) # label(axiom). [clausify(4461)]. 11.98/11.95 5276 -class_Orderings_Olinorder(A) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__neq__iff) # label(axiom). [clausify(4461)]. 11.98/11.95 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5276,a,5154,a)]. 11.98/11.95 5277 -class_Orderings_Olinorder(A) | B != C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__neq__iff) # label(axiom). [clausify(4461)]. 11.98/11.95 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(5277,a,4908,a)]. 11.98/11.95 5278 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Big__Operators_Olinorder__class_OMax(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_Max_Oin__idem) # label(axiom). [clausify(4492)]. 11.98/11.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)). [resolve(5278,a,4908,a)]. 11.98/11.95 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)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5278,a,5154,a)]. 11.98/11.97 5279 -class_Orderings_Olinorder(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) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),C)) # label(fact_min__max_OSup__fin_Ounion__disjoint) # label(axiom). [clausify(4494)]. 11.98/11.97 5280 -class_Orderings_Olinorder(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_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__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_COMBB(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool),A),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_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),A),c_COMBS(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(A,tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool))),A),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_fun(A,tc_HOL_Obool)),tc_fun(A,tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),A),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,tc_fun(A,A)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_fun(A,A)),tc_fun(A,tc_fun(A,tc_HOL_Obool))),A),c_COMBB(tc_fun(A,A),tc_fun(A,tc_HOL_Obool),A)),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))),c_Orderings_Oord__class_Omax(A)))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,A)),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))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),C)) # label(fact_min__max_Osup__Inf2__distrib) # label(axiom). [clausify(4505)]. 11.98/11.97 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_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__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_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),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_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_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),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_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),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_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_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Nat_Onat),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_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool))),tc_Nat_Onat),c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat)),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))),c_Orderings_Oord__class_Omax(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_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_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat)),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)),A)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),B)). [resolve(5280,a,4908,a)]. 11.98/11.97 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_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__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_COMBB(tc_fun(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),tc_Code__Numeral_Ocode__numeral),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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(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_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBS(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_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_Code__Numeral_Ocode__numeral,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),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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),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_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(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_fun(tc_fun(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_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBB(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)),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))),c_Orderings_Oord__class_Omax(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_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_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),hAPP(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_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A)))),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)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5280,a,5154,a)]. 12.07/11.99 5281 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B) = hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omax(A)),B) # label(fact_min__max_OSup__fin_OF__eq) # label(axiom). [clausify(4518)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A) = hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),A). [resolve(5281,a,4908,a)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5281,a,5154,a)]. 12.07/11.99 5282 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f531(C,D,B,A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),C)),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),C),B))) # label(fact_min__max_Oinf__le__fold__inf) # label(axiom). [clausify(4529)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f531(B,C,A,tc_Nat_Onat)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),B),A))). [resolve(5282,a,4908,a)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f531(B,C,A,tc_Code__Numeral_Ocode__numeral)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),B)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),B),A))). [resolve(5282,a,5154,a)]. 12.07/11.99 5283 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f531(D,C,B,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),D),B))) # label(fact_min__max_Oinf__le__fold__inf) # label(axiom). [clausify(4529)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f531(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),C),A))). [resolve(5283,a,4908,a)]. 12.07/11.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f531(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_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),C),A))). [resolve(5283,a,5154,a)]. 12.07/11.99 5284 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C)) # label(fact_min__max_Osup__least) # label(axiom). [clausify(4533)]. 12.07/12.04 5285 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(A,B,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,C)),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))) # label(fact_ivl__disj__un_I4_J) # label(axiom). [clausify(4536)]. 12.07/12.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,B)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))). [resolve(5285,a,4908,a)]. 12.07/12.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,B)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))). [resolve(5285,a,5154,a)]. 12.07/12.04 5286 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)) # label(fact_min__max_Oinf__left__commute) # label(axiom). [clausify(4566)]. 12.07/12.04 5287 -class_Orderings_Olinorder(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_Orderings_Oord__class_Oless__eq(A),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B))) # label(fact_min__max_OInf__le__Sup) # label(axiom). [clausify(4578)]. 12.07/12.04 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_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A))). [resolve(5287,a,4908,a)]. 12.07/12.04 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_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5287,a,5154,a)]. 12.07/12.04 5288 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osort__key(A,A,c_COMBI(A),B) = hAPP(c_List_Ofoldl(tc_List_Olist(A),A,hAPP(c_COMBC(A,tc_List_Olist(A),tc_List_Olist(A)),c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A))),c_List_Olist_ONil(A)),B) # label(fact_sort__foldl__insort) # label(axiom). [clausify(4585)]. 12.07/12.04 Derived: c_List_Olinorder__class_Osort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),A) = hAPP(c_List_Ofoldl(tc_List_Olist(tc_Nat_Onat),tc_Nat_Onat,hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(tc_Nat_Onat),tc_List_Olist(tc_Nat_Onat)),c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat))),c_List_Olist_ONil(tc_Nat_Onat)),A). [resolve(5288,a,4908,a)]. 12.07/12.04 Derived: c_List_Olinorder__class_Osort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_List_Ofoldl(tc_List_Olist(tc_Code__Numeral_Ocode__numeral),tc_Code__Numeral_Ocode__numeral,hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_List_Olist(tc_Code__Numeral_Ocode__numeral),tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral))),c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5288,a,5154,a)]. 12.19/12.12 5289 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),E),D))) # label(fact_sorted__insort__key) # label(axiom). [clausify(4597)]. 12.19/12.12 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),D),C))). [resolve(5289,a,4908,a)]. 12.19/12.12 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),D),C))). [resolve(5289,a,5154,a)]. 12.19/12.12 5290 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),D)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,C),E),D))) # label(fact_sorted__insort__key) # label(axiom). [clausify(4597)]. 12.19/12.12 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),C)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),D),C))). [resolve(5290,a,4908,a)]. 12.19/12.12 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),C)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),D),C))). [resolve(5290,a,5154,a)]. 12.19/12.12 5291 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),B) # label(fact_min__max_Osup__commute) # label(axiom). [clausify(4613)]. 12.19/12.12 5292 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatMost(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)) # label(fact_Compl__greaterThan) # label(axiom). [clausify(4643)]. 12.19/12.12 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A)). [resolve(5292,a,4908,a)]. 12.19/12.12 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5292,a,5154,a)]. 12.19/12.12 5293 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B = D # label(fact_atLeastLessThan__inj_I1_J) # label(axiom). [clausify(4650)]. 12.19/12.12 5294 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__less__linear) # label(axiom). [clausify(4667)]. 12.19/12.12 5295 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C) # label(fact_min__max_Oinf__left__idem) # label(axiom). [clausify(4680)]. 12.19/12.15 5296 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted__list__of__set(A,hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),C),c_List_Olinorder__class_Osorted__list__of__set(A,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)))))) # label(fact_sorted__list__of__set__insert) # label(axiom). [clausify(4682)]. 12.19/12.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat)),B),c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))). [resolve(5296,a,4908,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral)),B),c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))). [resolve(5296,a,5154,a)]. 12.19/12.15 5297 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),c_Big__Operators_Olattice_OInf__fin(A,c_Orderings_Oord__class_Omin(A),B)) # label(fact_min__max_OInf__fin_Oinsert__idem) # label(axiom). [clausify(4698)]. 12.19/12.15 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_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A)). [resolve(5297,a,4908,a)]. 12.19/12.15 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_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5297,a,5154,a)]. 12.19/12.15 5298 class_Orderings_Olinorder(tc_Int_Oint) # label(arity_Int__Oint__Orderings_Olinorder) # label(axiom). [assumption]. 12.19/12.15 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(A,f1(B,A,tc_Int_Oint))),hAPP(A,f2(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),f1(B,A,tc_Int_Oint)),f2(B,A,tc_Int_Oint))) | -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_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B)). [resolve(5298,a,4909,a)]. 12.19/12.15 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B). [resolve(5298,a,4910,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Int_Oint)),B),A). [resolve(5298,a,4911,a)]. 12.19/12.15 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olist_ONil(tc_Int_Oint)). [resolve(5298,a,4912,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)). [resolve(5298,a,4913,a)]. 12.19/12.15 Derived: A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,4914,a)]. 12.19/12.15 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_OdropWhile(tc_Int_Oint,B,A)). [resolve(5298,a,4915,a)]. 12.19/12.15 Derived: -c_List_Olinorder__class_Osorted(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(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),B)),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),C))). [resolve(5298,a,4916,a)]. 12.19/12.15 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) = A. [resolve(5298,a,4917,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Int_Oint,C,D,B))). [resolve(5298,a,4918,a)]. 12.19/12.15 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,B,C)). [resolve(5298,a,4919,a)]. 12.19/12.15 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Otl(tc_Int_Oint),A)). [resolve(5298,a,4920,a)]. 12.19/12.15 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),A). [resolve(5298,a,4921,a)]. 12.19/12.15 Derived: hBOOL(hAPP(A,B)) | hAPP(c_List_Ofilter(C,A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Int_Oint,D),B),E)) = hAPP(c_List_Ofilter(C,A),E). [resolve(5298,a,4922,a)]. 12.19/12.15 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_OtakeWhile(tc_Int_Oint,B,A)). [resolve(5298,a,4923,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),A) = A. [resolve(5298,a,4924,a)]. 12.19/12.15 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B),C) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,C)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B)). [resolve(5298,a,4925,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5298,a,4926,a)]. 12.19/12.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5298,a,4927,a)]. 12.19/12.16 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(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5298,a,4928,a)]. 12.19/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Complete__Lattice_OInf__class_OInf(tc_Int_Oint,B)),C)). [resolve(5298,a,4929,b)]. 12.19/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f43(A,B,tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Complete__Lattice_OInf__class_OInf(tc_Int_Oint,B)),A)). [resolve(5298,a,4930,b)]. 12.19/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),f43(A,B,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Complete__Lattice_OInf__class_OInf(tc_Int_Oint,B)),A)). [resolve(5298,a,4931,b)]. 12.19/12.16 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),A),B)) | B = A. [resolve(5298,a,4932,a)]. 12.19/12.16 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),A),B)) | B != A. [resolve(5298,a,4933,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),B),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),B),A)). [resolve(5298,a,4934,a)]. 12.19/12.16 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)). [resolve(5298,a,4935,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)). [resolve(5298,a,4936,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),B),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),C),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A)). [resolve(5298,a,4937,a)]. 12.19/12.16 Derived: c_List_Oremove1(tc_Int_Oint,A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),A),B)) = B. [resolve(5298,a,4938,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)). [resolve(5298,a,4939,a)]. 12.19/12.16 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_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B)). [resolve(5298,a,4940,a)]. 12.19/12.16 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)). [resolve(5298,a,4941,a)]. 12.19/12.16 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),A))). [resolve(5298,a,4942,a)]. 12.19/12.16 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) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B)). [resolve(5298,a,4943,a)]. 12.19/12.16 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),A). [resolve(5298,a,4944,a)]. 12.19/12.16 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_Orderings_Oord__class_Omin(tc_Int_Oint),C),A)),B)). [resolve(5298,a,4946,a)]. 12.19/12.16 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B)). [resolve(5298,a,4947,a)]. 12.19/12.16 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),C)). [resolve(5298,a,4948,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),C),A)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))). [resolve(5298,a,4949,a)]. 12.19/12.16 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_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B)). [resolve(5298,a,4950,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = B. [resolve(5298,a,4951,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,4952,a)]. 12.19/12.16 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)). [resolve(5298,a,4953,a)]. 12.19/12.16 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_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,B)). [resolve(5298,a,4954,a)]. 12.19/12.16 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)) = A. [resolve(5298,a,4955,a)]. 12.19/12.16 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Orev(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C))) | c_List_OtakeWhile(A,hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),D)),B),C) = hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,A),hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),D)),B)),C). [resolve(5298,a,4956,a)]. 12.19/12.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5298,a,4957,a)]. 12.19/12.16 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,C)). [resolve(5298,a,4958,a)]. 12.19/12.16 Derived: -class_Lattices_Osemilattice__sup(tc_Int_Oint) | c_Lattices_Osemilattice__sup__class_Osup(tc_Int_Oint) = c_Orderings_Oord__class_Omax(tc_Int_Oint). [resolve(5298,a,4959,b)]. 12.19/12.16 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_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__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_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),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_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),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_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),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_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Int_Oint),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_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint)),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))),c_Orderings_Oord__class_Omin(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_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(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_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),B)). [resolve(5298,a,4960,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A) = hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A). [resolve(5298,a,4961,a)]. 12.19/12.17 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,B,C)). [resolve(5298,a,4962,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A). [resolve(5298,a,4963,a)]. 12.19/12.17 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_Orderings_Oord__class_Omin(tc_Int_Oint),C),A)),B)). [resolve(5298,a,4964,a)]. 12.19/12.17 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),A)),B)). [resolve(5298,a,4966,a)]. 12.19/12.17 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_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B))). [resolve(5298,a,4967,a)]. 12.19/12.17 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_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B))). [resolve(5298,a,4968,a)]. 12.19/12.17 Derived: c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(c_List_Oset(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B))) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B). [resolve(5298,a,4969,a)]. 12.19/12.17 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) = A. [resolve(5298,a,4971,a)]. 12.19/12.17 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) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)). [resolve(5298,a,4972,a)]. 12.19/12.17 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),B),A)). [resolve(5298,a,4973,a)]. 12.19/12.17 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f84(B,A,tc_Int_Oint))),hAPP(A,f85(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f84(B,A,tc_Int_Oint)),f85(B,A,tc_Int_Oint))) | -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_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,A),B)) = hAPP(A,c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,B)). [resolve(5298,a,4974,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),B)),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5298,a,4975,a)]. 12.19/12.17 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)). [resolve(5298,a,4976,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B)),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5298,a,4977,a)]. 12.19/12.17 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Int_Oint),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(C,f99(D,C,B,A,tc_Int_Oint))),D)). [resolve(5298,a,4978,a)]. 12.19/12.17 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Int_Oint),B),C)),D)) | hBOOL(hAPP(hAPP(c_member(A),f99(D,C,B,A,tc_Int_Oint)),B)). [resolve(5298,a,4979,a)]. 12.19/12.17 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Int_Oint),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(C,E)),D)) | -hBOOL(hAPP(hAPP(c_member(A),E),B)). [resolve(5298,a,4980,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)). [resolve(5298,a,4981,a)]. 12.19/12.17 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f102(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)). [resolve(5298,a,4982,a)]. 12.19/12.17 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f103(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),B))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)). [resolve(5298,a,4983,a)]. 12.19/12.17 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f102(B,A,tc_Int_Oint)),f103(B,A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)). [resolve(5298,a,4984,a)]. 12.19/12.17 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)). [resolve(5298,a,4985,a)]. 12.19/12.17 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),B),A)). [resolve(5298,a,4986,a)]. 12.19/12.17 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),hAPP(c_List_Oset(tc_Int_Oint),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),B),D)). [resolve(5298,a,4987,a)]. 12.19/12.18 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),A),c_fequal),B)),hAPP(C,D))),D))). [resolve(5298,a,4989,a)]. 12.19/12.18 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),B),A)). [resolve(5298,a,4990,a)]. 12.19/12.18 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),B),A)). [resolve(5298,a,4991,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,C))) | hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(5298,a,4992,a)]. 12.19/12.18 Derived: hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,C))) | -hBOOL(hAPP(c_List_Odistinct(A),C)). [resolve(5298,a,4993,a)]. 12.19/12.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,B))). [resolve(5298,a,4994,b)]. 12.19/12.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f111(A,B,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,A))). [resolve(5298,a,4995,b)]. 12.19/12.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),f111(B,A,tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,B))). [resolve(5298,a,4996,b)]. 12.19/12.18 Derived: c_Orderings_Oord_Omin(tc_Int_Oint,hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Int_Oint))) = c_Orderings_Oord__class_Omax(tc_Int_Oint). [resolve(5298,a,4997,a)]. 12.19/12.18 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),B),A)). [resolve(5298,a,4999,a)]. 12.19/12.18 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),D))). [resolve(5298,a,5000,a)]. 12.19/12.18 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B). [resolve(5298,a,5001,a)]. 12.19/12.18 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(A,f119(B,A,tc_Int_Oint))),hAPP(A,f120(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),f119(B,A,tc_Int_Oint)),f120(B,A,tc_Int_Oint))) | -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_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,A),B)) = hAPP(A,c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),B)). [resolve(5298,a,5002,a)]. 12.19/12.18 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | -hBOOL(hAPP(D,E)) | hAPP(c_List_Ofilter(A,D),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),E),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),E),hAPP(c_List_Ofilter(A,D),C)). [resolve(5298,a,5003,a)]. 12.19/12.18 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)). [resolve(5298,a,5004,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5005,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = B. [resolve(5298,a,5006,a)]. 12.19/12.18 Derived: hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,B),D)). [resolve(5298,a,5007,a)]. 12.19/12.18 Derived: hAPP(c_List_Oset(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D)) = hAPP(hAPP(c_Set_Oinsert(A),C),hAPP(c_List_Oset(A),D)). [resolve(5298,a,5008,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)). [resolve(5298,a,5009,a)]. 12.19/12.18 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),A)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),B)) | hAPP(c_List_Oset(tc_Int_Oint),B) != hAPP(c_List_Oset(tc_Int_Oint),A) | B = A. [resolve(5298,a,5010,a)]. 12.19/12.18 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),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))). [resolve(5298,a,5011,a)]. 12.19/12.18 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f138(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5298,a,5012,a)]. 12.19/12.18 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f138(A,B,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5298,a,5013,a)]. 12.19/12.18 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5298,a,5014,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),A)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),C),B)). [resolve(5298,a,5015,a)]. 12.19/12.18 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,B,C)). [resolve(5298,a,5017,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),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(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B)). [resolve(5298,a,5019,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f147(B,C,A,tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B))). [resolve(5298,a,5020,a)]. 12.19/12.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f147(B,C,A,tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B))). [resolve(5298,a,5021,a)]. 12.19/12.19 Derived: hBOOL(hAPP(hAPP(c_member(A),f150(B,C,D,A,tc_Int_Oint)),hAPP(c_List_Oset(A),D))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,C),B),D) = hAPP(hAPP(c_List_Olist_OCons(A),B),D). [resolve(5298,a,5022,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,B)),hAPP(A,f150(B,A,C,D,tc_Int_Oint)))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),C) = hAPP(hAPP(c_List_Olist_OCons(D),B),C). [resolve(5298,a,5023,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(c_List_Oset(tc_Int_Oint),c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,A)) = A. [resolve(5298,a,5024,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,A)). [resolve(5298,a,5025,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,A))). [resolve(5298,a,5026,a)]. 12.19/12.19 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B))). [resolve(5298,a,5027,a)]. 12.19/12.19 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(c_List_Ofilter(A,D),C))). [resolve(5298,a,5028,a)]. 12.19/12.19 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_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(5298,a,5029,a)]. 12.19/12.19 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_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f160(B,A,tc_Int_Oint)),A)). [resolve(5298,a,5030,a)]. 12.19/12.19 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_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),f160(B,A,tc_Int_Oint))). [resolve(5298,a,5031,a)]. 12.19/12.19 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)))). [resolve(5298,a,5032,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B))). [resolve(5298,a,5033,a)]. 12.19/12.19 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,C)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C). [resolve(5298,a,5034,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5035,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A) = B. [resolve(5298,a,5036,a)]. 12.19/12.19 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),A,B) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),A),B). [resolve(5298,a,5037,a)]. 12.19/12.19 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)) | B != D | C != A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B). [resolve(5298,a,5038,a)]. 12.19/12.19 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)) | B = D | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B). [resolve(5298,a,5039,a)]. 12.19/12.19 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)) | C = A | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B). [resolve(5298,a,5040,a)]. 12.19/12.19 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),B),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),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),C)). [resolve(5298,a,5041,a)]. 12.19/12.19 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),f166(B,A,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),B)). [resolve(5298,a,5042,a)]. 12.19/12.19 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f166(B,A,tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),B)). [resolve(5298,a,5043,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),A). [resolve(5298,a,5044,a)]. 12.19/12.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5045,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = B. [resolve(5298,a,5046,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool))))) = c_List_Oremove1(tc_Int_Oint,B,c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,A)). [resolve(5298,a,5047,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),C),A))). [resolve(5298,a,5048,a)]. 12.19/12.20 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),B),C)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),A),C)). [resolve(5298,a,5049,a)]. 12.19/12.20 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),A) = A. [resolve(5298,a,5050,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A). [resolve(5298,a,5051,a)]. 12.19/12.20 Derived: c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,C)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,B,D)). [resolve(5298,a,5052,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5298,a,5053,a)]. 12.19/12.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5054,a)]. 12.19/12.20 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5055,a)]. 12.19/12.20 Derived: c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(c_List_Oset(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B))) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B). [resolve(5298,a,5057,a)]. 12.28/12.20 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_Orderings_Oord__class_Omin(tc_Int_Oint),f181(A,tc_Int_Oint)),f182(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f181(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f182(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),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),A)). [resolve(5298,a,5058,a)]. 12.28/12.20 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B) | A = B. [resolve(5298,a,5059,a)]. 12.28/12.20 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B) | A != B. [resolve(5298,a,5060,a)]. 12.28/12.20 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),B),A)). [resolve(5298,a,5061,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A). [resolve(5298,a,5062,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),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(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,B)). [resolve(5298,a,5063,a)]. 12.28/12.20 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_Orderings_Oord__class_Omax(tc_Int_Oint),f200(A,tc_Int_Oint)),f201(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f200(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f201(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),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),A)). [resolve(5298,a,5064,a)]. 12.28/12.20 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Orev(tc_Int_Oint),A)) | c_List_Olist_ONil(tc_Int_Oint) = A | c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5298,a,5065,a)]. 12.28/12.20 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Orev(tc_Int_Oint),A)) | c_List_Olist_ONil(tc_Int_Oint) != A | c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A,B) = B. [resolve(5298,a,5066,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = B. [resolve(5298,a,5067,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5068,a)]. 12.28/12.20 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f210(B,A,tc_Int_Oint))),hAPP(A,f211(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f210(B,A,tc_Int_Oint)),f211(B,A,tc_Int_Oint))) | -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_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,A),B)) = hAPP(A,c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)). [resolve(5298,a,5069,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A) = B. [resolve(5298,a,5070,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5071,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5072,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = B. [resolve(5298,a,5073,a)]. 12.28/12.20 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),c_List_Olist_ONil(tc_Int_Oint))). [resolve(5298,a,5075,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B)). [resolve(5298,a,5076,a)]. 12.28/12.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),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(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)). [resolve(5298,a,5077,a)]. 12.28/12.21 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)) = A. [resolve(5298,a,5078,a)]. 12.28/12.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | B != A. [resolve(5298,a,5080,a)]. 12.28/12.21 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),B),A)). [resolve(5298,a,5081,a)]. 12.28/12.21 Derived: hAPP(A,B) = hAPP(A,C) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),C),E)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),E)). [resolve(5298,a,5082,a)]. 12.28/12.21 Derived: c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,C)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,B,D)). [resolve(5298,a,5083,a)]. 12.28/12.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)). [resolve(5298,a,5084,a)]. 12.28/12.21 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A). [resolve(5298,a,5085,a)]. 12.28/12.21 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | c_List_Olinorder__class_Oinsort__insert__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),A,B) = B. [resolve(5298,a,5086,a)]. 12.28/12.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B)). [resolve(5298,a,5087,a)]. 12.28/12.21 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f243(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5298,a,5088,a)]. 12.28/12.21 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f243(A,B,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5298,a,5089,a)]. 12.28/12.21 Derived: c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5298,a,5092,a)]. 12.28/12.21 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),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,B,C)). [resolve(5298,a,5093,a)]. 12.28/12.22 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)). [resolve(5298,a,5094,a)]. 12.28/12.22 Derived: c_List_Olist_ONil(tc_Int_Oint) = A | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Obutlast(tc_Int_Oint,A)). [resolve(5298,a,5098,a)]. 12.28/12.22 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),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,B),C)). [resolve(5298,a,5099,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),A) = B. [resolve(5298,a,5101,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B) = A. [resolve(5298,a,5102,a)]. 12.28/12.22 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))). [resolve(5298,a,5103,a)]. 12.28/12.22 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olinorder__class_Osort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),A)). [resolve(5298,a,5104,a)]. 12.28/12.22 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)). [resolve(5298,a,5105,a)]. 12.28/12.22 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),A)). [resolve(5298,a,5106,a)]. 12.28/12.22 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))). [resolve(5298,a,5107,a)]. 12.28/12.22 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B))). [resolve(5298,a,5108,a)]. 12.28/12.22 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),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))). [resolve(5298,a,5109,a)]. 12.28/12.22 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(c_List_Oset(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B))) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B). [resolve(5298,a,5110,a)]. 12.28/12.22 Derived: hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),c_List_Olist_ONil(A)) = hAPP(hAPP(c_List_Olist_OCons(A),C),c_List_Olist_ONil(A)). [resolve(5298,a,5111,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5298,a,5112,a)]. 12.28/12.22 Derived: c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,D)). [resolve(5298,a,5114,a)]. 12.28/12.22 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5298,a,5115,a)]. 12.28/12.22 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f282(B,A,tc_Int_Oint)),A)). [resolve(5298,a,5116,a)]. 12.28/12.22 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f282(B,A,tc_Int_Oint))). [resolve(5298,a,5117,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A))). [resolve(5298,a,5118,a)]. 12.28/12.22 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool))) = c_List_Olist_ONil(tc_Int_Oint). [resolve(5298,a,5121,a)]. 12.28/12.22 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A)). [resolve(5298,a,5123,a)]. 12.28/12.22 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B) != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5298,a,5124,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olinorder__class_OMin(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_Orderings_Oord__class_Omin(tc_Int_Oint)),B),A). [resolve(5298,a,5126,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5127,a)]. 12.28/12.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = B. [resolve(5298,a,5128,a)]. 12.28/12.22 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Oremdups(tc_Int_Oint,A)). [resolve(5298,a,5129,a)]. 12.28/12.22 Derived: c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) = A. [resolve(5298,a,5130,a)]. 12.28/12.23 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),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),A)),B)). [resolve(5298,a,5131,a)]. 12.28/12.23 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_Orderings_Oord__class_Omax(tc_Int_Oint),C),A)),B)). [resolve(5298,a,5132,a)]. 12.28/12.23 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_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),B)). [resolve(5298,a,5133,a)]. 12.28/12.23 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_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,B)). [resolve(5298,a,5134,a)]. 12.28/12.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)))). [resolve(5298,a,5135,a)]. 12.28/12.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B)). [resolve(5298,a,5136,a)]. 12.28/12.23 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))) = A. [resolve(5298,a,5137,a)]. 12.28/12.23 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f296(A,tc_Int_Oint)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))). [resolve(5298,a,5139,a)]. 12.28/12.23 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f297(A,tc_Int_Oint)),f296(A,tc_Int_Oint))). [resolve(5298,a,5140,a)]. 12.28/12.23 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f297(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f296(A,tc_Int_Oint)))). [resolve(5298,a,5141,a)]. 12.28/12.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))). [resolve(5298,a,5142,a)]. 12.28/12.23 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Otake(tc_Int_Oint),B),A)). [resolve(5298,a,5143,a)]. 12.28/12.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)). [resolve(5298,a,5144,a)]. 12.28/12.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,B)),hAPP(A,C))) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),hAPP(hAPP(c_List_Olist_OCons(D),C),E)) = hAPP(hAPP(c_List_Olist_OCons(D),C),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(D,tc_Int_Oint,A),B),E)). [resolve(5298,a,5145,a)]. 12.28/12.23 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_Orderings_Oord__class_Omin(tc_Int_Oint),f311(A,tc_Int_Oint)),f312(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f311(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f312(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),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),A)). [resolve(5298,a,5146,a)]. 12.28/12.23 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,C))). [resolve(5298,a,5147,a)]. 12.28/12.23 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_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),B)). [resolve(5298,a,5148,a)]. 12.28/12.23 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B) | A = B. [resolve(5298,a,5149,a)]. 12.28/12.23 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B) | A != B. [resolve(5298,a,5150,a)]. 12.28/12.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),B)). [resolve(5298,a,5151,a)]. 12.28/12.23 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)). [resolve(5298,a,5153,a)]. 12.28/12.23 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_Orderings_Oord__class_Omax(tc_Int_Oint),f326(A,tc_Int_Oint)),f327(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f326(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f327(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),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)),A)). [resolve(5298,a,5155,a)]. 12.28/12.23 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),A) = A. [resolve(5298,a,5158,a)]. 12.28/12.23 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Oremove1(tc_Int_Oint,B,A)). [resolve(5298,a,5159,a)]. 12.28/12.23 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)). [resolve(5298,a,5160,a)]. 12.28/12.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B))). [resolve(5298,a,5161,a)]. 12.28/12.23 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Orev(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(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),C)),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),B))). [resolve(5298,a,5162,a)]. 12.28/12.24 Derived: c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),B)),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))). [resolve(5298,a,5163,a)]. 12.28/12.24 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),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,B,C)). [resolve(5298,a,5164,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),D),C)) | hAPP(c_List_Ohd(A),hAPP(c_List_Ofilter(A,hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,A),hAPP(c_fequal,hAPP(D,B))),D)),C)) != B | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,D),B),c_List_Oremove1(A,B,C)) = C. [resolve(5298,a,5165,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f348(B,A,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = B. [resolve(5298,a,5166,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f348(B,A,tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = B. [resolve(5298,a,5167,a)]. 12.28/12.24 Derived: c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,hAPP(c_List_Oset(tc_Int_Oint),A)) = c_List_Olinorder__class_Osort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),c_List_Oremdups(tc_Int_Oint,A)). [resolve(5298,a,5168,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),A),c_List_Oremove1(tc_Int_Oint,A,B)) = B. [resolve(5298,a,5169,a)]. 12.28/12.24 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_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),C)). [resolve(5298,a,5170,a)]. 12.28/12.24 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_Orderings_Oord__class_Oless(tc_Int_Oint),f354(B,A,tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),B)). [resolve(5298,a,5171,a)]. 12.28/12.24 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),f354(B,A,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),B)). [resolve(5298,a,5172,a)]. 12.28/12.24 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Int_Oint,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Int_Oint,A,B,D) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(C,tc_Int_Oint,A),B),D). [resolve(5298,a,5174,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(A,B)),hAPP(c_Set_Oimage(C,tc_Int_Oint,A),hAPP(c_List_Oset(C),D)))) | c_List_Olinorder__class_Oinsort__insert__key(C,tc_Int_Oint,A,B,D) = D. [resolve(5298,a,5175,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D))) | hBOOL(hAPP(c_List_Odistinct(A),D)). [resolve(5298,a,5176,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D))) | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5298,a,5177,a)]. 12.28/12.24 Derived: hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D))) | -hBOOL(hAPP(c_List_Odistinct(A),D)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),D))). [resolve(5298,a,5178,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olinorder__class_OMax(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_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A). [resolve(5298,a,5179,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f368(B,A,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = B. [resolve(5298,a,5180,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f368(B,A,tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = B. [resolve(5298,a,5181,a)]. 12.28/12.24 Derived: c_Finite__Set_Ofun__left__comm(tc_Int_Oint,tc_List_Olist(tc_Int_Oint),c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint))). [resolve(5298,a,5182,a)]. 12.28/12.24 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,B,C)). [resolve(5298,a,5183,a)]. 12.28/12.24 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f381(A,tc_Int_Oint)),f382(A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5184,a)]. 12.28/12.24 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f382(A,tc_Int_Oint)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5185,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f381(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f382(A,tc_Int_Oint)))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5186,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B)). [resolve(5298,a,5187,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),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(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5298,a,5188,a)]. 12.28/12.24 Derived: c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,hAPP(c_List_Ofilter(A,C),D)) = hAPP(c_List_Ofilter(A,C),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,D)). [resolve(5298,a,5190,a)]. 12.28/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),B)). [resolve(5298,a,5193,a)]. 12.28/12.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5298,a,5194,a)]. 12.28/12.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5298,a,5195,a)]. 12.28/12.25 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),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(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),B)). [resolve(5298,a,5196,a)]. 12.28/12.25 Derived: c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(c_List_Oset(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B))) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B). [resolve(5298,a,5197,a)]. 12.28/12.25 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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),D))). [resolve(5298,a,5198,a)]. 12.28/12.25 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D). [resolve(5298,a,5201,a)]. 12.28/12.25 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)). [resolve(5298,a,5204,a)]. 12.28/12.25 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(c_List_Ofilter(tc_Int_Oint,hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),hAPP(A,B))),B)). [resolve(5298,a,5205,a)]. 12.28/12.25 Derived: hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),D)). [resolve(5298,a,5206,a)]. 12.28/12.25 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5298,a,5207,a)]. 12.28/12.25 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5298,a,5208,a)]. 12.28/12.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5209,a)]. 12.28/12.25 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5211,a)]. 12.28/12.26 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),c_List_Olinorder__class_Oinsort__insert__key(A,tc_Int_Oint,B,D,C))). [resolve(5298,a,5213,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)). [resolve(5298,a,5214,a)]. 12.28/12.26 Derived: -class_Lattices_Osemilattice__inf(tc_Int_Oint) | c_Orderings_Oord__class_Omin(tc_Int_Oint) = c_Lattices_Osemilattice__inf__class_Oinf(tc_Int_Oint). [resolve(5298,a,5215,b)]. 12.28/12.26 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f429(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),B))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)). [resolve(5298,a,5216,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f429(A,B,tc_Int_Oint))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)). [resolve(5298,a,5217,a)]. 12.28/12.26 Derived: hAPP(c_List_Oset(A),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,C)) = hAPP(c_List_Oset(A),C). [resolve(5298,a,5218,a)]. 12.28/12.26 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Odrop(tc_Int_Oint),B),A)). [resolve(5298,a,5219,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5298,a,5223,a)]. 12.28/12.26 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),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),A)). [resolve(5298,a,5224,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)) = B. [resolve(5298,a,5225,a)]. 12.28/12.26 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),D),hAPP(E,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Int_Oint),C),E))). [resolve(5298,a,5226,b)]. 12.28/12.26 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f454(B,C,A,D,tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),D),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_Int_Oint),C),B))). [resolve(5298,a,5227,b)]. 12.28/12.26 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(B,f454(B,C,D,A,tc_Int_Oint)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(D,tc_Int_Oint),C),B))). [resolve(5298,a,5228,b)]. 12.28/12.26 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_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A))). [resolve(5298,a,5229,a)]. 12.28/12.26 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_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A))). [resolve(5298,a,5230,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A)),B)). [resolve(5298,a,5231,a)]. 12.28/12.26 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B). [resolve(5298,a,5232,a)]. 12.28/12.26 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B). [resolve(5298,a,5233,a)]. 12.28/12.26 Derived: -c_Fun_Oinj__on(A,tc_Int_Oint,B,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),hAPP(c_List_Oset(A),C)),hAPP(c_List_Oset(A),D))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)) | -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D))) | hAPP(c_List_Oset(A),D) != hAPP(c_List_Oset(A),C) | D = C. [resolve(5298,a,5234,a)]. 12.28/12.26 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_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__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_Orderings_Oord__class_Omax(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_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)). [resolve(5298,a,5236,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,B)),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A))). [resolve(5298,a,5237,a)]. 12.28/12.26 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),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),A)). [resolve(5298,a,5238,a)]. 12.28/12.26 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),c_List_Oremove1(A,D,C))). [resolve(5298,a,5239,a)]. 12.28/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5298,a,5241,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5298,a,5242,a)]. 12.28/12.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C)) | -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),A),C)). [resolve(5298,a,5243,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(c_List_Oset(tc_Int_Oint),f474(A,tc_Int_Oint)) = A. [resolve(5298,a,5244,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),f474(A,tc_Int_Oint))). [resolve(5298,a,5245,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hAPP(c_List_Oset(tc_Int_Oint),B) != A | B = f474(A,tc_Int_Oint). [resolve(5298,a,5246,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,f474(A,tc_Int_Oint)). [resolve(5298,a,5247,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),C)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),C))). [resolve(5298,a,5248,a)]. 12.28/12.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),C))). [resolve(5298,a,5249,a)]. 12.28/12.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)). [resolve(5298,a,5251,a)]. 12.28/12.27 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),B),C)) | c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,B),C)). [resolve(5298,a,5252,a)]. 12.28/12.27 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A)). [resolve(5298,a,5253,a)]. 12.28/12.27 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_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__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_Orderings_Oord__class_Omin(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_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5298,a,5254,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)). [resolve(5298,a,5255,a)]. 12.28/12.27 Derived: c_Orderings_Oord_Omax(tc_Int_Oint,hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Int_Oint))) = c_Orderings_Oord__class_Omin(tc_Int_Oint). [resolve(5298,a,5256,a)]. 12.28/12.27 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(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5298,a,5257,a)]. 12.28/12.27 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),B),C)) | c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,C) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,B,C)). [resolve(5298,a,5259,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5260,a)]. 12.28/12.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5261,a)]. 12.28/12.27 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)) = B. [resolve(5298,a,5262,a)]. 12.28/12.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B). [resolve(5298,a,5263,a)]. 12.28/12.27 Derived: c_List_Olist_ONil(tc_Int_Oint) != A | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5267,a)]. 12.28/12.27 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f513(C,tc_Int_Oint,B,A)),hAPP(c_List_Oset(tc_Int_Oint),B))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,C). [resolve(5298,a,5268,a)]. 12.28/12.27 Derived: hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B) != C | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f513(C,tc_Int_Oint,B,A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,C). [resolve(5298,a,5269,a)]. 12.28/12.27 Derived: c_List_Olist_ONil(tc_Int_Oint) = A | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),f515(A,tc_Int_Oint)),f514(A,tc_Int_Oint)) = A | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5270,a)]. 12.28/12.27 Derived: c_List_Olist_ONil(tc_Int_Oint) = A | c_List_Olinorder__class_Osorted(tc_Int_Oint,f514(A,tc_Int_Oint)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5271,a)]. 12.28/12.28 Derived: c_List_Olist_ONil(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),hAPP(c_List_Oset(tc_Int_Oint),f514(A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f515(A,tc_Int_Oint)),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5298,a,5272,a)]. 12.28/12.28 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_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)). [resolve(5298,a,5274,a)]. 12.28/12.28 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5298,a,5276,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5298,a,5278,a)]. 12.28/12.28 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_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__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_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),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_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),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_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),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_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Int_Oint),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_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool))),tc_Int_Oint),c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint)),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))),c_Orderings_Oord__class_Omax(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_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(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_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),B))))))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),B)). [resolve(5298,a,5280,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A) = hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),A). [resolve(5298,a,5281,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f531(B,C,A,tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),B)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),B),A))). [resolve(5298,a,5282,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f531(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)),hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),C),A))). [resolve(5298,a,5283,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,B)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))). [resolve(5298,a,5285,a)]. 12.28/12.28 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_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A))). [resolve(5298,a,5287,a)]. 12.28/12.28 Derived: c_List_Olinorder__class_Osort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),A) = hAPP(c_List_Ofoldl(tc_List_Olist(tc_Int_Oint),tc_Int_Oint,hAPP(c_COMBC(tc_Int_Oint,tc_List_Olist(tc_Int_Oint),tc_List_Olist(tc_Int_Oint)),c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint))),c_List_Olist_ONil(tc_Int_Oint)),A). [resolve(5298,a,5288,a)]. 12.28/12.28 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),D),C))). [resolve(5298,a,5289,a)]. 12.28/12.28 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),C)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),D),C))). [resolve(5298,a,5290,a)]. 12.28/12.28 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A)). [resolve(5298,a,5292,a)]. 12.28/12.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint)),B),c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))). [resolve(5298,a,5296,a)]. 12.47/12.39 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_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olattice_OInf__fin(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint),A)). [resolve(5298,a,5297,a)]. 12.47/12.39 5299 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,c_List_Olinorder__class_Oinsort__insert__key(A,A,c_COMBI(A),C,B)) # label(fact_sorted__insort__insert) # label(axiom). [clausify(4705)]. 12.47/12.39 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olinorder__class_Oinsort__insert__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),B,A)). [resolve(5299,a,4908,a)]. 12.47/12.39 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olinorder__class_Oinsort__insert__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),B,A)). [resolve(5299,a,5154,a)]. 12.47/12.39 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olinorder__class_Oinsort__insert__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),B,A)). [resolve(5299,a,5298,a)]. 12.47/12.39 5300 -class_Orderings_Olinorder(A) | hAPP(c_List_Oset(A),c_List_Olinorder__class_Oinsort__insert__key(A,A,c_COMBI(A),B,C)) = hAPP(hAPP(c_Set_Oinsert(A),B),hAPP(c_List_Oset(A),C)) # label(fact_set__insort__insert) # label(axiom). [clausify(4718)]. 12.47/12.39 Derived: hAPP(c_List_Oset(tc_Nat_Onat),c_List_Olinorder__class_Oinsort__insert__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),A,B)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B)). [resolve(5300,a,4908,a)]. 12.47/12.39 Derived: hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),c_List_Olinorder__class_Oinsort__insert__key(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_COMBI(tc_Code__Numeral_Ocode__numeral),A,B)) = hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5300,a,5154,a)]. 12.47/12.39 Derived: hAPP(c_List_Oset(tc_Int_Oint),c_List_Olinorder__class_Oinsort__insert__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),A,B)) = hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B)). [resolve(5300,a,5298,a)]. 12.47/12.39 5301 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) # label(fact_sorted__many) # label(axiom). [clausify(4728)]. 12.47/12.39 5302 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),D) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C)) # label(fact_ivl__diff) # label(axiom). [clausify(4801)]. 12.47/12.39 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)). [resolve(5302,a,4908,a)]. 12.49/12.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,B),C) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,A),B)). [resolve(5302,a,5154,a)]. 12.49/12.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,B),C) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),C)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B)). [resolve(5302,a,5298,a)]. 12.49/12.44 5303 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__supE) # label(axiom). [clausify(4837)]. 12.49/12.44 5304 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__max_Ole__supE) # label(axiom). [clausify(4837)]. 12.49/12.44 5305 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatMost(A),B) != hAPP(c_SetInterval_Oord__class_OatMost(A),C) | B = C # label(fact_atMost__eq__iff) # label(axiom). [clausify(4844)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A = B. [resolve(5305,a,4908,a)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),B) | A = B. [resolve(5305,a,5154,a)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A = B. [resolve(5305,a,5298,a)]. 12.49/12.44 5306 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatMost(A),B) = hAPP(c_SetInterval_Oord__class_OatMost(A),C) | B != C # label(fact_atMost__eq__iff) # label(axiom). [clausify(4844)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A != B. [resolve(5306,a,4908,a)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),B) | A != B. [resolve(5306,a,5154,a)]. 12.49/12.44 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A != B. [resolve(5306,a,5298,a)]. 12.49/12.44 5307 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Big__Operators_Olinorder__class_OMin(A,B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),c_Big__Operators_Olinorder__class_OMin(A,C)),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_Min_Osubset__idem) # label(axiom). [clausify(4845)]. 12.49/12.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,B)),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5307,a,4908,a)]. 12.49/12.44 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)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,B)),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5307,a,5154,a)]. 12.49/12.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Int_Oint,tc_HOL_Obool)),B),A)) | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,B)),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5307,a,5298,a)]. 12.49/12.47 5308 -class_Orderings_Olinorder(A) | c_SetInterval_Oord__class_OgreaterThanLessThan(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),E)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,B,D)),c_SetInterval_Oord__class_OgreaterThanLessThan(A,C,E)) # label(fact_Int__greaterThanLessThan) # label(axiom). [clausify(4855)]. 12.49/12.47 Derived: c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,A,C)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,B,D)). [resolve(5308,a,4908,a)]. 12.49/12.47 Derived: c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,A,C)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Code__Numeral_Ocode__numeral,B,D)). [resolve(5308,a,5154,a)]. 12.49/12.47 Derived: c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),D)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,A,C)),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,B,D)). [resolve(5308,a,5298,a)]. 12.49/12.47 5309 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C)) # label(fact_min__max_Ole__supI) # label(axiom). [clausify(4856)]. 12.49/12.47 5310 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),c_Big__Operators_Olattice_OSup__fin(A,c_Orderings_Oord__class_Omax(A),B)) # label(fact_min__max_OSup__fin_Oinsert__idem) # label(axiom). [clausify(4869)]. 12.49/12.47 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_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(5310,a,4908,a)]. 12.49/12.47 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_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olattice_OSup__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5310,a,5154,a)]. 13.88/13.88 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_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5310,a,5298,a)]. 13.88/13.88 5311 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_Min_Oidem) # label(axiom). [clausify(4886)]. 13.88/13.88 5312 -class_Rings_Osemiring__0(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(A,A,B),hAPP(c_Groups_Otimes__class_Otimes(A),C)),D)),E) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)) # label(fact_setsum__right__distrib) # label(axiom). [clausify(1515)]. 13.88/13.88 5313 class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring__0) # label(axiom). [assumption]. 13.88/13.88 5314 class_Rings_Osemiring__0(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__0) # label(axiom). [assumption]. 13.88/13.88 5315 class_Rings_Osemiring__0(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__0) # label(axiom). [assumption]. 13.88/13.88 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)). [resolve(5312,a,5313,a)]. 13.88/13.88 Derived: 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),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)). [resolve(5312,a,5314,a)]. 13.88/13.88 Derived: 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),hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)). [resolve(5312,a,5315,a)]. 13.88/13.88 5316 -class_Rings_Osemiring__0(A) | -class_Power_Opower(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(2054)]. 13.88/13.88 Derived: -class_Power_Opower(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(5316,a,5313,a)]. 13.88/13.88 Derived: -class_Power_Opower(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(5316,a,5314,a)]. 13.88/13.88 5317 -class_Rings_Osemiring__0(A) | -class_Power_Opower(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(2054)]. 14.58/14.51 Derived: -class_Power_Opower(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(5317,a,5313,a)]. 14.58/14.51 Derived: -class_Power_Opower(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(5317,a,5314,a)]. 14.58/14.51 Derived: -class_Power_Opower(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(5317,a,5315,a)]. 14.58/14.51 5318 -class_Rings_Osemiring__0(A) | -class_Power_Opower(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(2752)]. 14.58/14.51 Derived: -class_Power_Opower(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(5318,a,5313,a)]. 14.58/14.51 Derived: -class_Power_Opower(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(5318,a,5314,a)]. 14.58/14.51 Derived: -class_Power_Opower(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(5318,a,5315,a)]. 14.58/14.51 5319 -class_Rings_Osemiring__0(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(hAPP(c_COMBB(A,A,B),hAPP(c_Groups_Otimes__class_Otimes(A),C)),D)),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))) # label(fact_listsum__const__mult) # label(axiom). [clausify(2829)]. 14.58/14.51 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_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,A),hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),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),D))). [resolve(5319,a,5313,a)]. 14.58/14.51 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))). [resolve(5319,a,5314,a)]. 14.58/14.51 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,A),hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))). [resolve(5319,a,5315,a)]. 14.58/14.51 5320 -class_Rings_Osemiring__0(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),C)),D)),E)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E))),D) # label(fact_listsum__mult__const) # label(axiom). [clausify(3630)]. 14.58/14.51 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_COMBC(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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))),C). [resolve(5320,a,5313,a)]. 15.08/15.00 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(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_Otimes__class_Otimes(tc_Int_Oint)),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D))),C). [resolve(5320,a,5314,a)]. 15.08/15.00 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(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_Otimes__class_Otimes(tc_Nat_Onat)),B)),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D))),C). [resolve(5320,a,5315,a)]. 15.08/15.00 5321 -class_Rings_Osemiring__0(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),C)),D)),E) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),E)),D) # label(fact_setsum__left__distrib) # label(axiom). [clausify(4112)]. 15.08/15.00 Derived: 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_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),D)),C). [resolve(5321,a,5313,a)]. 15.08/15.00 Derived: 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_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),D)),C). [resolve(5321,a,5314,a)]. 15.08/15.00 Derived: 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_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),D)),C). [resolve(5321,a,5315,a)]. 15.08/15.00 5322 -class_Rings_Osemiring__0(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(hAPP(c_COMBC(B,tc_fun(C,A),tc_fun(C,A)),hAPP(hAPP(c_COMBB(tc_fun(A,A),tc_fun(tc_fun(C,A),tc_fun(C,A)),B),c_COMBB(A,A,C)),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),D))),E))),F)),V6) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),V6)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),E),F)) # label(fact_setsum__product) # label(axiom). [clausify(4328)]. 15.08/15.00 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(hAPP(c_COMBC(A,tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),A),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,B)),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C))),D))),E)),F) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),F)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),D),E)). [resolve(5322,a,5313,a)]. 15.09/15.03 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(hAPP(c_COMBC(A,tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),A),c_COMBB(tc_Int_Oint,tc_Int_Oint,B)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C))),D))),E)),F) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),F)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),D),E)). [resolve(5322,a,5314,a)]. 15.09/15.03 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(hAPP(c_COMBC(A,tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),A),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,B)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C))),D))),E)),F) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),F)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),D),E)). [resolve(5322,a,5315,a)]. 15.09/15.03 5323 -class_Rings_Osemiring__0(A) | -c_Fun_Oinj__on(tc_prod(B,C),A,hAPP(c_Product__Type_Oprod_Oprod__case(B,C,A),hAPP(hAPP(c_COMBC(B,tc_fun(C,A),tc_fun(C,A)),hAPP(hAPP(c_COMBB(tc_fun(A,A),tc_fun(tc_fun(C,A),tc_fun(C,A)),B),c_COMBB(A,A,C)),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),D))),E)),hAPP(hAPP(c_Product__Type_OSigma(B,C),F),hAPP(c_COMBK(tc_fun(C,tc_HOL_Obool),B),V6))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_Fun_Oid(A)),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_fun(C,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool),A),hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_HOL_Obool,B),c_HOL_OEx(C))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_fun(C,tc_HOL_Obool)),tc_fun(B,tc_fun(C,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_fun(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool))),tc_fun(tc_fun(B,tc_fun(C,tc_HOL_Obool)),tc_fun(B,tc_fun(C,tc_HOL_Obool))),A),c_COMBS(B,tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_fun(C,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(B,tc_fun(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool))),A),hAPP(c_COMBB(tc_fun(C,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool)),B),c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_fun(C,tc_HOL_Obool)),tc_fun(B,tc_fun(C,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),A),hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),B),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_fun(C,A)),tc_fun(B,tc_fun(C,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(C,A),tc_fun(C,tc_HOL_Obool)),tc_fun(tc_fun(B,tc_fun(C,A)),tc_fun(B,tc_fun(C,tc_HOL_Obool))),A),c_COMBB(tc_fun(C,A),tc_fun(C,tc_HOL_Obool),B)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(C,A),tc_fun(C,tc_HOL_Obool)),A),c_COMBB(A,tc_HOL_Obool,C)),c_fequal))),hAPP(hAPP(c_COMBC(B,tc_fun(C,A),tc_fun(C,A)),hAPP(hAPP(c_COMBB(tc_fun(A,A),tc_fun(tc_fun(C,A),tc_fun(C,A)),B),c_COMBB(A,A,C)),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),D))),E)))))),hAPP(hAPP(c_COMBC(B,tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool)),B),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,C)),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)),F)))),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),V6))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),F)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),E),V6)) # label(fact_setsum__mult__setsum__if__inj) # label(axiom). [clausify(4377)]. 15.09/15.03 Derived: -c_Fun_Oinj__on(tc_prod(A,B),tc_Code__Numeral_Ocode__numeral,hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),A),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,B)),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C))),D)),hAPP(hAPP(c_Product__Type_OSigma(A,B),E),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),F))) | 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_OCollect(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(B))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_fun(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBS(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(A,tc_fun(B,tc_Code__Numeral_Ocode__numeral)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_fun(B,tc_Code__Numeral_Ocode__numeral)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_HOL_Obool),A)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,B)),c_fequal))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(B,tc_Code__Numeral_Ocode__numeral)),A),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,B)),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C))),D)))))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,B)),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)),E)))),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),F))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),D),F)). [resolve(5323,a,5313,a)]. 15.09/15.03 Derived: -c_Fun_Oinj__on(tc_prod(A,B),tc_Int_Oint,hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),A),c_COMBB(tc_Int_Oint,tc_Int_Oint,B)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C))),D)),hAPP(hAPP(c_Product__Type_OSigma(A,B),E),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),F))) | 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_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_Int_Oint),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(B))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_fun(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Int_Oint),c_COMBS(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_Int_Oint),hAPP(c_COMBB(tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Int_Oint),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,tc_fun(B,tc_Int_Oint)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_fun(B,tc_Int_Oint)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Int_Oint),c_COMBB(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_HOL_Obool),A)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_HOL_Obool,B)),c_fequal))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_fun(B,tc_Int_Oint),tc_fun(B,tc_Int_Oint)),A),c_COMBB(tc_Int_Oint,tc_Int_Oint,B)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C))),D)))))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,B)),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)),E)))),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),F))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),D),F)). [resolve(5323,a,5314,a)]. 15.09/15.03 Derived: -c_Fun_Oinj__on(tc_prod(A,B),tc_Nat_Onat,hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),A),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,B)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C))),D)),hAPP(hAPP(c_Product__Type_OSigma(A,B),E),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),F))) | 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_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(B))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_fun(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Nat_Onat),c_COMBS(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(A,tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool))),tc_Nat_Onat),hAPP(c_COMBB(tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(B,tc_HOL_Obool)),tc_fun(A,tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_Nat_Onat),hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(A,tc_fun(B,tc_Nat_Onat)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_fun(B,tc_Nat_Onat)),tc_fun(A,tc_fun(B,tc_HOL_Obool))),tc_Nat_Onat),c_COMBB(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_HOL_Obool),A)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_HOL_Obool)),tc_Nat_Onat),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,B)),c_fequal))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(B,tc_Nat_Onat),tc_fun(B,tc_Nat_Onat)),A),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,B)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C))),D)))))),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),A),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,B)),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)),E)))),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),F))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),D),F)). [resolve(5323,a,5315,a)]. 15.39/15.36 5324 -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,f574(C,B,A))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(4866)]. 15.39/15.36 Derived: -class_Rings_Odvd(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,f574(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5324,a,5313,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(tc_Int_Oint) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))) | hBOOL(hAPP(A,f574(B,A,tc_Int_Oint))). [resolve(5324,a,5314,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(tc_Nat_Onat) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))) | hBOOL(hAPP(A,f574(B,A,tc_Nat_Onat))). [resolve(5324,a,5315,a)]. 15.39/15.36 5325 -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),f574(C,B,A)),c_Groups_Ozero__class_Ozero(A)))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(4866)]. 15.39/15.36 Derived: -class_Rings_Odvd(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),f574(B,A,tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))). [resolve(5325,a,5313,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(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),f574(B,A,tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))). [resolve(5325,a,5314,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(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),f574(B,A,tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))). [resolve(5325,a,5315,a)]. 15.39/15.36 5326 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),f575(C,B,A)))) | -hBOOL(hAPP(B,D)) | -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)))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(4866)]. 15.39/15.36 Derived: -class_Rings_Odvd(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),f575(B,A,tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(A,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),C),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))). [resolve(5326,a,5313,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(tc_Int_Oint) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),f575(B,A,tc_Int_Oint)))) | -hBOOL(hAPP(A,C)) | -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)))). [resolve(5326,a,5314,a)]. 15.39/15.36 Derived: -class_Rings_Odvd(tc_Nat_Onat) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),f575(B,A,tc_Nat_Onat)))) | -hBOOL(hAPP(A,C)) | -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)))). [resolve(5326,a,5315,a)]. 23.08/23.02 5327 class_Int_Onumber__ring(tc_Int_Oint) # label(arity_Int__Oint__Int_Onumber__ring) # label(axiom). [assumption]. 23.08/23.02 5328 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(11)]. 23.08/23.02 5329 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(11)]. 23.08/23.02 5330 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)),B) = B # label(fact_add__numeral__0) # label(axiom). [clausify(25)]. 23.08/23.02 5331 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) # label(fact_add__number__of__left) # label(axiom). [clausify(31)]. 23.08/23.02 5332 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) # label(fact_number__of__Min) # label(axiom). [clausify(38)]. 23.08/23.02 5333 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))),B) = B # label(fact_mult__numeral__1) # label(axiom). [clausify(114)]. 23.08/23.02 5334 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) # label(fact_of__int__m1) # label(axiom). [clausify(173)]. 23.08/23.02 5335 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) # label(fact_divide__Numeral0) # label(axiom). [clausify(211)]. 23.08/23.02 5336 -class_Int_Onumber__ring(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),B))) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd) # label(axiom). [clausify(277)]. 23.08/23.02 5337 -class_Int_Onumber__ring(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),B))) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),hAPP(hAPP(c_COMBS(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Groups_Otimes__class_Otimes(A),C))),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd) # label(axiom). [clausify(277)]. 23.08/23.02 5338 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))) # label(fact_not__iszero__Numeral1) # label(axiom). [clausify(361)]. 23.08/23.02 5339 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) # label(fact_mult__number__of__left) # label(axiom). [clausify(423)]. 23.08/23.02 5340 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_divide__minus1) # label(axiom). [clausify(481)]. 23.08/23.02 5341 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_diff__number__of__eq) # label(axiom). [clausify(506)]. 23.08/23.02 5342 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) # label(fact_arith__simps_I31_J) # label(axiom). [clausify(533)]. 23.08/23.02 5343 -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(559)]. 23.08/23.02 5344 -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(559)]. 23.08/23.02 5345 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(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))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_double__number__of__Bit0) # label(axiom). [clausify(787)]. 23.08/23.02 5346 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Otimes__class_Otimes(A)),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(C))) # label(fact_power__number__of__even__number__of) # label(axiom). [clausify(1067)]. 23.08/23.02 5347 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) = B # label(fact_add__numeral__0__right) # label(axiom). [clausify(1283)]. 23.08/23.02 5348 -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(1351)]. 23.08/23.02 5349 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(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)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)),C)) # label(fact_power2__diff) # label(axiom). [clausify(1486)]. 23.08/23.02 5350 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber__ring(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_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_half__gt__zero__iff) # label(axiom). [clausify(1504)]. 23.08/23.02 5351 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber__ring(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_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_half__gt__zero__iff) # label(axiom). [clausify(1504)]. 23.08/23.02 5352 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(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)))),B))) # label(fact_power__m1__odd) # label(axiom). [clausify(1509)]. 23.08/23.02 5353 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) # label(fact_eq__special_I3_J) # label(axiom). [clausify(1551)]. 23.08/23.02 5354 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) # label(fact_eq__special_I3_J) # label(axiom). [clausify(1551)]. 23.08/23.02 5355 -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(1613)]. 23.08/23.02 5356 -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(1613)]. 23.08/23.02 5357 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | 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))) # label(fact_le__special_I4_J) # label(axiom). [clausify(1644)]. 23.08/23.02 5358 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | -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))) # label(fact_le__special_I4_J) # label(axiom). [clausify(1644)]. 23.08/23.02 5359 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_number__of__eq) # label(axiom). [clausify(1710)]. 23.08/23.02 5360 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_arith__simps_I32_J) # label(axiom). [clausify(1949)]. 23.08/23.02 5361 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__Bit1) # label(axiom). [clausify(1958)]. 23.08/23.02 5362 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls) # label(fact_number__of__Pls) # label(axiom). [clausify(2025)]. 23.08/23.04 5363 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) | B = C # label(fact_eq__number__of) # label(axiom). [clausify(2167)]. 23.08/23.04 5364 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | B != C # label(fact_eq__number__of) # label(axiom). [clausify(2167)]. 23.08/23.04 5365 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_add__number__of__eq) # label(axiom). [clausify(2174)]. 23.08/23.04 5366 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__diff) # label(axiom). [clausify(2177)]. 23.08/23.04 Derived: -class_Rings_Olinordered__idom(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(5327,a,5328,a)]. 23.08/23.04 Derived: -class_Rings_Olinordered__idom(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(5327,a,5329,a)]. 23.08/23.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)),A) = A. [resolve(5327,a,5330,a)]. 23.08/23.04 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)). [resolve(5327,a,5331,a)]. 23.08/23.04 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin). [resolve(5327,a,5332,a)]. 23.08/23.04 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))),A) = A. [resolve(5327,a,5333,a)]. 23.08/23.04 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),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). [resolve(5327,a,5334,a)]. 23.08/23.04 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)). [resolve(5327,a,5335,b)]. 23.08/23.04 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))). [resolve(5327,a,5338,a)]. 23.08/23.04 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)). [resolve(5327,a,5339,a)]. 23.08/23.04 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5327,a,5340,b)]. 23.08/23.06 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(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))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5327,a,5345,a)]. 23.08/23.06 Derived: hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),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))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(B))). [resolve(5327,a,5346,a)]. 23.08/23.06 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)) = A. [resolve(5327,a,5347,a)]. 23.08/23.06 Derived: -class_Rings_Olinordered__idom(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(5327,a,5348,b)]. 23.08/23.06 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),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_Ominus__class_Ominus(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)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)))),A)),B)). [resolve(5327,a,5349,a)]. 23.08/23.06 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))). [resolve(5327,a,5350,b)]. 23.08/23.06 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))). [resolve(5327,a,5351,b)]. 23.08/23.06 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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)))),A))). [resolve(5327,a,5352,a)]. 23.08/23.06 Derived: c_Groups_Ozero__class_Ozero(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))). [resolve(5327,a,5353,a)]. 23.08/23.06 Derived: c_Groups_Ozero__class_Ozero(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))). [resolve(5327,a,5354,a)]. 23.19/23.12 Derived: -class_Rings_Olinordered__idom(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(5327,a,5355,a)]. 23.19/23.12 Derived: -class_Rings_Olinordered__idom(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(5327,a,5356,a)]. 23.19/23.12 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | 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))). [resolve(5327,a,5357,a)]. 23.19/23.12 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | -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))). [resolve(5327,a,5358,a)]. 23.19/23.12 Derived: hAPP(c_Int_Onumber__class_Onumber__of(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_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5327,a,5360,a)]. 23.19/23.12 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(A)) = 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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5327,a,5361,a)]. 23.19/23.12 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls). [resolve(5327,a,5362,a)]. 23.19/23.12 Derived: -class_Int_Oring__char__0(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) | A = B. [resolve(5327,a,5363,a)]. 23.19/23.12 Derived: -class_Int_Oring__char__0(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) | A != B. [resolve(5327,a,5364,a)]. 23.19/23.12 Derived: hAPP(c_Int_Onumber__class_Onumber__of(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_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5327,a,5365,a)]. 23.19/23.12 Derived: hAPP(c_Int_Onumber__class_Onumber__of(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_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5327,a,5366,a)]. 23.19/23.12 5367 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A)) # label(fact_diff__special_I2_J) # label(axiom). [clausify(2359)]. 23.19/23.12 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5367,a,5327,a)]. 23.19/23.12 5368 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))),D) # label(fact_add__number__of__diff1) # label(axiom). [clausify(2409)]. 23.28/23.21 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))),C). [resolve(5368,a,5327,a)]. 23.28/23.21 5369 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) # label(fact_iszero__Numeral0) # label(axiom). [clausify(2416)]. 23.28/23.21 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)). [resolve(5369,a,5327,a)]. 23.28/23.21 5370 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__number__of__eq) # label(axiom). [clausify(2474)]. 23.28/23.21 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5370,a,5327,a)]. 23.28/23.21 5371 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__number__of__eq) # label(axiom). [clausify(2474)]. 23.28/23.21 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5371,a,5327,a)]. 23.28/23.21 5372 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_add__number__of__diff2) # label(axiom). [clausify(2491)]. 23.28/23.21 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5372,a,5327,a)]. 23.28/23.21 5373 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) # label(fact_number__of__minus) # label(axiom). [clausify(2613)]. 23.28/23.21 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5373,a,5327,a)]. 23.28/23.21 5374 -class_Int_Onumber__ring(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),B))) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),hAPP(hAPP(c_COMBS(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd__number__of) # label(axiom). [clausify(2615)]. 23.37/23.34 5375 -class_Int_Onumber__ring(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),B))) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd__number__of) # label(axiom). [clausify(2615)]. 23.37/23.34 5376 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))),C) # label(fact_minus__number__of__mult) # label(axiom). [clausify(2637)]. 23.37/23.34 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))),B). [resolve(5376,a,5327,a)]. 23.37/23.34 5377 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Otimes__class_Otimes(A)),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(C))) # label(fact_power__number__of__even) # label(axiom). [clausify(2688)]. 23.37/23.34 Derived: hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),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))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(B))). [resolve(5377,a,5327,a)]. 23.37/23.34 5378 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls)) # label(fact_semiring__norm_I110_J) # label(axiom). [clausify(2743)]. 23.37/23.34 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)). [resolve(5378,a,5327,a)]. 23.37/23.34 5379 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_Osucc(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__succ) # label(axiom). [clausify(2820)]. 23.37/23.34 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Osucc(A)) = 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),A)). [resolve(5379,a,5327,a)]. 23.37/23.34 5380 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_mult__2) # label(axiom). [clausify(2875)]. 23.37/23.34 Derived: 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),A),A). [resolve(5380,a,5327,a)]. 23.49/23.42 5381 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(2879)]. 23.49/23.42 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | 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))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5381,b,5327,a)]. 23.49/23.42 5382 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_abs__number__of) # label(axiom). [clausify(2879)]. 23.49/23.42 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -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))) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5382,b,5327,a)]. 23.49/23.42 5383 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(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(2934)]. 23.49/23.42 Derived: -class_Int_Oring__char__0(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(5383,a,5327,a)]. 23.49/23.42 5384 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(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(2934)]. 23.49/23.42 Derived: -class_Int_Oring__char__0(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(5384,a,5327,a)]. 23.49/23.42 5385 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_Opred(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__pred) # label(axiom). [clausify(2959)]. 23.49/23.42 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5385,a,5327,a)]. 23.49/23.42 5386 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__mult) # label(axiom). [clausify(2978)]. 23.49/23.42 5387 -class_Int_Onumber__ring(A) | hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_of__nat__number__of__eq) # label(axiom). [clausify(3036)]. 23.59/23.52 5388 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__eq) # label(axiom). [clausify(3036)]. 23.59/23.52 5389 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) # label(fact_le__special_I2_J) # label(axiom). [clausify(3141)]. 23.59/23.52 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5389,a,5327,a)]. 23.59/23.52 5390 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) # label(fact_le__special_I2_J) # label(axiom). [clausify(3141)]. 23.59/23.52 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5390,a,5327,a)]. 23.59/23.52 5391 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__Bit0) # label(axiom). [clausify(3178)]. 23.59/23.52 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5391,a,5327,a)]. 23.59/23.52 5392 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(3200)]. 23.59/23.52 Derived: -class_Rings_Olinordered__idom(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(5392,a,5327,a)]. 23.59/23.52 5393 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(3200)]. 23.59/23.52 Derived: -class_Rings_Olinordered__idom(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(5393,a,5327,a)]. 23.59/23.52 5394 -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(3233)]. 23.99/23.96 Derived: -class_Rings_Olinordered__idom(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(5394,a,5327,a)]. 23.99/23.96 5395 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_nonzero__number__of__Min) # label(axiom). [clausify(3235)]. 23.99/23.96 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5395,a,5327,a)]. 23.99/23.96 5396 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls)) # label(fact_numeral__1__eq__1) # label(axiom). [clausify(3282)]. 23.99/23.96 5397 -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(3336)]. 23.99/23.96 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(A))). [resolve(5397,a,5327,a)]. 23.99/23.96 5398 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) # label(fact_less__number__of) # label(axiom). [clausify(3502)]. 23.99/23.96 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5398,b,5327,a)]. 23.99/23.96 5399 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) # label(fact_less__number__of) # label(axiom). [clausify(3502)]. 23.99/23.96 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5399,b,5327,a)]. 23.99/23.96 5400 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A)) # label(fact_one__add__one__is__two) # label(axiom). [clausify(3553)]. 23.99/23.96 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = 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(5400,a,5327,a)]. 23.99/23.96 5401 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) # label(fact_arith__simps_I30_J) # label(axiom). [clausify(3598)]. 23.99/23.96 5402 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls) # label(fact_semiring__norm_I112_J) # label(axiom). [clausify(3784)]. 23.99/23.96 5403 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__add) # label(axiom). [clausify(3793)]. 23.99/23.96 5404 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) # label(fact_eq__special_I4_J) # label(axiom). [clausify(3928)]. 24.10/24.07 Derived: c_Groups_Oone__class_Oone(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))). [resolve(5404,a,5327,a)]. 24.10/24.07 5405 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) # label(fact_eq__special_I4_J) # label(axiom). [clausify(3928)]. 24.10/24.07 Derived: c_Groups_Oone__class_Oone(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))). [resolve(5405,a,5327,a)]. 24.10/24.07 5406 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_mult__2__right) # label(axiom). [clausify(4025)]. 24.10/24.07 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A). [resolve(5406,a,5327,a)]. 24.10/24.07 5407 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(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)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)),C)) # label(fact_power2__sum) # label(axiom). [clausify(4050)]. 24.10/24.07 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),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_Oplus__class_Oplus(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)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)))),A)),B)). [resolve(5407,a,5327,a)]. 24.10/24.07 5408 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),c_Groups_Oone__class_Oone(A)),B)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),B) # label(fact_minus1__divide) # label(axiom). [clausify(4059)]. 24.10/24.07 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A). [resolve(5408,b,5327,a)]. 24.10/24.07 5409 -class_Int_Onumber__ring(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),B))) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_of__nat__number__of__lemma) # label(axiom). [clausify(4084)]. 24.40/24.30 Derived: -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))) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5409,a,5327,a)]. 24.40/24.30 5410 -class_Int_Onumber__ring(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),B))) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__lemma) # label(axiom). [clausify(4084)]. 24.40/24.30 Derived: 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))) | 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),A)). [resolve(5410,a,5327,a)]. 24.40/24.30 5411 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_add__special_I2_J) # label(axiom). [clausify(4086)]. 24.40/24.30 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) = 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),A)). [resolve(5411,a,5327,a)]. 24.40/24.30 5412 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(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)))),B)) # label(fact_power__m1__even) # label(axiom). [clausify(4156)]. 24.40/24.30 Derived: c_Groups_Oone__class_Oone(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)),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(5412,a,5327,a)]. 24.40/24.30 5413 -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__number__of) # label(axiom). [clausify(4157)]. 24.40/24.30 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5413,a,5327,a)]. 24.40/24.30 5414 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))) = B # label(fact_mult__numeral__1__right) # label(axiom). [clausify(4242)]. 24.40/24.30 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))) = A. [resolve(5414,a,5327,a)]. 24.40/24.30 5415 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_of__int__number__of__eq) # label(axiom). [clausify(4314)]. 24.40/24.30 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5415,a,5327,a)]. 24.40/24.30 5416 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I1_J) # label(axiom). [clausify(4333)]. 24.59/24.51 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)) | 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))). [resolve(5416,a,5327,a)]. 24.59/24.51 5417 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I1_J) # label(axiom). [clausify(4333)]. 24.59/24.51 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)) | -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))). [resolve(5417,a,5327,a)]. 24.59/24.51 5418 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_mult__minus1__right) # label(axiom). [clausify(4392)]. 24.59/24.51 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5418,a,5327,a)]. 24.59/24.51 5419 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),B) # label(fact_mult__minus1) # label(axiom). [clausify(4497)]. 24.59/24.51 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A). [resolve(5419,a,5327,a)]. 24.59/24.51 5420 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A)) # label(fact_add__special_I3_J) # label(axiom). [clausify(4542)]. 24.59/24.51 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5420,a,5327,a)]. 24.59/24.51 5421 -class_Int_Onumber__ring(A) | -class_Fields_Olinordered__field__inverse__zero(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_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_half__gt__zero) # label(axiom). [clausify(4615)]. 24.59/24.51 5422 -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(4633)]. 24.59/24.51 Derived: -class_Rings_Olinordered__idom(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(5422,a,5327,a)]. 24.59/24.51 5423 -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(4633)]. 24.79/24.74 Derived: -class_Rings_Olinordered__idom(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(5423,a,5327,a)]. 24.79/24.74 5424 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__number__of) # label(axiom). [clausify(4663)]. 24.79/24.74 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5424,a,5327,a)]. 24.79/24.74 5425 -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(4769)]. 24.79/24.74 Derived: -class_Rings_Olinordered__idom(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(5425,b,5327,a)]. 24.79/24.74 5426 -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(4769)]. 24.79/24.74 Derived: -class_Rings_Olinordered__idom(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(5426,b,5327,a)]. 24.79/24.74 5427 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_eq__special_I1_J) # label(axiom). [clausify(4831)]. 24.79/24.74 Derived: c_Groups_Ozero__class_Ozero(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(5427,a,5327,a)]. 24.79/24.74 5428 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_eq__special_I1_J) # label(axiom). [clausify(4831)]. 24.79/24.74 Derived: c_Groups_Ozero__class_Ozero(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(5428,a,5327,a)]. 24.79/24.74 5429 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(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),B))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_diff__special_I1_J) # label(axiom). [clausify(4894)]. 25.99/25.92 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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),A))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5429,a,5327,a)]. 25.99/25.92 5430 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(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),B)))) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I2_J) # label(axiom). [clausify(4902)]. 25.99/25.92 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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),A)))) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5430,a,5327,a)]. 25.99/25.92 5431 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(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),B)))) | c_Groups_Oone__class_Oone(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I2_J) # label(axiom). [clausify(4902)]. 25.99/25.92 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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),A)))) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5431,a,5327,a)]. 25.99/25.92 5432 class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 25.99/25.92 5433 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_mult_Ocomm__neutral) # label(axiom). [clausify(12)]. 25.99/25.92 5434 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)) # label(fact_setprod_Oinsert) # label(axiom). [clausify(51)]. 25.99/25.92 5435 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f7(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod__Un__one) # label(axiom). [clausify(89)]. 25.99/25.92 5436 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Oone__class_Oone(A) != hAPP(E,f7(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod__Un__one) # label(axiom). [clausify(89)]. 25.99/25.92 5437 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_mult__1) # label(axiom). [clausify(137)]. 25.99/25.92 5438 -class_Groups_Ocomm__monoid__mult(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod_Ounion__disjoint) # label(axiom). [clausify(204)]. 25.99/25.92 5439 -class_Groups_Ocomm__monoid__mult(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod_Oinfinite) # label(axiom). [clausify(259)]. 25.99/25.92 5440 -class_Groups_Ocomm__monoid__mult(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hAPP(c_Fun_Ocomp(C,A,B,V6),D) != V7 | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),V7),E) # label(fact_setprod__reindex__cong) # label(axiom). [clausify(280)]. 25.99/25.92 5441 -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_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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),C),B) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom). [clausify(282)]. 25.99/25.92 5442 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)) # label(fact_setprod__insert) # label(axiom). [clausify(287)]. 25.99/25.92 5443 -class_Groups_Ocomm__monoid__mult(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod__Un__disjoint) # label(axiom). [clausify(397)]. 25.99/25.92 5444 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(A))),C) # label(fact_setprod__delta_H) # label(axiom). [clausify(474)]. 25.99/25.92 5445 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(A))),C) = hAPP(E,D) # label(fact_setprod__delta_H) # label(axiom). [clausify(474)]. 25.99/25.92 5446 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),B),C) = 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)) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom). [clausify(639)]. 25.99/25.93 5447 -class_Groups_Ocomm__monoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setprod__empty) # label(axiom). [clausify(684)]. 25.99/25.93 5448 -class_Groups_Ocomm__monoid__mult(A) | -c_Fun_Oinj__on(B,A,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,A),c_Fun_Oid(A)),hAPP(c_Set_Oimage(B,A,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),D) # label(fact_setprod__reindex__id) # label(axiom). [clausify(781)]. 25.99/25.93 5449 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f104(D,E,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),E),B) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(982)]. 25.99/25.93 5450 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f104(D,E,C,B,A)) != hAPP(E,f104(D,E,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),E),B) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom). [clausify(982)]. 25.99/25.93 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5432,a,5433,a)]. 25.99/25.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)). [resolve(5432,a,5434,a)]. 25.99/25.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f7(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5432,a,5435,a)]. 25.99/25.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) != hAPP(D,f7(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5432,a,5436,a)]. 25.99/25.93 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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5432,a,5438,a)]. 25.99/25.95 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5432,a,5439,a)]. 25.99/25.95 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,F),C) != V6 | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),V6),D). [resolve(5432,a,5440,a)]. 25.99/25.95 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),B),A). [resolve(5432,a,5441,a)]. 25.99/25.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Nat_Onat))),B). [resolve(5432,a,5444,a)]. 25.99/25.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(5432,a,5445,a)]. 25.99/25.95 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Nat_Onat),A),B) = 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)). [resolve(5432,a,5446,a)]. 25.99/25.95 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5432,a,5447,a)]. 25.99/25.95 Derived: -c_Fun_Oinj__on(A,tc_Nat_Onat,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(A,tc_Nat_Onat),B),C). [resolve(5432,a,5448,a)]. 25.99/25.95 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | A != B | -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)),f104(C,D,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),D),A). [resolve(5432,a,5449,b)]. 25.99/25.95 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f104(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f104(C,D,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),D),A). [resolve(5432,a,5450,b)]. 25.99/25.95 5451 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),C) # label(fact_setprod_Oeq__fold) # label(axiom). [clausify(1178)]. 26.09/26.00 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5451,a,5432,a)]. 26.09/26.00 5452 -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_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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),C),B) # label(fact_transfer__int__nat__sum__prod_I1_J) # label(axiom). [clausify(1238)]. 26.09/26.00 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),B),A). [resolve(5452,b,5432,a)]. 26.09/26.00 5453 -class_Groups_Ocomm__monoid__mult(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),f135(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__mult__class_Osetprod(B,A),E),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) # label(fact_setprod__mono__one__right) # label(axiom). [clausify(1255)]. 26.09/26.00 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),f135(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__mult__class_Osetprod(A,tc_Nat_Onat),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B). [resolve(5453,a,5432,a)]. 26.09/26.00 5454 -class_Groups_Ocomm__monoid__mult(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_Oone__class_Oone(A) != hAPP(E,f135(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) # label(fact_setprod__mono__one__right) # label(axiom). [clausify(1255)]. 26.09/26.00 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_Oone__class_Oone(tc_Nat_Onat) != hAPP(D,f135(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B). [resolve(5454,a,5432,a)]. 26.09/26.00 5455 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D))),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),D)) # label(fact_fold__image__Un__Int) # label(axiom). [clausify(1395)]. 26.09/26.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Nat_Onat)),C)). [resolve(5455,a,5432,a)]. 26.09/26.02 5456 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f156(D,E,F,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f157(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(F,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(F,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f157(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5456,a,5432,a)]. 26.09/26.02 5457 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f156(D,E,F,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f158(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(F,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(F,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f158(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5457,a,5432,a)]. 26.09/26.02 5458 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f156(D,E,F,C,B,A)),C)) | f158(D,E,F,C,B,A) != f157(D,E,F,C,B,A) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(F,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(F,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Nat_Onat)),B)) | f158(C,D,E,B,A,tc_Nat_Onat) != f157(C,D,E,B,A,tc_Nat_Onat) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5458,a,5432,a)]. 26.09/26.02 5459 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f156(D,E,F,C,B,A)),C)) | c_Orderings_Obot__class_Obot(tc_fun(F,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(F,tc_HOL_Obool)),hAPP(E,f157(D,E,F,C,B,A))),hAPP(E,f158(D,E,F,C,B,A))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(F,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(F,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.02 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Nat_Onat)),B)) | c_Orderings_Obot__class_Obot(tc_fun(E,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(E,tc_HOL_Obool)),hAPP(D,f157(C,D,E,B,A,tc_Nat_Onat))),hAPP(D,f158(C,D,E,B,A,tc_Nat_Onat))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5459,a,5432,a)]. 26.09/26.02 5460 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f156(F,E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_member(B),f157(F,E,D,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(D,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_member(A),f157(E,D,C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5460,a,5432,a)]. 26.09/26.03 5461 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f156(F,E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_member(B),f158(F,E,D,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(D,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_member(A),f158(E,D,C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5461,a,5432,a)]. 26.09/26.03 5462 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f156(F,E,D,C,B,A)))) | f158(F,E,D,C,B,A) != f157(F,E,D,C,B,A) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(D,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Nat_Onat)))) | f158(E,D,C,B,A,tc_Nat_Onat) != f157(E,D,C,B,A,tc_Nat_Onat) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5462,a,5432,a)]. 26.09/26.03 5463 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f156(F,E,D,C,B,A)))) | c_Orderings_Obot__class_Obot(tc_fun(D,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(D,tc_HOL_Obool)),hAPP(E,f157(F,E,D,C,B,A))),hAPP(E,f158(F,E,D,C,B,A))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_fun(D,tc_HOL_Obool)),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),A,B),hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__UN__disjoint) # label(axiom). [clausify(1472)]. 26.09/26.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Nat_Onat)))) | c_Orderings_Obot__class_Obot(tc_fun(C,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(C,tc_HOL_Obool)),hAPP(D,f157(E,D,C,B,A,tc_Nat_Onat))),hAPP(D,f158(E,D,C,B,A,tc_Nat_Onat))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5463,a,5432,a)]. 26.09/26.06 5464 class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 26.09/26.06 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(5464,a,5433,a)]. 26.09/26.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)). [resolve(5464,a,5434,a)]. 26.09/26.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f7(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5464,a,5435,a)]. 26.09/26.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f7(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5464,a,5436,a)]. 26.09/26.06 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(5464,a,5437,a)]. 26.09/26.08 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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5464,a,5438,a)]. 26.09/26.08 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5464,a,5439,a)]. 26.09/26.08 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,F),C) != V6 | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5464,a,5440,a)]. 26.09/26.08 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5464,a,5441,a)]. 26.09/26.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5464,a,5444,a)]. 26.09/26.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(5464,a,5445,a)]. 26.09/26.08 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B) = 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)). [resolve(5464,a,5446,a)]. 26.09/26.08 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5464,a,5447,a)]. 26.09/26.09 Derived: -c_Fun_Oinj__on(A,tc_Code__Numeral_Ocode__numeral,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5464,a,5448,a)]. 26.09/26.09 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | A != B | -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)),f104(C,D,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),A). [resolve(5464,a,5449,b)]. 26.09/26.09 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f104(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f104(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),A). [resolve(5464,a,5450,b)]. 26.09/26.09 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),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)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5451,a)]. 26.09/26.09 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5464,a,5452,b)]. 26.09/26.09 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),f135(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__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5464,a,5453,a)]. 26.09/26.09 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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f135(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5464,a,5454,a)]. 26.09/26.09 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)),D),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),C)). [resolve(5464,a,5455,a)]. 26.19/26.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f157(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5456,a)]. 26.19/26.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f158(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5457,a)]. 26.19/26.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | f158(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) != f157(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5458,a)]. 26.19/26.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Orderings_Obot__class_Obot(tc_fun(E,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(E,tc_HOL_Obool)),hAPP(D,f157(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f158(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5459,a)]. 26.19/26.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_member(A),f157(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5460,a)]. 26.19/26.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_member(A),f158(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5461,a)]. 26.19/26.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | f158(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral) != f157(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral) | 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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5462,a)]. 26.19/26.11 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | c_Orderings_Obot__class_Obot(tc_fun(C,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(C,tc_HOL_Obool)),hAPP(D,f157(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f158(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral))) | 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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = 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(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5464,a,5463,a)]. 26.20/26.12 5465 class_Groups_Ocomm__monoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 26.20/26.12 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)). [resolve(5465,a,5434,a)]. 26.20/26.12 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f7(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5465,a,5435,a)]. 26.20/26.12 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(D,f7(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5465,a,5436,a)]. 26.20/26.12 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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5465,a,5438,a)]. 26.20/26.12 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5465,a,5439,a)]. 26.20/26.12 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,F),C) != V6 | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),V6),D). [resolve(5465,a,5440,a)]. 26.20/26.12 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),B),A). [resolve(5465,a,5441,a)]. 26.20/26.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Int_Oint))),B). [resolve(5465,a,5444,a)]. 26.20/26.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Int_Oint))),B) = hAPP(D,C). [resolve(5465,a,5445,a)]. 26.20/26.14 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Int_Oint),A),B) = 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)). [resolve(5465,a,5446,a)]. 26.20/26.14 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(5465,a,5447,a)]. 26.20/26.14 Derived: -c_Fun_Oinj__on(A,tc_Int_Oint,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(A,tc_Int_Oint),B),C). [resolve(5465,a,5448,a)]. 26.20/26.14 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | A != B | -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)),f104(C,D,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),D),A). [resolve(5465,a,5449,b)]. 26.20/26.14 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f104(C,D,B,A,tc_Int_Oint)) != hAPP(D,f104(C,D,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),D),A). [resolve(5465,a,5450,b)]. 26.20/26.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5451,a)]. 26.20/26.14 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),B),A). [resolve(5465,a,5452,b)]. 26.20/26.14 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),f135(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__mult__class_Osetprod(A,tc_Int_Oint),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B). [resolve(5465,a,5453,a)]. 26.20/26.15 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_Oone__class_Oone(tc_Int_Oint) != hAPP(D,f135(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B). [resolve(5465,a,5454,a)]. 26.20/26.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Int_Oint)),C)). [resolve(5465,a,5455,a)]. 26.20/26.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f157(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5456,a)]. 26.20/26.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f158(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5457,a)]. 26.20/26.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Int_Oint)),B)) | f158(C,D,E,B,A,tc_Int_Oint) != f157(C,D,E,B,A,tc_Int_Oint) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5458,a)]. 26.20/26.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f156(C,D,E,B,A,tc_Int_Oint)),B)) | c_Orderings_Obot__class_Obot(tc_fun(E,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(E,tc_HOL_Obool)),hAPP(D,f157(C,D,E,B,A,tc_Int_Oint))),hAPP(D,f158(C,D,E,B,A,tc_Int_Oint))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(E,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5459,a)]. 26.20/26.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_member(A),f157(E,D,C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5460,a)]. 26.20/26.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_member(A),f158(E,D,C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5461,a)]. 26.20/26.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Int_Oint)))) | f158(E,D,C,B,A,tc_Int_Oint) != f157(E,D,C,B,A,tc_Int_Oint) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5462,a)]. 26.20/26.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f156(E,D,C,B,A,tc_Int_Oint)))) | c_Orderings_Obot__class_Obot(tc_fun(C,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(C,tc_HOL_Obool)),hAPP(D,f157(E,D,C,B,A,tc_Int_Oint))),hAPP(D,f158(E,D,C,B,A,tc_Int_Oint))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(A,tc_fun(C,tc_HOL_Obool)),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_Int_Oint,A),hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5465,a,5463,a)]. 26.30/26.25 5466 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_setprod_Oremove) # label(axiom). [clausify(1858)]. 26.30/26.25 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5466,a,5432,a)]. 26.30/26.25 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5466,a,5464,a)]. 26.30/26.25 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5466,a,5465,a)]. 26.30/26.25 5467 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),C)),D)),E) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),E)) # label(fact_setprod__timesf) # label(axiom). [clausify(1919)]. 26.30/26.25 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),D)). [resolve(5467,a,5432,a)]. 26.30/26.25 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),D)). [resolve(5467,a,5464,a)]. 26.30/26.25 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint)),B)),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),D)). [resolve(5467,a,5465,a)]. 26.39/26.32 5468 -class_Rings_Oring__1(A) | -class_Groups_Ocomm__monoid__mult(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(2078)]. 26.39/26.32 Derived: -class_Rings_Oring__1(tc_Nat_Onat) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Nat_Onat),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_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,A),c_Int_Oring__1__class_Oof__int(tc_Nat_Onat)),B)),C). [resolve(5468,b,5432,a)]. 26.39/26.32 Derived: -class_Rings_Oring__1(tc_Code__Numeral_Ocode__numeral) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Code__Numeral_Ocode__numeral),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_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral,A),c_Int_Oring__1__class_Oof__int(tc_Code__Numeral_Ocode__numeral)),B)),C). [resolve(5468,b,5464,a)]. 26.39/26.32 Derived: -class_Rings_Oring__1(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(5468,b,5465,a)]. 26.39/26.32 5469 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),hAPP(hAPP(c_Set_Oinsert(B),E),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(D,E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_setprod_Oinsert__remove) # label(axiom). [clausify(2135)]. 26.39/26.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5469,a,5432,a)]. 26.39/26.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5469,a,5464,a)]. 26.39/26.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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(5469,a,5465,a)]. 26.50/26.46 5470 -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(C,A)),D)),F)),E) # label(fact_setprod__cartesian__product) # label(axiom). [clausify(2490)]. 26.50/26.46 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(B,tc_Nat_Onat)),C)),E)),D). [resolve(5470,a,5432,a)]. 26.50/26.46 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(B,tc_Code__Numeral_Ocode__numeral)),C)),E)),D). [resolve(5470,a,5464,a)]. 26.50/26.46 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(B,tc_Int_Oint)),C)),E)),D). [resolve(5470,a,5465,a)]. 26.50/26.46 5471 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),C) # label(fact_setprod_OF__eq) # label(axiom). [clausify(2535)]. 26.50/26.46 5472 -class_Groups_Ocomm__monoid__mult(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod_OF__eq) # label(axiom). [clausify(2535)]. 26.50/26.46 5473 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f287(E,D,C,B,A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) # label(fact_fold__image__Un__one) # label(axiom). [clausify(2541)]. 26.50/26.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f287(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_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5473,a,5432,a)]. 26.50/26.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f287(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_Groups_Otimes__class_Otimes(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)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5473,a,5464,a)]. 26.50/26.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f287(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_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Int_Oint)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5473,a,5465,a)]. 26.50/26.46 5474 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Oone__class_Oone(A) != hAPP(E,f287(E,D,C,B,A)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) # label(fact_fold__image__Un__one) # label(axiom). [clausify(2541)]. 26.50/26.46 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) != hAPP(D,f287(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5474,a,5432,a)]. 26.60/26.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f287(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(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)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5474,a,5464,a)]. 26.60/26.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(D,f287(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Int_Oint)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5474,a,5465,a)]. 26.60/26.56 5475 -class_Groups_Ocomm__monoid__mult(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_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) # label(fact_fold__Un__disjoint) # label(axiom). [clausify(2799)]. 26.60/26.56 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Nat_Onat)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5475,a,5432,a)]. 26.60/26.56 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_Groups_Otimes__class_Otimes(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)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5475,a,5464,a)]. 26.60/26.56 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_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Int_Oint)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(5475,a,5465,a)]. 26.60/26.56 5476 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f321(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,tc_prod(B,F),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Product__Type_Oprod_Oprod__case(B,F,A),D)),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Product__Type_OSigma(B,F),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBS(B,tc_fun(F,tc_HOL_Obool),A),hAPP(hAPP(c_COMBC(B,A,tc_fun(tc_fun(F,tc_HOL_Obool),A)),hAPP(hAPP(c_COMBB(tc_fun(F,A),tc_fun(A,tc_fun(tc_fun(F,tc_HOL_Obool),A)),B),c_Finite__Set_Ofold__image(A,F,c_Groups_Otimes__class_Otimes(A))),D)),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__Sigma) # label(axiom). [clausify(2800)]. 26.60/26.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f321(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,tc_prod(A,E),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Nat_Onat),C)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_fun(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_fun(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat)),A),c_Finite__Set_Ofold__image(tc_Nat_Onat,E,c_Groups_Otimes__class_Otimes(tc_Nat_Onat))),C)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5476,a,5432,a)]. 26.60/26.56 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f321(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,tc_prod(A,E),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Code__Numeral_Ocode__numeral),C)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = 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(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_fun(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_Code__Numeral_Ocode__numeral,tc_fun(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral)),A),c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,E,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral))),C)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5476,a,5464,a)]. 26.60/26.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f321(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,tc_prod(A,E),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Int_Oint),C)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_fun(tc_fun(E,tc_HOL_Obool),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_fun(E,tc_HOL_Obool),tc_Int_Oint)),A),c_Finite__Set_Ofold__image(tc_Int_Oint,E,c_Groups_Otimes__class_Otimes(tc_Int_Oint))),C)),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5476,a,5465,a)]. 26.60/26.57 5477 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f321(F,E,D,C,B,A)))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,tc_prod(B,D),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Product__Type_Oprod_Oprod__case(B,D,A),F)),c_Groups_Oone__class_Oone(A)),hAPP(hAPP(c_Product__Type_OSigma(B,D),C),E)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBC(B,A,tc_fun(tc_fun(D,tc_HOL_Obool),A)),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(A,tc_fun(tc_fun(D,tc_HOL_Obool),A)),B),c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A))),F)),c_Groups_Oone__class_Oone(A))),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__Sigma) # label(axiom). [clausify(2800)]. 26.60/26.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f321(E,D,C,B,A,tc_Nat_Onat)))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,tc_prod(A,C),c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Nat_Onat),E)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat)),A),c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat))),E)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5477,a,5432,a)]. 26.60/26.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f321(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,tc_prod(A,C),c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Code__Numeral_Ocode__numeral),E)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = 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(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_fun(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_Code__Numeral_Ocode__numeral,tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral)),A),c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral))),E)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5477,a,5464,a)]. 26.60/26.57 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f321(E,D,C,B,A,tc_Int_Oint)))) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,tc_prod(A,C),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Int_Oint),E)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint)),A),c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint))),E)),c_Groups_Oone__class_Oone(tc_Int_Oint))),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5477,a,5465,a)]. 26.69/26.65 5478 -class_Groups_Ocomm__monoid__mult(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_power__mult__distrib) # label(axiom). [clausify(2939)]. 26.69/26.65 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(5478,a,5432,a)]. 26.69/26.65 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(5478,a,5464,a)]. 26.69/26.65 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(5478,a,5465,a)]. 26.69/26.65 5479 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),D)),E)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)) # label(fact_setprod_Odistrib) # label(axiom). [clausify(2962)]. 26.69/26.65 5480 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod_Ounion__inter) # label(axiom). [clausify(2979)]. 26.69/26.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5480,a,5432,a)]. 26.79/26.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5480,a,5464,a)]. 26.79/26.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5480,a,5465,a)]. 26.79/26.69 5481 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),c_Groups_Oone__class_Oone(A)),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Otimes__class_Otimes(A)),D)),E)),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__distrib) # label(axiom). [clausify(3099)]. 26.79/26.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Nat_Onat)),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_Nat_Onat)),C)),D)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5481,a,5432,a)]. 26.79/26.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(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),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(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)),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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C)),D)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5481,a,5464,a)]. 26.79/26.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(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_Groups_Oone__class_Oone(tc_Int_Oint)),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_Int_Oint)),C)),D)),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5481,a,5465,a)]. 26.79/26.70 5482 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(A))),C) # label(fact_setprod__delta) # label(axiom). [clausify(3101)]. 26.79/26.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Nat_Onat))),B). [resolve(5482,a,5432,a)]. 26.79/26.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5482,a,5464,a)]. 26.79/26.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Int_Oint))),B). [resolve(5482,a,5465,a)]. 26.79/26.70 5483 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(A))),C) = hAPP(E,D) # label(fact_setprod__delta) # label(axiom). [clausify(3101)]. 26.79/26.70 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(5483,a,5432,a)]. 26.80/26.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(5483,a,5464,a)]. 26.80/26.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Oone__class_Oone(tc_Int_Oint))),B) = hAPP(D,C). [resolve(5483,a,5465,a)]. 26.80/26.78 5484 -class_Groups_Ocomm__monoid__mult(A) | -class_Rings_Odvd(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_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C))) # label(fact_setprod__dvd__setprod__subset) # label(axiom). [clausify(3159)]. 26.80/26.78 Derived: -class_Rings_Odvd(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_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B))). [resolve(5484,a,5432,a)]. 26.80/26.78 Derived: -class_Rings_Odvd(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_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5484,a,5464,a)]. 26.80/26.78 Derived: -class_Rings_Odvd(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_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B))). [resolve(5484,a,5465,a)]. 26.80/26.78 5485 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -c_Fun_Oinj__on(B,B,D,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(c_Fun_Ocomp(B,A,B,E),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),hAPP(c_Set_Oimage(B,B,D),C)) # label(fact_setprod_Oreindex) # label(axiom). [clausify(3258)]. 26.80/26.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(A,tc_Nat_Onat,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5485,a,5432,a)]. 26.80/26.78 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5485,a,5464,a)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(A,tc_Int_Oint,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(5485,a,5465,a)]. 26.89/26.82 5486 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f374(D,C,B,A)),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__1) # label(axiom). [clausify(3264)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f374(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5486,a,5432,a)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f374(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Oone__class_Oone(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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5486,a,5464,a)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f374(C,B,A,tc_Int_Oint)),B)) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5486,a,5465,a)]. 26.89/26.82 5487 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Oone__class_Oone(A) != hAPP(D,f374(D,C,B,A)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),c_Groups_Oone__class_Oone(A)),C) # label(fact_fold__image__1) # label(axiom). [clausify(3264)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) != hAPP(C,f374(C,B,A,tc_Nat_Onat)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),B). [resolve(5487,a,5432,a)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f374(C,B,A,tc_Code__Numeral_Ocode__numeral)) | c_Groups_Oone__class_Oone(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),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),B). [resolve(5487,a,5464,a)]. 26.89/26.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(C,f374(C,B,A,tc_Int_Oint)) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),c_Groups_Oone__class_Oone(tc_Int_Oint)),B). [resolve(5487,a,5465,a)]. 26.89/26.82 5488 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),C) = 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)) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom). [clausify(3345)]. 26.89/26.82 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),B) = 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)). [resolve(5488,a,5432,a)]. 27.10/27.07 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B) = 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)). [resolve(5488,a,5464,a)]. 27.10/27.07 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),B) = 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)). [resolve(5488,a,5465,a)]. 27.10/27.07 5489 -class_Groups_Ocomm__monoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setprod_Oempty) # label(axiom). [clausify(3349)]. 27.10/27.07 5490 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(c_COMBK(A,B),D)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Finite__Set_Ocard(B),C)) # label(fact_setprod__constant) # label(axiom). [clausify(3490)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),C)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5490,a,5432,a)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),C)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5490,a,5464,a)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),C)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5490,a,5465,a)]. 27.10/27.07 5491 -class_Groups_Ocomm__monoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(c_COMBK(A,B),c_Groups_Oone__class_Oone(A))),C) # label(fact_setprod__1) # label(axiom). [clausify(3750)]. 27.10/27.07 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),c_Groups_Oone__class_Oone(tc_Nat_Onat))),B). [resolve(5491,a,5432,a)]. 27.10/27.07 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5491,a,5464,a)]. 27.10/27.07 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),c_Groups_Oone__class_Oone(tc_Int_Oint))),B). [resolve(5491,a,5465,a)]. 27.10/27.07 5492 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f425(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod_Ounion__inter__neutral) # label(axiom). [clausify(3811)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f425(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5492,a,5432,a)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f425(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5492,a,5464,a)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f425(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__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5492,a,5465,a)]. 27.10/27.07 5493 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Oone__class_Oone(A) != hAPP(E,f425(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod_Ounion__inter__neutral) # label(axiom). [clausify(3811)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Nat_Onat) != hAPP(D,f425(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)). [resolve(5493,a,5432,a)]. 27.10/27.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f425(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5493,a,5464,a)]. 27.19/27.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(D,f425(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)). [resolve(5493,a,5465,a)]. 27.19/27.18 5494 -class_Groups_Ocomm__monoid__mult(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hBOOL(hAPP(hAPP(c_member(B),f442(V6,V7,F,E,D,C,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),V7),E) # label(fact_strong__setprod__reindex__cong) # label(axiom). [clausify(3953)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f442(F,V6,E,D,C,B,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),V6),D). [resolve(5494,a,5432,a)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f442(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5494,a,5464,a)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f442(F,V6,E,D,C,B,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),V6),D). [resolve(5494,a,5465,a)]. 27.19/27.18 5495 -class_Groups_Ocomm__monoid__mult(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hAPP(hAPP(c_Fun_Ocomp(C,A,B,V6),D),f442(V6,V7,F,E,D,C,B,A)) != hAPP(V7,f442(V6,V7,F,E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),V7),E) # label(fact_strong__setprod__reindex__cong) # label(axiom). [clausify(3953)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,F),C),f442(F,V6,E,D,C,B,A,tc_Nat_Onat)) != hAPP(V6,f442(F,V6,E,D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),V6),D). [resolve(5495,a,5432,a)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,F),C),f442(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(V6,f442(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5495,a,5464,a)]. 27.19/27.18 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,F),C),f442(F,V6,E,D,C,B,A,tc_Int_Oint)) != hAPP(V6,f442(F,V6,E,D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),V6),D). [resolve(5495,a,5465,a)]. 27.19/27.18 5496 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f451(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(F,A)),D)),E)),C) # label(fact_setprod__Sigma) # label(axiom). [clausify(3981)]. 27.19/27.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f451(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(E,tc_Nat_Onat)),C)),D)),B). [resolve(5496,a,5432,a)]. 27.19/27.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f451(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(E,tc_Code__Numeral_Ocode__numeral)),C)),D)),B). [resolve(5496,a,5464,a)]. 27.19/27.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f451(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(E,tc_Int_Oint)),C)),D)),B). [resolve(5496,a,5465,a)]. 27.19/27.19 5497 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f451(F,E,D,C,B,A)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(D,A)),F)),E)),C) # label(fact_setprod__Sigma) # label(axiom). [clausify(3981)]. 27.19/27.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f451(E,D,C,B,A,tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(C,tc_Nat_Onat)),E)),D)),B). [resolve(5497,a,5432,a)]. 27.19/27.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f451(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(C,tc_Code__Numeral_Ocode__numeral)),E)),D)),B). [resolve(5497,a,5464,a)]. 27.40/27.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f451(E,D,C,B,A,tc_Int_Oint)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(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__mult__class_Osetprod(C,tc_Int_Oint)),E)),D)),B). [resolve(5497,a,5465,a)]. 27.40/27.32 5498 -class_Groups_Ocomm__monoid__mult(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(C,A),F),hAPP(c_Set_Oimage(B,C,D),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(c_Fun_Ocomp(C,A,B,F),D)),E) # label(fact_setprod__reindex) # label(axiom). [clausify(4062)]. 27.40/27.32 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,E),C)),D). [resolve(5498,a,5432,a)]. 27.40/27.32 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,E),C)),D). [resolve(5498,a,5464,a)]. 27.40/27.32 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,E),C)),D). [resolve(5498,a,5465,a)]. 27.40/27.32 5499 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),F)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(c_Power_Opower__class_Opower(A),F),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(B),C)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_setprod__gen__delta) # label(axiom). [clausify(4175)]. 27.40/27.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),B)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5499,a,5432,a)]. 27.40/27.32 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),B)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5499,a,5464,a)]. 27.49/27.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),B)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))). [resolve(5499,a,5465,a)]. 27.49/27.42 5500 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),F)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),F),hAPP(c_Finite__Set_Ocard(B),C)) # label(fact_setprod__gen__delta) # label(axiom). [clausify(4175)]. 27.49/27.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),E),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5500,a,5432,a)]. 27.49/27.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5500,a,5464,a)]. 27.49/27.42 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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)),E)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),E),hAPP(c_Finite__Set_Ocard(A),B)). [resolve(5500,a,5465,a)]. 27.49/27.42 5501 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),f499(D,E,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),E),B) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4321)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | A != B | -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)),f499(C,D,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),D),A). [resolve(5501,a,5432,a)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | A != B | -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)),f499(C,D,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),A). [resolve(5501,a,5464,a)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | A != B | -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)),f499(C,D,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),D),A). [resolve(5501,a,5465,a)]. 27.49/27.49 5502 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f499(D,E,C,B,A)) != hAPP(E,f499(D,E,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),E),B) # label(fact_transfer__nat__int__sum__prod__cong_I2_J) # label(axiom). [clausify(4321)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f499(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f499(C,D,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),D),A). [resolve(5502,a,5432,a)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f499(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f499(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),D),A). [resolve(5502,a,5464,a)]. 27.49/27.49 Derived: -class_Groups_Ocomm__monoid__add(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f499(C,D,B,A,tc_Int_Oint)) != hAPP(D,f499(C,D,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),D),A). [resolve(5502,a,5465,a)]. 27.49/27.49 5503 -class_Groups_Ocomm__monoid__mult(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),f518(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__mult__class_Osetprod(B,A),E),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) # label(fact_setprod__mono__one__left) # label(axiom). [clausify(4430)]. 27.58/27.54 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),f518(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__mult__class_Osetprod(A,tc_Nat_Onat),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B). [resolve(5503,a,5432,a)]. 27.58/27.54 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),f518(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__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5503,a,5464,a)]. 27.58/27.54 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),f518(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__mult__class_Osetprod(A,tc_Int_Oint),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B). [resolve(5503,a,5465,a)]. 27.58/27.54 5504 -class_Groups_Ocomm__monoid__mult(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_Oone__class_Oone(A) != hAPP(E,f518(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C) # label(fact_setprod__mono__one__left) # label(axiom). [clausify(4430)]. 27.58/27.54 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_Oone__class_Oone(tc_Nat_Onat) != hAPP(D,f518(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B). [resolve(5504,a,5432,a)]. 27.58/27.54 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_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f518(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5504,a,5464,a)]. 27.58/27.54 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_Oone__class_Oone(tc_Int_Oint) != hAPP(D,f518(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B). [resolve(5504,a,5465,a)]. 27.58/27.54 5505 -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(4503)]. 27.58/27.54 Derived: -class_Rings_Osemiring__1(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(5505,b,5432,a)]. 27.58/27.54 Derived: -class_Rings_Osemiring__1(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(5505,b,5464,a)]. 31.18/31.14 Derived: -class_Rings_Osemiring__1(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(5505,b,5465,a)]. 31.18/31.14 5506 -class_Groups_Ocomm__monoid__mult(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__infinite) # label(axiom). [clausify(4676)]. 31.18/31.14 5507 -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(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__mult__class_Osetprod(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)) # label(fact_setprod__Un__Int) # label(axiom). [clausify(4827)]. 31.18/31.14 5508 class_Groups_Oab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__group__add) # label(axiom). [assumption]. 31.18/31.14 5509 -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(16)]. 31.18/31.14 5510 -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(146)]. 31.18/31.14 5511 -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) | B != C | E = D # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(676)]. 31.18/31.14 5512 -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) | B = C | E != D # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(676)]. 31.18/31.14 5513 -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(725)]. 31.18/31.14 5514 -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(725)]. 31.18/31.14 5515 -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(748)]. 31.18/31.14 5516 -class_Groups_Oab__group__add(A) | B != C | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B) # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(846)]. 31.18/31.14 5517 -class_Groups_Oab__group__add(A) | B = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B) # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(846)]. 31.18/31.14 5518 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_ab__diff__minus) # label(axiom). [clausify(889)]. 31.18/31.14 5519 -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(1780)]. 31.18/31.14 5520 -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(1968)]. 31.18/31.14 5521 -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(2176)]. 31.18/31.14 5522 -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(3279)]. 31.18/31.14 5523 -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(3485)]. 31.18/31.14 5524 -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(3540)]. 31.18/31.14 5525 -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(3660)]. 31.18/31.14 5526 -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(3962)]. 31.18/31.14 5527 -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(3962)]. 31.18/31.14 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(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_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5508,a,5509,a)]. 31.18/31.14 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(5508,a,5510,a)]. 31.18/31.14 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) | A != B | D = C. [resolve(5508,a,5511,a)]. 31.18/31.14 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) | A = B | D != C. [resolve(5508,a,5512,a)]. 31.18/31.14 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(5508,a,5513,a)]. 31.18/31.14 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(5508,a,5514,a)]. 31.18/31.14 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(5508,a,5515,a)]. 31.18/31.15 Derived: A != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A). [resolve(5508,a,5516,a)]. 31.18/31.15 Derived: A = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A). [resolve(5508,a,5517,a)]. 31.18/31.15 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B). [resolve(5508,a,5518,a)]. 31.18/31.15 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(5508,a,5519,a)]. 31.18/31.15 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(5508,a,5520,a)]. 31.18/31.15 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(5508,a,5521,a)]. 31.18/31.15 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A). [resolve(5508,a,5522,a)]. 31.18/31.15 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(5508,a,5523,a)]. 31.18/31.15 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(5508,a,5524,a)]. 31.18/31.15 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(5508,a,5525,a)]. 31.18/31.15 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(5508,a,5526,a)]. 32.49/32.49 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(5508,a,5527,a)]. 32.49/32.49 5528 class_Rings_Ocomm__semiring__1(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 32.49/32.49 5529 -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(19)]. 32.49/32.49 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(5528,a,5529,a)]. 32.49/32.49 5530 -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(319)]. 32.49/32.49 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5530,a,5528,a)]. 32.49/32.49 5531 -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(648)]. 32.49/32.49 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(5531,a,5528,a)]. 32.49/32.49 5532 -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(659)]. 32.49/32.49 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(5532,a,5528,a)]. 32.49/32.49 5533 -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),C),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__trans) # label(axiom). [clausify(1055)]. 32.59/32.56 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),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5533,a,5528,a)]. 32.59/32.56 5534 -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(1183)]. 32.59/32.56 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(5534,a,5528,a)]. 32.59/32.56 5535 -class_Rings_Ocomm__semiring__1(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) # label(axiom). [clausify(1376)]. 32.59/32.56 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(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(5535,a,5528,a)]. 32.59/32.56 5536 -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(1402)]. 32.59/32.56 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(5536,a,5528,a)]. 32.59/32.56 5537 -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_dvd__0__left) # label(axiom). [clausify(1404)]. 32.59/32.56 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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) = A. [resolve(5537,a,5528,a)]. 32.59/32.56 5538 -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(1415)]. 32.59/32.56 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(5538,a,5528,a)]. 32.69/32.65 5539 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(1479)]. 32.69/32.65 Derived: 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) = 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),B)). [resolve(5539,a,5528,a)]. 32.69/32.65 5540 -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(1480)]. 32.69/32.65 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(5540,a,5528,a)]. 32.69/32.65 5541 -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(1625)]. 32.69/32.65 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(5541,a,5528,a)]. 32.69/32.65 5542 -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(1793)]. 32.69/32.65 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(5542,a,5528,a)]. 32.69/32.65 5543 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) = 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)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(1805)]. 32.69/32.65 Derived: 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)) = 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)). [resolve(5543,a,5528,a)]. 32.91/32.87 5544 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(1829)]. 32.91/32.87 Derived: 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),B)),C) = 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),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5544,a,5528,a)]. 32.91/32.87 5545 -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(1847)]. 32.91/32.87 5546 -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(1985)]. 32.91/32.87 5547 -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(2060)]. 32.91/32.87 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(5547,a,5528,a)]. 32.91/32.87 5548 -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(2398)]. 32.91/32.87 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(5548,a,5528,a)]. 32.91/32.87 5549 -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(2414)]. 32.91/32.87 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(5549,a,5528,a)]. 32.91/32.87 5550 -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(2423)]. 32.91/32.87 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(5550,a,5528,a)]. 32.91/32.87 5551 -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(2539)]. 33.12/33.08 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(5551,a,5528,a)]. 33.12/33.08 5552 -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(2547)]. 33.12/33.08 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(5552,a,5528,a)]. 33.12/33.08 5553 -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(2595)]. 33.12/33.08 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(5553,a,5528,a)]. 33.12/33.08 5554 -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(2603)]. 33.12/33.08 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(5554,a,5528,a)]. 33.12/33.08 5555 -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(2651)]. 33.12/33.08 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(5555,a,5528,a)]. 33.12/33.08 5556 -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(2775)]. 33.12/33.08 5557 -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(2868)]. 33.12/33.08 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(5557,a,5528,a)]. 33.12/33.08 5558 -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(3102)]. 33.21/33.15 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(5558,a,5528,a)]. 33.21/33.15 5559 -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(3145)]. 33.21/33.15 5560 -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(3153)]. 33.21/33.15 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(5560,a,5528,a)]. 33.21/33.15 5561 -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(3181)]. 33.21/33.15 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(5561,a,5528,a)]. 33.21/33.15 5562 -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(3247)]. 33.21/33.15 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(5562,a,5528,a)]. 33.21/33.15 5563 -class_Rings_Ocomm__semiring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),B)) # label(fact_dvd__refl) # label(axiom). [clausify(3257)]. 33.30/33.27 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),A)). [resolve(5563,a,5528,a)]. 33.30/33.27 5564 -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(3375)]. 33.30/33.27 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(5564,a,5528,a)]. 33.30/33.27 5565 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Oone__class_Oone(A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),C) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(3381)]. 33.30/33.27 Derived: 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) = 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). [resolve(5565,a,5528,a)]. 33.30/33.27 5566 -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(3412)]. 33.30/33.27 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(5566,a,5528,a)]. 33.30/33.27 5567 -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(3425)]. 33.30/33.27 5568 -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(3475)]. 33.30/33.27 5569 class_Rings_Ocomm__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 33.30/33.27 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(5569,a,5529,a)]. 33.30/33.27 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5569,a,5530,a)]. 33.30/33.27 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(5569,a,5531,a)]. 33.30/33.27 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(5569,a,5532,a)]. 33.30/33.27 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),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5569,a,5533,a)]. 33.30/33.28 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)). [resolve(5569,a,5534,a)]. 33.30/33.28 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(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(5569,a,5535,a)]. 33.30/33.28 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(5569,a,5536,a)]. 33.30/33.28 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5569,a,5537,a)]. 33.30/33.28 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(5569,a,5538,a)]. 33.30/33.28 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)). [resolve(5569,a,5539,a)]. 33.30/33.28 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),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(5569,a,5540,a)]. 33.30/33.28 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(5569,a,5541,a)]. 33.30/33.28 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(5569,a,5542,a)]. 33.30/33.28 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5569,a,5547,a)]. 33.30/33.28 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(5569,a,5548,a)]. 33.30/33.28 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(5569,a,5549,a)]. 33.30/33.28 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(5569,a,5550,a)]. 33.30/33.28 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(5569,a,5551,a)]. 33.30/33.28 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(5569,a,5552,a)]. 33.43/33.33 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(5569,a,5553,a)]. 33.43/33.33 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(5569,a,5554,a)]. 33.43/33.33 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5569,a,5555,a)]. 33.43/33.33 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(5569,a,5557,a)]. 33.43/33.33 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(5569,a,5558,a)]. 33.43/33.33 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(5569,a,5560,a)]. 33.43/33.33 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(5569,a,5561,a)]. 33.43/33.33 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(5569,a,5562,a)]. 33.43/33.33 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),A)). [resolve(5569,a,5563,a)]. 33.43/33.33 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),B). [resolve(5569,a,5565,a)]. 33.43/33.33 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(5569,a,5566,a)]. 33.43/33.33 5570 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | c_Groups_Ozero__class_Ozero(A) = hAPP(D,f404(D,C,B,A)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(3601)]. 33.43/33.33 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(C,f404(C,B,A,tc_Code__Numeral_Ocode__numeral)). [resolve(5570,a,5528,a)]. 33.53/33.46 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(C,f404(C,B,A,tc_Int_Oint)). [resolve(5570,a,5569,a)]. 33.53/33.46 5571 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | hBOOL(hAPP(hAPP(c_member(B),f404(D,C,B,A)),C)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(3601)]. 33.53/33.46 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B) | hBOOL(hAPP(hAPP(c_member(A),f404(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5571,a,5528,a)]. 33.53/33.46 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) | hBOOL(hAPP(hAPP(c_member(A),f404(C,B,A,tc_Int_Oint)),B)). [resolve(5571,a,5569,a)]. 33.53/33.46 5572 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,E) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(3601)]. 33.53/33.46 5573 -class_Rings_Ocomm__semiring__1(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))),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(3647)]. 33.53/33.46 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),A). [resolve(5573,a,5528,a)]. 33.53/33.46 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A). [resolve(5573,a,5569,a)]. 33.53/33.46 5574 -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(3656)]. 33.53/33.46 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(5574,a,5528,a)]. 33.53/33.46 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(5574,a,5569,a)]. 33.53/33.46 5575 -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(3803)]. 33.53/33.46 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(5575,a,5569,a)]. 33.73/33.70 5576 -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(3836)]. 33.73/33.70 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A) = A. [resolve(5576,a,5569,a)]. 33.73/33.70 5577 class_Rings_Ocomm__semiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 33.73/33.70 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(5577,a,5529,a)]. 33.73/33.70 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5577,a,5530,a)]. 33.73/33.70 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(5577,a,5531,a)]. 33.73/33.70 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(5577,a,5532,a)]. 33.73/33.70 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),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5577,a,5533,a)]. 33.73/33.70 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(5577,a,5534,a)]. 33.73/33.70 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(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(5577,a,5535,a)]. 33.73/33.70 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(5577,a,5536,a)]. 33.73/33.70 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5577,a,5537,a)]. 33.73/33.70 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(5577,a,5538,a)]. 33.73/33.70 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)). [resolve(5577,a,5539,a)]. 33.73/33.70 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(5577,a,5541,a)]. 33.83/33.71 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(5577,a,5542,a)]. 33.83/33.71 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5577,a,5544,a)]. 33.83/33.71 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5577,a,5547,a)]. 33.83/33.71 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(5577,a,5548,a)]. 33.83/33.71 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(5577,a,5549,a)]. 33.83/33.71 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(5577,a,5550,a)]. 33.83/33.71 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(5577,a,5551,a)]. 33.83/33.71 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(5577,a,5552,a)]. 33.83/33.71 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(5577,a,5553,a)]. 33.83/33.71 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(5577,a,5554,a)]. 33.83/33.71 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(5577,a,5555,a)]. 33.83/33.71 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(5577,a,5557,a)]. 33.83/33.71 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(5577,a,5558,a)]. 33.83/33.71 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(5577,a,5560,a)]. 34.03/33.97 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(5577,a,5561,a)]. 34.03/33.97 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(5577,a,5562,a)]. 34.03/33.97 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),A)). [resolve(5577,a,5563,a)]. 34.03/33.97 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(5577,a,5564,a)]. 34.03/33.97 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),B). [resolve(5577,a,5565,a)]. 34.03/33.97 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(5577,a,5566,a)]. 34.03/33.97 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(C,f404(C,B,A,tc_Nat_Onat)). [resolve(5577,a,5570,a)]. 34.03/33.97 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) | hBOOL(hAPP(hAPP(c_member(A),f404(C,B,A,tc_Nat_Onat)),B)). [resolve(5577,a,5571,a)]. 34.03/33.97 Derived: 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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),A). [resolve(5577,a,5573,a)]. 34.03/33.97 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(5577,a,5574,a)]. 34.03/33.97 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A) = A. [resolve(5577,a,5576,a)]. 34.03/33.97 5578 -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(4304)]. 34.03/33.97 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(5578,a,5528,a)]. 34.03/33.97 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A). [resolve(5578,a,5569,a)]. 34.03/33.97 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(5578,a,5577,a)]. 34.03/33.97 5579 -class_Rings_Ocomm__semiring__1(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(4324)]. 34.03/33.97 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(5579,a,5569,a)]. 34.03/33.97 5580 -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(4595)]. 34.12/34.08 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(5580,a,5528,a)]. 34.12/34.08 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = A. [resolve(5580,a,5569,a)]. 34.12/34.08 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A) = A. [resolve(5580,a,5577,a)]. 34.12/34.08 5581 -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(4694)]. 34.12/34.08 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(5581,a,5528,a)]. 34.12/34.08 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(5581,a,5569,a)]. 34.12/34.08 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(5581,a,5577,a)]. 34.12/34.08 5582 -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(4725)]. 34.12/34.08 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(5582,a,5528,a)]. 34.12/34.08 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(5582,a,5569,a)]. 34.12/34.08 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(5582,a,5577,a)]. 34.12/34.08 5583 -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(4757)]. 34.12/34.08 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(5583,a,5528,a)]. 44.83/44.76 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(5583,a,5569,a)]. 44.83/44.76 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(5583,a,5577,a)]. 44.83/44.76 5584 -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(4849)]. 44.83/44.76 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(5584,a,5528,a)]. 44.83/44.76 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(5584,a,5569,a)]. 44.83/44.76 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(5584,a,5577,a)]. 44.83/44.76 5585 class_Groups_Oabs__if(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oabs__if) # label(axiom). [assumption]. 44.83/44.76 5586 -class_Groups_Oabs__if(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) = B # label(fact_abs__if) # label(axiom). [clausify(26)]. 44.83/44.76 5587 -class_Groups_Oabs__if(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__if) # label(axiom). [clausify(26)]. 44.83/44.76 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = A. [resolve(5585,a,5586,a)]. 44.83/44.76 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5585,a,5587,a)]. 44.83/44.76 5588 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 44.83/44.76 5589 -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(48)]. 44.91/44.79 5590 -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(99)]. 44.91/44.79 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(5588,a,5590,a)]. 44.91/44.79 5591 -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(316)]. 44.91/44.79 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(5591,a,5588,a)]. 44.91/44.79 5592 -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(409)]. 44.91/44.79 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(5592,a,5588,a)]. 44.91/44.79 5593 -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(488)]. 44.91/44.79 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(5593,a,5588,a)]. 44.91/44.79 5594 -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__increasing__iff) # label(axiom). [clausify(547)]. 44.91/44.79 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),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__eq(tc_Nat_Onat),B),C)). [resolve(5594,a,5588,a)]. 44.91/44.79 5595 -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__increasing__iff) # label(axiom). [clausify(547)]. 44.91/44.82 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),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__eq(tc_Nat_Onat),B),C)). [resolve(5595,a,5588,a)]. 44.91/44.82 5596 class_Rings_Olinordered__semidom(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semidom) # label(axiom). [assumption]. 44.91/44.82 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)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5596,a,5589,a)]. 44.91/44.82 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(5596,a,5590,a)]. 44.91/44.82 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(5596,a,5591,a)]. 44.91/44.82 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(5596,a,5592,a)]. 44.91/44.82 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(5596,a,5593,a)]. 44.91/44.82 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_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__eq(tc_Nat_Onat),B),C)). [resolve(5596,a,5594,a)]. 44.91/44.82 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_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__eq(tc_Nat_Onat),B),C)). [resolve(5596,a,5595,a)]. 44.91/44.88 5597 -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(670)]. 44.91/44.88 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(5597,a,5596,a)]. 44.91/44.88 5598 -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(959)]. 44.91/44.88 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(5598,a,5588,a)]. 44.91/44.88 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(5598,a,5596,a)]. 44.91/44.88 5599 -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(1007)]. 44.91/44.88 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(5599,a,5588,a)]. 44.91/44.88 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(5599,a,5596,a)]. 44.91/44.88 5600 -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(1101)]. 44.91/44.88 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(5600,a,5596,a)]. 45.03/44.91 5601 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 45.03/44.91 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(5601,a,5590,a)]. 45.03/44.91 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(5601,a,5591,a)]. 45.03/44.91 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(5601,a,5592,a)]. 45.03/44.91 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(5601,a,5593,a)]. 45.03/44.91 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_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__eq(tc_Nat_Onat),B),C)). [resolve(5601,a,5594,a)]. 45.03/44.91 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_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__eq(tc_Nat_Onat),B),C)). [resolve(5601,a,5595,a)]. 45.03/44.91 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(5601,a,5597,a)]. 45.03/44.91 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(5601,a,5598,a)]. 45.03/44.91 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(5601,a,5599,a)]. 45.03/44.91 5602 -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(1208)]. 45.03/44.91 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_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A))). [resolve(5602,a,5588,a)]. 45.03/44.96 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(5602,a,5596,a)]. 45.03/44.96 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(5602,a,5601,a)]. 45.03/44.96 5603 -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(1208)]. 45.03/44.96 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(5603,a,5588,a)]. 45.03/44.96 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(5603,a,5596,a)]. 45.03/44.96 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(5603,a,5601,a)]. 45.03/44.96 5604 -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(1261)]. 45.03/44.96 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(5604,a,5596,a)]. 45.03/44.96 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(5604,a,5601,a)]. 45.03/44.96 5605 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f144(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(1372)]. 45.03/44.96 Derived: hBOOL(hAPP(hAPP(c_member(A),f144(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(5605,a,5596,a)]. 45.03/44.96 Derived: hBOOL(hAPP(hAPP(c_member(A),f144(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(5605,a,5601,a)]. 45.03/44.99 5606 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f144(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(1372)]. 45.03/44.99 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,f144(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(5606,a,5596,a)]. 45.03/44.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f144(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(5606,a,5601,a)]. 45.03/44.99 5607 -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(1414)]. 45.03/44.99 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(5607,a,5588,a)]. 45.03/44.99 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(5607,a,5596,a)]. 45.03/44.99 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(5607,a,5601,a)]. 45.03/44.99 5608 -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(1466)]. 45.03/44.99 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(5608,a,5588,a)]. 45.03/44.99 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(5608,a,5596,a)]. 45.03/44.99 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(5608,a,5601,a)]. 45.12/45.04 5609 -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(1512)]. 45.12/45.04 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(5609,a,5588,a)]. 45.12/45.04 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(5609,a,5596,a)]. 45.12/45.04 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(5609,a,5601,a)]. 45.12/45.04 5610 -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(1512)]. 45.12/45.04 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(5610,a,5588,a)]. 45.12/45.04 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(5610,a,5596,a)]. 45.12/45.04 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(5610,a,5601,a)]. 45.12/45.04 5611 -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(1658)]. 45.12/45.04 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(5611,a,5588,a)]. 45.21/45.17 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),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(5611,a,5596,a)]. 45.21/45.17 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),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(5611,a,5601,a)]. 45.21/45.17 5612 -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(1733)]. 45.21/45.17 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(5612,a,5588,a)]. 45.21/45.17 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(5612,a,5596,a)]. 45.21/45.17 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(5612,a,5601,a)]. 45.21/45.17 5613 -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(2144)]. 45.21/45.17 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(5613,a,5588,a)]. 45.21/45.17 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(5613,a,5596,a)]. 45.42/45.39 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(5613,a,5601,a)]. 45.42/45.39 5614 -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(2347)]. 45.51/45.39 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(5614,a,5588,a)]. 45.51/45.39 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(5614,a,5596,a)]. 45.51/45.39 5615 -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(2347)]. 45.51/45.39 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(5615,a,5588,a)]. 45.51/45.39 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(5615,a,5596,a)]. 45.51/45.39 5616 -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(2548)]. 45.51/45.39 5617 -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(2627)]. 45.51/45.39 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(5617,a,5596,a)]. 45.51/45.39 5618 -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(2793)]. 45.63/45.60 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(5618,a,5588,a)]. 45.63/45.60 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(5618,a,5596,a)]. 45.63/45.60 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(5618,a,5601,a)]. 45.63/45.60 5619 -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(3107)]. 45.63/45.60 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(5619,a,5596,a)]. 45.63/45.60 5620 -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(3157)]. 45.63/45.60 5621 -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(3331)]. 45.63/45.60 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(5621,a,5588,a)]. 45.63/45.60 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(5621,a,5596,a)]. 45.81/45.73 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(5621,a,5601,a)]. 45.81/45.73 5622 -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(3406)]. 45.81/45.73 5623 -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(3539)]. 45.81/45.73 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(5623,a,5588,a)]. 45.81/45.73 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(5623,a,5596,a)]. 45.81/45.73 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(5623,a,5601,a)]. 45.81/45.73 5624 -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(3631)]. 45.81/45.73 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(5624,a,5588,a)]. 45.81/45.73 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(5624,a,5596,a)]. 45.81/45.73 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(5624,a,5601,a)]. 45.81/45.73 5625 -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(3631)]. 45.93/45.89 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(5625,a,5588,a)]. 45.93/45.89 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(5625,a,5596,a)]. 45.93/45.89 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(5625,a,5601,a)]. 45.93/45.89 5626 -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)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | B = D # label(fact_power__inject__base) # label(axiom). [clausify(3643)]. 45.93/45.89 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | A = C. [resolve(5626,a,5588,a)]. 45.93/45.89 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)),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)) | A = C. [resolve(5626,a,5596,a)]. 45.93/45.89 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | A = C. [resolve(5626,a,5601,a)]. 45.93/45.89 5627 -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(3687)]. 45.93/45.89 5628 -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(3818)]. 45.93/45.89 5629 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C = D # label(fact_power__inject__exp) # label(axiom). [clausify(3894)]. 45.93/45.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(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),C) | B = C. [resolve(5629,a,5588,a)]. 46.12/46.08 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)) | 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) | B = C. [resolve(5629,a,5596,a)]. 46.12/46.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(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),C) | B = C. [resolve(5629,a,5601,a)]. 46.12/46.08 5630 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C != D # label(fact_power__inject__exp) # label(axiom). [clausify(3894)]. 46.12/46.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(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),C) | B != C. [resolve(5630,a,5588,a)]. 46.12/46.08 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)) | 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) | B != C. [resolve(5630,a,5596,a)]. 46.12/46.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(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),C) | B != C. [resolve(5630,a,5601,a)]. 46.12/46.08 5631 -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(3914)]. 46.12/46.08 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(5631,a,5596,a)]. 46.12/46.08 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(5631,a,5601,a)]. 46.12/46.08 5632 -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(4171)]. 46.12/46.08 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(5632,a,5588,a)]. 46.12/46.08 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(5632,a,5596,a)]. 46.32/46.23 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(5632,a,5601,a)]. 46.32/46.23 5633 -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(4195)]. 46.32/46.23 5634 -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(4371)]. 46.32/46.23 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(5634,a,5588,a)]. 46.32/46.23 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(5634,a,5596,a)]. 46.32/46.23 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(5634,a,5601,a)]. 46.32/46.23 5635 -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(4415)]. 46.32/46.23 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(5635,a,5588,a)]. 46.32/46.23 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(5635,a,5596,a)]. 46.42/46.34 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(5635,a,5601,a)]. 46.42/46.34 5636 -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)),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(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | B = D # label(fact_power__eq__imp__eq__base) # label(axiom). [clausify(4490)]. 46.42/46.34 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)),C)) | -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)),B)) | A = C. [resolve(5636,a,5588,a)]. 46.42/46.34 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)),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_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5636,a,5596,a)]. 46.42/46.34 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)),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_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5636,a,5601,a)]. 46.42/46.34 5637 -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(4519)]. 46.42/46.34 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(5637,a,5588,a)]. 46.42/46.34 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(5637,a,5596,a)]. 46.42/46.34 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(5637,a,5601,a)]. 46.42/46.34 5638 -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(4599)]. 46.42/46.34 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(5638,a,5601,a)]. 46.42/46.34 5639 -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(4610)]. 46.51/46.46 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(5639,a,5588,a)]. 46.51/46.46 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(5639,a,5596,a)]. 46.51/46.46 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(5639,a,5601,a)]. 46.51/46.46 5640 -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(4635)]. 46.51/46.46 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(5640,a,5588,a)]. 46.51/46.46 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(5640,a,5596,a)]. 46.51/46.46 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(5640,a,5601,a)]. 46.51/46.46 5641 -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(4780)]. 46.51/46.46 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(5641,a,5588,a)]. 48.33/48.24 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(5641,a,5596,a)]. 48.33/48.24 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(5641,a,5601,a)]. 48.33/48.24 5642 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f569(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(4825)]. 48.33/48.24 Derived: hBOOL(hAPP(hAPP(c_member(A),f569(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(5642,a,5588,a)]. 48.33/48.24 Derived: hBOOL(hAPP(hAPP(c_member(A),f569(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(5642,a,5596,a)]. 48.33/48.24 Derived: hBOOL(hAPP(hAPP(c_member(A),f569(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(5642,a,5601,a)]. 48.33/48.24 5643 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f569(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(4825)]. 48.33/48.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(A,f569(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(5643,a,5588,a)]. 48.33/48.24 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,f569(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(5643,a,5596,a)]. 48.33/48.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f569(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(5643,a,5601,a)]. 48.33/48.24 5644 class_Rings_Ocomm__ring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__ring__1) # label(axiom). [assumption]. 48.33/48.24 5645 -class_Rings_Ocomm__ring__1(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom). [clausify(50)]. 48.33/48.24 5646 -class_Rings_Ocomm__ring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_minus__dvd__iff) # label(axiom). [clausify(479)]. 51.41/51.32 5647 -class_Rings_Ocomm__ring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_minus__dvd__iff) # label(axiom). [clausify(479)]. 51.41/51.32 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(5644,a,5645,a)]. 51.41/51.32 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(5644,a,5646,a)]. 51.41/51.32 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(5644,a,5647,a)]. 51.41/51.32 5648 -class_Rings_Ocomm__ring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_dvd__minus__iff) # label(axiom). [clausify(3432)]. 51.41/51.32 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(5648,a,5644,a)]. 51.41/51.32 5649 -class_Rings_Ocomm__ring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_dvd__minus__iff) # label(axiom). [clausify(3432)]. 51.41/51.32 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(5649,a,5644,a)]. 51.41/51.32 5650 -class_Rings_Ocomm__ring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom). [clausify(4692)]. 51.41/51.32 5651 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]. 51.41/51.32 5652 -c_Typedef_Otype__definition(A,B,C,D,E) | F != V6 | hAPP(C,F) = hAPP(C,V6) # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(56)]. 51.41/51.32 5653 -c_Typedef_Otype__definition(A,B,C,D,E) | F = V6 | hAPP(C,F) != hAPP(C,V6) # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(56)]. 51.41/51.32 5654 -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(187)]. 51.41/51.32 5655 -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(556)]. 51.41/51.32 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(5655,a,5651,a)]. 51.41/51.32 5656 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]. 51.41/51.32 Derived: A != B | hAPP(c_Int_ORep__Integ,A) = hAPP(c_Int_ORep__Integ,B). [resolve(5656,a,5652,a)]. 51.41/51.32 Derived: A = B | hAPP(c_Int_ORep__Integ,A) != hAPP(c_Int_ORep__Integ,B). [resolve(5656,a,5653,a)]. 51.41/51.32 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(5656,a,5654,a)]. 51.41/51.32 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(5656,a,5655,a)]. 54.42/54.32 5657 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(D,hAPP(C,F)) = F # label(fact_type__definition_ORep__inverse) # label(axiom). [clausify(2224)]. 54.42/54.32 5658 -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(2340)]. 54.42/54.32 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(5658,a,5651,a)]. 54.42/54.32 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(5658,a,5656,a)]. 54.42/54.32 5659 -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)) | hAPP(D,V6) != hAPP(D,F) | V6 = F # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(2632)]. 54.42/54.32 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)) | hAPP(c_Int_OAbs__Integ,B) != hAPP(c_Int_OAbs__Integ,A) | B = A. [resolve(5659,a,5656,a)]. 54.42/54.32 5660 -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)) | hAPP(D,V6) = hAPP(D,F) | V6 != F # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(2632)]. 54.42/54.32 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)) | hAPP(c_Int_OAbs__Integ,B) = hAPP(c_Int_OAbs__Integ,A) | B != A. [resolve(5660,a,5656,a)]. 54.42/54.32 5661 -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(3048)]. 54.42/54.32 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,A)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))). [resolve(5661,a,5651,a)]. 54.42/54.32 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),hAPP(c_Int_ORep__Integ,A)),c_Int_OInteg)). [resolve(5661,a,5656,a)]. 54.42/54.32 5662 c_Predicate_Otransp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_transp__def) # label(axiom). [clausify(59)]. 54.42/54.32 5663 -c_Predicate_Otransp(A,B) | c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_transp__def) # label(axiom). [clausify(59)]. 54.42/54.32 5664 -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)) # label(fact_transpE) # label(axiom). [clausify(1022)]. 54.42/54.32 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),D)) | hBOOL(hAPP(hAPP(A,B),D)) | -c_Relation_Otrans(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))). [resolve(5664,a,5662,a)]. 54.42/54.32 5665 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_equivpE) # label(axiom). [clausify(1298)]. 54.42/54.32 Derived: -c_Equiv__Relations_Oequivp(A,B) | c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5665,b,5663,a)]. 54.42/54.32 5666 -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -c_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(1598)]. 54.42/54.32 Derived: -c_Predicate_Osymp(A,B) | -c_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5666,a,5662,a)]. 54.42/54.32 5667 c_Predicate_Otransp(A,B) | -c_Equiv__Relations_Oequivp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(1598)]. 57.53/57.47 Derived: -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)). [resolve(5667,a,5664,a)]. 57.53/57.47 5668 -c_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | c_Equiv__Relations_Opart__equivp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(3150)]. 57.53/57.47 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | c_Equiv__Relations_Opart__equivp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5668,b,5662,a)]. 57.53/57.47 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | c_Equiv__Relations_Opart__equivp(A,B) | -c_Equiv__Relations_Oequivp(A,B). [resolve(5668,b,5665,b)]. 57.53/57.47 5669 c_Predicate_Otransp(A,B) | -c_Equiv__Relations_Opart__equivp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(3150)]. 57.53/57.47 Derived: -c_Equiv__Relations_Opart__equivp(A,B) | c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5669,a,5663,a)]. 57.53/57.47 Derived: -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)). [resolve(5669,a,5664,a)]. 57.53/57.47 Derived: -c_Equiv__Relations_Opart__equivp(A,B) | -c_Predicate_Osymp(A,B) | -c_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(5669,a,5666,a)]. 57.53/57.47 5670 -c_Predicate_Oreflp(A,B) | -c_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) # label(fact_equivpI) # label(axiom). [clausify(3442)]. 57.53/57.47 5671 -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Osymp(C,A) | -c_Predicate_Otransp(C,A) | c_Equiv__Relations_Opart__equivp(C,A) # label(fact_part__equivpI) # label(axiom). [clausify(3711)]. 57.53/57.47 5672 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(3919)]. 57.53/57.47 5673 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_minus__le__self__iff) # label(axiom). [clausify(616)]. 57.53/57.47 5674 class_Groups_Olinordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 57.53/57.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5673,a,5674,a)]. 57.53/57.47 5675 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_minus__le__self__iff) # label(axiom). [clausify(616)]. 57.53/57.47 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5675,a,5674,a)]. 57.53/57.47 5676 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__min__eq__max) # label(axiom). [clausify(1049)]. 57.53/57.47 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5676,a,5674,a)]. 57.53/57.47 5677 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__minus__self__iff) # label(axiom). [clausify(1250)]. 57.82/57.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5677,a,5674,a)]. 57.82/57.73 5678 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__minus__self__iff) # label(axiom). [clausify(1250)]. 57.82/57.73 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5678,a,5674,a)]. 57.82/57.73 5679 -class_Groups_Olinordered__ab__group__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_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_minus__Max__eq__Min) # label(axiom). [clausify(1702)]. 57.82/57.73 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_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5679,a,5674,a)]. 57.82/57.73 5680 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__nonneg) # label(axiom). [clausify(2209)]. 57.82/57.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5680,a,5674,a)]. 57.82/57.73 5681 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__nonneg) # label(axiom). [clausify(2209)]. 57.82/57.73 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5681,a,5674,a)]. 57.82/57.73 5682 -class_Groups_Olinordered__ab__group__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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(2383)]. 57.82/57.73 5683 -class_Groups_Olinordered__ab__group__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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(2383)]. 57.82/57.73 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_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A))). [resolve(5683,a,5674,a)]. 58.53/58.40 5684 -class_Groups_Olinordered__ab__group__add(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),B),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(2755)]. 58.53/58.40 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),A),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5684,a,5674,a)]. 58.53/58.40 5685 -class_Groups_Olinordered__ab__group__add(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),B),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(2755)]. 58.53/58.40 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),A),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5685,a,5674,a)]. 58.53/58.40 5686 -class_Groups_Olinordered__ab__group__add(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_double__eq__0__iff) # label(axiom). [clausify(3352)]. 58.53/58.40 Derived: 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(5686,a,5674,a)]. 58.53/58.40 5687 -class_Groups_Olinordered__ab__group__add(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_double__eq__0__iff) # label(axiom). [clausify(3352)]. 58.53/58.40 Derived: 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(5687,a,5674,a)]. 58.53/58.40 5688 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__max__eq__min) # label(axiom). [clausify(3752)]. 58.53/58.40 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5688,a,5674,a)]. 58.53/58.40 5689 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != B | c_Groups_Ozero__class_Ozero(A) = B # label(fact_equal__neg__zero) # label(axiom). [clausify(3782)]. 58.53/58.40 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5689,a,5674,a)]. 58.53/58.40 5690 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = B | c_Groups_Ozero__class_Ozero(A) != B # label(fact_equal__neg__zero) # label(axiom). [clausify(3782)]. 58.53/58.40 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5690,a,5674,a)]. 58.53/58.40 5691 -class_Groups_Olinordered__ab__group__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_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_minus__Min__eq__Max) # label(axiom). [clausify(4015)]. 58.53/58.40 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_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5691,a,5674,a)]. 60.82/60.78 5692 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = B # label(fact_neg__equal__zero) # label(axiom). [clausify(4095)]. 60.82/60.78 5693 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != B # label(fact_neg__equal__zero) # label(axiom). [clausify(4095)]. 60.82/60.78 5694 -class_Groups_Olinordered__ab__group__add(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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(4106)]. 60.82/60.78 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5694,a,5674,a)]. 60.82/60.78 5695 -class_Groups_Olinordered__ab__group__add(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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(4106)]. 60.82/60.78 Derived: 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5695,a,5674,a)]. 60.82/60.78 5696 -class_Groups_Olinordered__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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(4132)]. 60.82/60.78 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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5696,a,5674,a)]. 60.82/60.78 5697 -class_Groups_Olinordered__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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(4132)]. 60.82/60.78 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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5697,a,5674,a)]. 60.82/60.78 5698 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_double__zero__sym) # label(axiom). [clausify(4544)]. 60.82/60.78 5699 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_double__zero__sym) # label(axiom). [clausify(4544)]. 60.82/60.78 5700 class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 60.82/60.78 5701 -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_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(70)]. 60.82/60.78 5702 -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_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(70)]. 60.82/60.78 5703 -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(84)]. 60.82/60.78 5704 -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(677)]. 60.82/60.78 5705 -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(989)]. 60.82/60.78 5706 -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(1105)]. 60.82/60.78 5707 -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(1134)]. 60.82/60.78 5708 -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(1242)]. 60.82/60.78 5709 -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(1319)]. 60.82/60.78 5710 -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(1475)]. 60.82/60.78 5711 -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_Ouminus__class_Ouminus(A),B) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__of__neg) # label(axiom). [clausify(1564)]. 60.82/60.78 5712 -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(1661)]. 60.82/60.78 5713 -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(1950)]. 60.92/60.80 5714 -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(2128)]. 60.92/60.80 5715 -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_Ouminus__class_Ouminus(A),B) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__of__nonpos) # label(axiom). [clausify(2319)]. 60.92/60.80 Derived: -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))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5700,a,5701,a)]. 60.92/60.80 Derived: 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))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5700,a,5702,a)]. 60.92/60.80 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(5700,a,5703,a)]. 60.92/60.80 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(5700,a,5704,a)]. 60.92/60.80 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(5700,a,5705,a)]. 60.92/60.80 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(5700,a,5706,a)]. 60.92/60.80 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(5700,a,5707,a)]. 60.92/60.80 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(5700,a,5708,a)]. 60.92/60.80 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(5700,a,5709,a)]. 60.92/60.80 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(5700,a,5710,a)]. 60.92/60.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5700,a,5712,a)]. 60.92/60.80 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(5700,a,5713,a)]. 61.12/61.04 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(5700,a,5714,a)]. 61.12/61.04 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_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5700,a,5715,a)]. 61.12/61.04 5716 -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(2512)]. 61.12/61.04 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(5716,a,5700,a)]. 61.12/61.04 5717 -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(2512)]. 61.12/61.04 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(5717,a,5700,a)]. 61.12/61.04 5718 -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(2525)]. 61.12/61.04 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(5718,a,5700,a)]. 61.12/61.04 5719 -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(2952)]. 61.12/61.04 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(5719,a,5700,a)]. 61.12/61.04 5720 -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__iff) # label(axiom). [clausify(3049)]. 61.12/61.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(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)). [resolve(5720,a,5700,a)]. 61.12/61.04 5721 -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__iff) # label(axiom). [clausify(3049)]. 61.12/61.04 5722 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(3049)]. 61.73/61.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(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),A),B)). [resolve(5722,a,5700,a)]. 61.73/61.69 5723 -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(3117)]. 61.73/61.69 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(5723,a,5700,a)]. 61.73/61.69 5724 -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(3580)]. 61.73/61.69 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(5724,a,5700,a)]. 61.73/61.69 5725 -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(3593)]. 61.73/61.69 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(5725,a,5700,a)]. 61.73/61.69 5726 -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(3648)]. 61.73/61.69 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(5726,a,5700,a)]. 61.73/61.69 5727 -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(3733)]. 61.73/61.69 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(5727,a,5700,a)]. 61.73/61.69 5728 -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(3930)]. 61.73/61.69 5729 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_zero__less__abs__iff) # label(axiom). [clausify(4126)]. 61.73/61.69 Derived: -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))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5729,a,5700,a)]. 61.73/61.69 5730 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_zero__less__abs__iff) # label(axiom). [clausify(4126)]. 61.73/61.69 Derived: 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))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5730,a,5700,a)]. 61.73/61.69 5731 -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(4173)]. 64.42/64.31 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(5731,a,5700,a)]. 64.42/64.31 5732 -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(4346)]. 64.42/64.31 5733 -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(4463)]. 64.42/64.31 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(5733,a,5700,a)]. 64.42/64.31 5734 -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(4762)]. 64.42/64.31 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(5734,a,5700,a)]. 64.42/64.31 5735 -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(4793)]. 64.42/64.31 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(5735,a,5700,a)]. 64.42/64.31 5736 -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(4896)]. 64.42/64.31 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(5736,a,5700,a)]. 64.42/64.31 5737 class_Groups_Ogroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ogroup__add) # label(axiom). [assumption]. 64.42/64.31 5738 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)) = C # label(fact_minus__add__cancel) # label(axiom). [clausify(73)]. 64.42/64.31 5739 -class_Groups_Ogroup__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),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_minus__add) # label(axiom). [clausify(135)]. 64.42/64.32 5740 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != C | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) = B # label(fact_equation__minus__iff) # label(axiom). [clausify(519)]. 64.42/64.32 5741 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = C | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) != B # label(fact_equation__minus__iff) # label(axiom). [clausify(519)]. 64.42/64.32 5742 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_right__minus) # label(axiom). [clausify(548)]. 64.42/64.32 5743 -class_Groups_Ogroup__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_left__minus) # label(axiom). [clausify(1407)]. 64.42/64.32 5744 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != C | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) = B # label(fact_minus__equation__iff) # label(axiom). [clausify(1427)]. 64.42/64.32 5745 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = C | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) != B # label(fact_minus__equation__iff) # label(axiom). [clausify(1427)]. 64.42/64.32 5746 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_diff__0__right) # label(axiom). [clausify(1449)]. 64.42/64.32 5747 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_neg__0__equal__iff__equal) # label(axiom). [clausify(1557)]. 64.42/64.32 5748 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_neg__0__equal__iff__equal) # label(axiom). [clausify(1557)]. 64.42/64.32 5749 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | C = B # label(fact_right__minus__eq) # label(axiom). [clausify(2380)]. 64.42/64.32 5750 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | C != B # label(fact_right__minus__eq) # label(axiom). [clausify(2380)]. 64.42/64.32 5751 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) = B # label(fact_eq__neg__iff__add__eq__0) # label(axiom). [clausify(2654)]. 64.42/64.32 5752 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | hAPP(c_Groups_Ouminus__class_Ouminus(A),C) != B # label(fact_eq__neg__iff__add__eq__0) # label(axiom). [clausify(2654)]. 64.42/64.32 5753 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) = C # label(fact_add__minus__cancel) # label(axiom). [clausify(2891)]. 64.42/64.32 5754 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_diff__0) # label(axiom). [clausify(3086)]. 64.42/64.32 5755 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = C # label(fact_minus__unique) # label(axiom). [clausify(3294)]. 64.42/64.32 5756 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Ouminus__class_Ouminus(A),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(3317)]. 64.42/64.32 5757 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(3317)]. 64.42/64.32 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)) = B. [resolve(5737,a,5738,a)]. 64.93/64.88 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(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_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5737,a,5739,a)]. 64.93/64.88 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) = A. [resolve(5737,a,5740,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5737,a,5742,a)]. 64.93/64.88 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5737,a,5746,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5737,a,5747,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5737,a,5748,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) = A. [resolve(5737,a,5751,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) != A. [resolve(5737,a,5752,a)]. 64.93/64.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) = B. [resolve(5737,a,5753,a)]. 64.93/64.88 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5737,a,5754,a)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = B. [resolve(5737,a,5755,a)]. 64.93/64.88 5758 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_diff__minus__eq__add) # label(axiom). [clausify(3491)]. 64.93/64.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5758,a,5737,a)]. 64.93/64.88 5759 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Ozero__class_Ozero(A)) # label(fact_minus__zero) # label(axiom). [clausify(3506)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5759,a,5737,a)]. 64.93/64.88 5760 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),C) = B # label(fact_diff__add__cancel) # label(axiom). [clausify(3576)]. 64.93/64.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),B) = A. [resolve(5760,a,5737,a)]. 64.93/64.88 5761 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),C) = B # label(fact_add__diff__cancel) # label(axiom). [clausify(3926)]. 64.93/64.88 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),B) = A. [resolve(5761,a,5737,a)]. 64.93/64.88 5762 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),B) # label(fact_diff__self) # label(axiom). [clausify(4419)]. 64.93/64.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),A). [resolve(5762,a,5737,a)]. 66.83/66.73 5763 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_diff__def) # label(axiom). [clausify(4457)]. 66.83/66.73 5764 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != hAPP(c_Groups_Ouminus__class_Ouminus(A),C) | B = C # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(4482)]. 66.83/66.73 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) | A = B. [resolve(5764,a,5737,a)]. 66.83/66.73 5765 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),C) | B != C # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(4482)]. 66.83/66.73 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) | A != B. [resolve(5765,a,5737,a)]. 66.83/66.73 5766 -class_Groups_Ogroup__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) = B # label(fact_minus__minus) # label(axiom). [clausify(4501)]. 66.83/66.73 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = A. [resolve(5766,a,5737,a)]. 66.83/66.73 5767 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 66.83/66.73 5768 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_SetInterval_Oord__class_OatMost(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_OatLeast(A),B)) # label(fact_image__uminus__atLeast) # label(axiom). [clausify(87)]. 66.83/66.73 5769 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)) # label(fact_max__diff__distrib__left) # label(axiom). [clausify(213)]. 66.83/66.73 5770 -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(647)]. 66.83/66.73 5771 -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(647)]. 66.83/66.73 5772 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_SetInterval_Oord__class_OlessThan(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_OgreaterThan(A),B)) # label(fact_image__uminus__greaterThan) # label(axiom). [clausify(661)]. 66.83/66.73 5773 -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(833)]. 66.83/66.73 5774 -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(833)]. 66.83/66.73 5775 -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(1070)]. 66.92/66.81 5776 -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(1251)]. 66.92/66.81 Derived: hAPP(c_SetInterval_Oord__class_OatMost(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_OatLeast(tc_Int_Oint),A)). [resolve(5767,a,5768,a)]. 66.92/66.81 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)). [resolve(5767,a,5769,a)]. 66.92/66.81 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(5767,a,5770,a)]. 66.92/66.81 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(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_OgreaterThan(tc_Int_Oint),A)). [resolve(5767,a,5772,a)]. 66.92/66.81 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(5767,a,5773,a)]. 66.92/66.81 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(5767,a,5774,a)]. 66.92/66.81 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(5767,a,5775,a)]. 66.92/66.81 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(5767,a,5776,a)]. 66.92/66.81 5777 -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(1490)]. 66.92/66.81 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(5777,a,5767,a)]. 66.92/66.81 5778 -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(1490)]. 66.92/66.81 5779 -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(1693)]. 67.04/66.97 5780 -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(1693)]. 67.04/66.97 5781 -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(1821)]. 67.04/66.97 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(5781,a,5767,a)]. 67.04/66.97 5782 -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(1852)]. 67.04/66.97 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(5782,a,5767,a)]. 67.04/66.97 5783 -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(1852)]. 67.04/66.97 5784 -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(1884)]. 67.04/66.97 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(5784,a,5767,a)]. 67.04/66.97 5785 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_SetInterval_Oord__class_OatLeast(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_OatMost(A),B)) # label(fact_image__uminus__atMost) # label(axiom). [clausify(2028)]. 67.04/66.97 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(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_OatMost(tc_Int_Oint),A)). [resolve(5785,a,5767,a)]. 67.04/66.97 5786 -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(2254)]. 67.04/66.97 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(5786,a,5767,a)]. 67.04/66.97 5787 -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(2254)]. 67.54/67.40 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(5787,a,5767,a)]. 67.54/67.40 5788 -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(2598)]. 67.54/67.40 Derived: c_Fun_Oinj__on(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5788,a,5767,a)]. 67.54/67.40 5789 -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(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(2732)]. 67.54/67.40 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_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5789,a,5767,a)]. 67.54/67.40 5790 -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(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(2732)]. 67.54/67.40 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_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5790,a,5767,a)]. 67.54/67.40 5791 -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(2801)]. 67.54/67.40 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(5791,a,5767,a)]. 67.54/67.40 5792 -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(2801)]. 67.54/67.40 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(5792,a,5767,a)]. 67.54/67.40 5793 -class_Groups_Oordered__ab__group__add(A) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(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_OgreaterThanAtMost(A,C,B)) # label(fact_image__uminus__greaterThanAtMost) # label(axiom). [clausify(2974)]. 67.54/67.40 Derived: hAPP(c_SetInterval_Oord__class_OatLeastLessThan(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_OgreaterThanAtMost(tc_Int_Oint,B,A)). [resolve(5793,a,5767,a)]. 67.54/67.40 5794 -class_Groups_Oordered__ab__group__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),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(3355)]. 68.14/68.01 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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5794,a,5767,a)]. 68.14/68.01 5795 -class_Groups_Oordered__ab__group__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),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__le__0__iff__le) # label(axiom). [clausify(3355)]. 68.14/68.01 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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5795,a,5767,a)]. 68.14/68.01 5796 -class_Groups_Oordered__ab__group__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_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(3372)]. 68.14/68.01 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)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5796,a,5767,a)]. 68.14/68.01 5797 -class_Groups_Oordered__ab__group__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_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(3372)]. 68.14/68.01 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)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5797,a,5767,a)]. 68.14/68.01 5798 -class_Groups_Oordered__ab__group__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),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3858)]. 68.14/68.01 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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5798,a,5767,a)]. 68.14/68.01 5799 -class_Groups_Oordered__ab__group__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),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3858)]. 68.14/68.01 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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5799,a,5767,a)]. 68.14/68.01 5800 -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(3929)]. 68.14/68.01 5801 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f496(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(4307)]. 68.34/68.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f496(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(5801,a,5767,a)]. 68.34/68.24 5802 -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,f496(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(4307)]. 68.34/68.24 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,f496(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(5802,a,5767,a)]. 68.34/68.24 5803 -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(4565)]. 68.34/68.24 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(5803,a,5767,a)]. 68.34/68.24 5804 -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(4586)]. 68.34/68.24 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(5804,a,5767,a)]. 68.34/68.24 5805 -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(4586)]. 68.34/68.24 5806 -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),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(4655)]. 68.34/68.24 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),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5806,a,5767,a)]. 68.34/68.24 5807 -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),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_diff__eq__diff__less) # label(axiom). [clausify(4655)]. 69.95/69.87 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),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5807,a,5767,a)]. 69.95/69.87 5808 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f554(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(4717)]. 69.95/69.87 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f554(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(5808,a,5767,a)]. 69.95/69.87 5809 -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,f554(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(4717)]. 69.95/69.87 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,f554(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(5809,a,5767,a)]. 69.95/69.87 5810 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(A),C),B)) # label(fact_neg__less__iff__less) # label(axiom). [clausify(4747)]. 69.95/69.87 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Int_Oint),B),A)). [resolve(5810,a,5767,a)]. 69.95/69.87 5811 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(A),C),B)) # label(fact_neg__less__iff__less) # label(axiom). [clausify(4747)]. 69.95/69.87 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Int_Oint),B),A)). [resolve(5811,a,5767,a)]. 69.95/69.87 5812 class_Rings_Oring__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 69.95/69.87 5813 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(101)]. 69.95/69.87 5814 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(101)]. 69.95/69.87 5815 -class_Rings_Oring__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_mult__eq__0__iff) # label(axiom). [clausify(101)]. 69.95/69.87 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(5812,a,5813,a)]. 73.05/72.94 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A). [resolve(5812,a,5814,a)]. 73.05/72.94 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(5812,a,5815,a)]. 73.05/72.94 5816 class_Rings_Olinordered__ring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 73.05/72.94 5817 -class_Rings_Olinordered__ring__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(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5818 -class_Rings_Olinordered__ring__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(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5819 -class_Rings_Olinordered__ring__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(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5820 -class_Rings_Olinordered__ring__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(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5821 -class_Rings_Olinordered__ring__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(A),D),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5822 -class_Rings_Olinordered__ring__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(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(102)]. 73.05/72.94 5823 -class_Rings_Olinordered__ring__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),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),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(751)]. 73.05/72.94 5824 -class_Rings_Olinordered__ring__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),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),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(751)]. 73.05/72.94 5825 -class_Rings_Olinordered__ring__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__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__eq(A),D),C)) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(841)]. 73.05/72.94 5826 -class_Rings_Olinordered__ring__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__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__eq(A),D),C)) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(841)]. 73.05/72.94 5827 -class_Rings_Olinordered__ring__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__eq(A),C),D)) | 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))) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(1048)]. 73.05/72.94 5828 -class_Rings_Olinordered__ring__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__eq(A),C),D)) | -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))) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(1048)]. 73.05/72.94 5829 -class_Rings_Olinordered__ring__strict(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_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.94 5830 -class_Rings_Olinordered__ring__strict(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),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.94 5831 -class_Rings_Olinordered__ring__strict(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),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.94 5832 -class_Rings_Olinordered__ring__strict(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),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.94 5833 -class_Rings_Olinordered__ring__strict(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),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.94 5834 -class_Rings_Olinordered__ring__strict(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),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(1089)]. 73.05/72.95 5835 -class_Rings_Olinordered__ring__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),c_Groups_Ozero__class_Ozero(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))) # label(fact_mult__neg__neg) # label(axiom). [clausify(1309)]. 73.05/72.95 5836 -class_Rings_Olinordered__ring__strict(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_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(1424)]. 73.05/72.95 5837 -class_Rings_Olinordered__ring__strict(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_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(1424)]. 73.05/72.95 5838 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != B | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom). [clausify(1424)]. 73.05/72.95 5839 -class_Rings_Olinordered__ring__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),D)) | 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))) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(1481)]. 73.05/72.95 5840 -class_Rings_Olinordered__ring__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),D)) | -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))) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(1481)]. 73.05/72.95 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(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5816,a,5817,a)]. 73.05/72.95 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(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5816,a,5818,a)]. 73.05/72.95 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(tc_Int_Oint),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)),B)). [resolve(5816,a,5819,a)]. 73.05/72.95 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(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5816,a,5820,a)]. 73.05/72.97 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(tc_Int_Oint),C),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5816,a,5821,a)]. 73.05/72.97 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(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(5816,a,5822,a)]. 73.05/72.97 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),B),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),A),B))). [resolve(5816,a,5823,a)]. 73.05/72.97 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),B),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),A),B))). [resolve(5816,a,5824,a)]. 73.05/72.97 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__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__eq(tc_Int_Oint),C),B)). [resolve(5816,a,5825,a)]. 73.05/72.97 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__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__eq(tc_Int_Oint),C),B)). [resolve(5816,a,5826,a)]. 73.05/72.97 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__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))). [resolve(5816,a,5827,a)]. 73.05/72.97 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__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))). [resolve(5816,a,5828,a)]. 73.05/72.97 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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5816,a,5830,a)]. 73.05/72.97 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_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_Otimes__class_Otimes(tc_Int_Oint),B),A))). [resolve(5816,a,5831,a)]. 73.14/73.07 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_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_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5816,a,5834,a)]. 73.14/73.07 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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(5816,a,5835,a)]. 73.14/73.07 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(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)))). [resolve(5816,a,5836,a)]. 73.14/73.07 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(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)))). [resolve(5816,a,5837,a)]. 73.14/73.07 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(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),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)))). [resolve(5816,a,5838,a)]. 73.14/73.07 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),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(5816,a,5840,a)]. 73.14/73.07 5841 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.07 5842 -class_Rings_Olinordered__ring__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),D)) | 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))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.07 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),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(5842,a,5816,a)]. 73.14/73.07 5843 -class_Rings_Olinordered__ring__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),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.09 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),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(5843,a,5816,a)]. 73.14/73.09 5844 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.09 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),B),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(5844,a,5816,a)]. 73.14/73.09 5845 -class_Rings_Olinordered__ring__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)),B)) | -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))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.09 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)),A)) | -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))). [resolve(5845,a,5816,a)]. 73.14/73.09 5846 -class_Rings_Olinordered__ring__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),D)) | -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))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1895)]. 73.14/73.09 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)) | -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))). [resolve(5846,a,5816,a)]. 73.14/73.09 5847 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | 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)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.14/73.09 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)),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)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5847,a,5816,a)]. 73.14/73.09 5848 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | 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_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.35/73.23 5849 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(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)),B)) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.35/73.23 5850 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(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))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.35/73.23 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)),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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5850,a,5816,a)]. 73.35/73.23 5851 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -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_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.35/73.23 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)),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)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5851,a,5816,a)]. 73.35/73.23 5852 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(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),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1925)]. 73.35/73.23 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)),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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5852,a,5816,a)]. 73.35/73.23 5853 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(2400)]. 73.35/73.23 Derived: 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),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5853,a,5816,a)]. 73.35/73.23 5854 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(2400)]. 74.64/74.51 Derived: 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),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5854,a,5816,a)]. 74.64/74.51 5855 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(2400)]. 74.64/74.51 Derived: 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),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5855,a,5816,a)]. 74.64/74.51 5856 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__strict__right__mono__neg) # label(axiom). [clausify(2417)]. 74.64/74.51 5857 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__strict__left__mono__neg) # label(axiom). [clausify(3287)]. 74.64/74.51 5858 -class_Rings_Olinordered__ring__strict(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_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(4778)]. 74.64/74.51 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_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5858,a,5816,a)]. 74.64/74.51 5859 -class_Rings_Olinordered__ring__strict(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_Groups_Otimes__class_Otimes(A),C),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(4778)]. 74.64/74.51 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_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5859,a,5816,a)]. 74.64/74.51 5860 -class_Rings_Olinordered__ring__strict(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_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(4778)]. 74.64/74.51 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5860,a,5816,a)]. 76.94/76.83 5861 class_Rings_Olinordered__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 76.94/76.83 5862 -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(106)]. 76.94/76.83 5863 -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(123)]. 76.94/76.83 5864 -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(601)]. 76.94/76.83 5865 -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(1350)]. 76.94/76.83 5866 -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(1374)]. 76.94/76.83 5867 -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(1671)]. 76.94/76.83 5868 -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(1877)]. 76.94/76.83 5869 -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(2681)]. 76.94/76.83 5870 -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(2882)]. 76.94/76.83 5871 -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(3385)]. 76.94/76.83 5872 -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(3518)]. 76.94/76.83 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),B),A)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5861,a,5862,a)]. 76.94/76.83 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(5861,a,5863,a)]. 76.94/76.83 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(5861,a,5864,a)]. 76.94/76.83 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(5861,a,5865,a)]. 76.94/76.83 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(5861,a,5866,a)]. 76.94/76.88 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(5861,a,5867,a)]. 76.94/76.88 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(5861,a,5868,a)]. 76.94/76.88 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(5861,a,5869,a)]. 76.94/76.88 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(5861,a,5870,a)]. 76.94/76.88 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(5861,a,5871,a)]. 76.94/76.88 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(5861,a,5872,a)]. 76.94/76.88 5873 -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(3744)]. 77.33/77.23 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(5873,a,5861,a)]. 77.33/77.23 5874 -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(3947)]. 77.33/77.23 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(5874,a,5861,a)]. 77.33/77.23 5875 -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(4047)]. 77.33/77.23 5876 -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(4319)]. 77.33/77.23 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(5876,a,5861,a)]. 77.33/77.23 5877 -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(4357)]. 77.33/77.23 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(5877,a,5861,a)]. 77.33/77.28 5878 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 77.33/77.28 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),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(5878,a,5862,a)]. 77.33/77.28 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(5878,a,5863,a)]. 77.33/77.28 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))). [resolve(5878,a,5864,a)]. 77.33/77.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5878,a,5865,a)]. 77.33/77.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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))) | -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),B),C)). [resolve(5878,a,5866,a)]. 77.33/77.28 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),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(5878,a,5867,a)]. 77.33/77.28 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(5878,a,5868,a)]. 77.33/77.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_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(5878,a,5869,a)]. 77.33/77.28 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(5878,a,5870,a)]. 77.45/77.35 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(5878,a,5871,a)]. 77.45/77.35 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)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)). [resolve(5878,a,5872,a)]. 77.45/77.35 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))). [resolve(5878,a,5873,a)]. 77.45/77.35 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(5878,a,5874,a)]. 77.45/77.36 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)),B)) | 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))). [resolve(5878,a,5876,a)]. 77.45/77.36 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),A),C))) | -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)). [resolve(5878,a,5877,a)]. 77.45/77.36 5879 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 77.45/77.36 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),B),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5879,a,5862,a)]. 77.45/77.36 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(5879,a,5863,a)]. 77.45/77.36 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(5879,a,5865,a)]. 77.45/77.36 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(5879,a,5866,a)]. 77.45/77.36 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(5879,a,5867,a)]. 77.45/77.36 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(5879,a,5868,a)]. 77.45/77.36 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(5879,a,5869,a)]. 77.45/77.36 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(5879,a,5870,a)]. 77.45/77.36 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(5879,a,5871,a)]. 77.45/77.36 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(5879,a,5872,a)]. 77.45/77.36 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(5879,a,5874,a)]. 79.44/79.33 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(5879,a,5876,a)]. 79.44/79.33 5880 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 79.44/79.33 5881 -class_Groups_Omonoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_List_Omonoid__add__class_Olistsum(A),C)) # label(fact_listsum__simps_I2_J) # label(axiom). [clausify(131)]. 79.44/79.33 5882 -class_Groups_Omonoid__add(A) | c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)) = c_List_Omonoid__add__class_Olistsum(A) # label(fact_listsum__foldl) # label(axiom). [clausify(202)]. 79.44/79.33 5883 -class_Groups_Omonoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_member(B),f176(C,D,E,B,A)),hAPP(c_List_Oset(B),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E)))) # label(fact_listsum__mono) # label(axiom). [clausify(1704)]. 79.44/79.33 5884 -class_Groups_Omonoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f176(C,B,D,E,A))),hAPP(C,f176(C,B,D,E,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),D)))) # label(fact_listsum__mono) # label(axiom). [clausify(1704)]. 79.44/79.33 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),B)). [resolve(5880,a,5881,a)]. 79.44/79.33 Derived: 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)) = c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat). [resolve(5880,a,5882,a)]. 79.44/79.33 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f176(B,C,D,A,tc_Nat_Onat)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)))). [resolve(5880,a,5883,a)]. 79.44/79.33 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f176(B,A,C,D,tc_Nat_Onat))),hAPP(B,f176(B,A,C,D,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),C)))). [resolve(5880,a,5884,a)]. 79.44/79.33 5885 class_Groups_Omonoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__add) # label(axiom). [assumption]. 79.44/79.33 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),B)). [resolve(5885,a,5881,a)]. 79.44/79.33 Derived: c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_List_Omonoid__add__class_Olistsum(tc_Int_Oint). [resolve(5885,a,5882,a)]. 79.94/79.83 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f176(B,C,D,A,tc_Int_Oint)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)))). [resolve(5885,a,5883,a)]. 79.94/79.83 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f176(B,A,C,D,tc_Int_Oint))),hAPP(B,f176(B,A,C,D,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),C)))). [resolve(5885,a,5884,a)]. 79.94/79.83 5886 -class_Groups_Omonoid__add(A) | hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_foldl__absorb0) # label(axiom). [clausify(2067)]. 79.94/79.83 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),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)),B)). [resolve(5886,a,5880,a)]. 79.94/79.83 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5886,a,5885,a)]. 79.94/79.83 5887 -class_Groups_Omonoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Oappend(A),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),B)),hAPP(c_List_Omonoid__add__class_Olistsum(A),C)) # label(fact_listsum__append) # label(axiom). [clausify(2214)]. 79.94/79.83 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),B)). [resolve(5887,a,5880,a)]. 79.94/79.83 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),B)). [resolve(5887,a,5885,a)]. 79.94/79.83 5888 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_add__0__left) # label(axiom). [clausify(2791)]. 79.94/79.83 5889 -class_Groups_Omonoid__add(A) | hBOOL(hAPP(hAPP(c_member(B),f390(C,D,E,B,A)),hAPP(c_List_Oset(B),E))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),hAPP(c_List_Ofilter(B,D),E))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E)) # label(fact_listsum__map__filter) # label(axiom). [clausify(3391)]. 79.94/79.83 Derived: hBOOL(hAPP(hAPP(c_member(A),f390(B,C,D,A,tc_Nat_Onat)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(c_List_Ofilter(A,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)). [resolve(5889,a,5880,a)]. 79.94/79.83 Derived: hBOOL(hAPP(hAPP(c_member(A),f390(B,C,D,A,tc_Int_Oint)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(c_List_Ofilter(A,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)). [resolve(5889,a,5885,a)]. 80.34/80.23 5890 -class_Groups_Omonoid__add(A) | -hBOOL(hAPP(B,f390(C,B,D,E,A))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),hAPP(c_List_Ofilter(E,B),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),D)) # label(fact_listsum__map__filter) # label(axiom). [clausify(3391)]. 80.34/80.23 Derived: -hBOOL(hAPP(A,f390(B,A,C,D,tc_Nat_Onat))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),C)). [resolve(5890,a,5880,a)]. 80.34/80.23 Derived: -hBOOL(hAPP(A,f390(B,A,C,D,tc_Int_Oint))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),C)). [resolve(5890,a,5885,a)]. 80.34/80.23 5891 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(B,f390(B,C,D,E,A)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),hAPP(c_List_Ofilter(E,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),D)) # label(fact_listsum__map__filter) # label(axiom). [clausify(3391)]. 80.34/80.23 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(A,f390(A,B,C,D,tc_Nat_Onat)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),C)). [resolve(5891,a,5880,a)]. 80.34/80.23 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(A,f390(A,B,C,D,tc_Int_Oint)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),C)). [resolve(5891,a,5885,a)]. 80.34/80.23 5892 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_List_Omonoid__add__class_Olistsum(A),c_List_Olist_ONil(A)) # label(fact_listsum__simps_I1_J) # label(axiom). [clausify(3920)]. 80.34/80.23 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),c_List_Olist_ONil(tc_Nat_Onat)). [resolve(5892,a,5880,a)]. 80.34/80.23 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),c_List_Olist_ONil(tc_Int_Oint)). [resolve(5892,a,5885,a)]. 80.34/80.23 5893 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add__0__right) # label(axiom). [clausify(3973)]. 80.34/80.23 5894 class_Groups_Omonoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Omonoid__add) # label(axiom). [assumption]. 80.34/80.23 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5894,a,5881,a)]. 80.34/80.23 Derived: c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral). [resolve(5894,a,5882,a)]. 80.34/80.23 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f176(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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),C),D))),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)))). [resolve(5894,a,5883,a)]. 80.34/80.23 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,f176(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),hAPP(B,f176(B,A,C,D,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),C)))). [resolve(5894,a,5884,a)]. 80.55/80.43 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,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),hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5894,a,5886,a)]. 80.55/80.43 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5894,a,5887,a)]. 80.55/80.43 Derived: hBOOL(hAPP(hAPP(c_member(A),f390(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(A,C),D))) = 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)). [resolve(5894,a,5889,a)]. 80.55/80.43 Derived: -hBOOL(hAPP(A,f390(B,A,C,D,tc_Code__Numeral_Ocode__numeral))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5894,a,5890,a)]. 80.55/80.43 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(A,f390(A,B,C,D,tc_Code__Numeral_Ocode__numeral)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5894,a,5891,a)]. 80.55/80.43 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)). [resolve(5894,a,5892,a)]. 80.55/80.43 5895 -class_Groups_Omonoid__add(A) | hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),C,c_Groups_Ozero__class_Ozero(A))) # label(fact_foldl__foldr1__lemma) # label(axiom). [clausify(4315)]. 80.55/80.43 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5895,a,5880,a)]. 80.55/80.43 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B,c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5895,a,5885,a)]. 80.55/80.43 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,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_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5895,a,5894,a)]. 88.63/88.49 5896 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(c_COMBK(A,B),c_Groups_Ozero__class_Ozero(A))),C)) # label(fact_listsum__0) # label(axiom). [clausify(4508)]. 88.63/88.49 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(5896,a,5880,a)]. 88.63/88.49 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = 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),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B)). [resolve(5896,a,5885,a)]. 88.63/88.49 Derived: 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(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(5896,a,5894,a)]. 88.63/88.49 5897 -class_Groups_Omonoid__add(A) | c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),B,c_Groups_Ozero__class_Ozero(A)) = hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_foldl__foldr1) # label(axiom). [clausify(4859)]. 88.63/88.49 Derived: c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A,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),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5897,a,5880,a)]. 88.63/88.49 Derived: c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5897,a,5885,a)]. 88.63/88.49 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,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5897,a,5894,a)]. 88.63/88.49 5898 -class_Groups_Ocancel__ab__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C = D # label(fact_add__imp__eq) # label(axiom). [clausify(2851)]. 88.63/88.49 5899 class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 88.63/88.49 5900 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 88.63/88.49 5901 class_Groups_Ocancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 88.63/88.49 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C) | B = C. [resolve(5898,a,5899,a)]. 88.63/88.49 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C) | B = C. [resolve(5898,a,5900,a)]. 88.63/88.49 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),A),C) | B = C. [resolve(5898,a,5901,a)]. 88.63/88.49 5902 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | C != D | E = F | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),F)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)) # label(fact_add__scale__eq__noteq) # label(axiom). [clausify(1369)]. 89.05/88.94 5903 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) # label(arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 89.05/88.94 5904 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Int_Oint) # label(arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 89.05/88.94 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)). [resolve(5902,a,5903,a)]. 89.05/88.94 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)). [resolve(5902,a,5904,a)]. 89.05/88.94 5905 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != B | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__0__iff) # label(axiom). [clausify(1694)]. 89.05/88.94 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5905,a,5904,a)]. 89.05/88.94 5906 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_add__0__iff) # label(axiom). [clausify(1694)]. 89.05/88.94 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5906,a,5903,a)]. 89.05/88.94 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5906,a,5904,a)]. 89.05/88.94 5907 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | C != E # label(fact_crossproduct__noteq) # label(axiom). [clausify(3089)]. 89.05/88.94 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) | B != D. [resolve(5907,a,5903,a)]. 89.05/88.94 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) | B != D. [resolve(5907,a,5904,a)]. 89.05/88.94 5908 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | D != B # label(fact_crossproduct__noteq) # label(axiom). [clausify(3089)]. 89.05/88.94 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) | C != A. [resolve(5908,a,5903,a)]. 89.34/89.25 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) | C != A. [resolve(5908,a,5904,a)]. 89.34/89.25 5909 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | C = E | D = B # label(fact_crossproduct__noteq) # label(axiom). [clausify(3089)]. 89.34/89.25 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),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) | B = D | C = A. [resolve(5909,a,5903,a)]. 89.34/89.25 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),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) | B = D | C = A. [resolve(5909,a,5904,a)]. 89.34/89.25 5910 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | B = D | C = E # label(fact_crossproduct__eq) # label(axiom). [clausify(3832)]. 89.34/89.25 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),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) | A = C | B = D. [resolve(5910,a,5903,a)]. 89.34/89.25 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),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) | A = C | B = D. [resolve(5910,a,5904,a)]. 89.34/89.25 5911 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | B != D # label(fact_crossproduct__eq) # label(axiom). [clausify(3832)]. 89.34/89.25 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)) | A != C. [resolve(5911,a,5903,a)]. 89.34/89.25 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),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)) | A != C. [resolve(5911,a,5904,a)]. 94.73/94.63 5912 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(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),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) | C != E # label(fact_crossproduct__eq) # label(axiom). [clausify(3832)]. 94.73/94.63 5913 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]. 94.73/94.63 5914 -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(165)]. 94.73/94.63 5915 -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(321)]. 94.73/94.63 5916 -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(646)]. 94.73/94.63 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),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(5913,a,5914,a)]. 94.73/94.63 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(5913,a,5915,a)]. 94.73/94.63 5917 -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) != B | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(903)]. 94.73/94.63 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),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A | 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). [resolve(5917,a,5913,a)]. 94.94/94.81 5918 -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) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(903)]. 94.94/94.81 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) = A | 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(5918,a,5913,a)]. 94.94/94.81 5919 -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) = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(903)]. 94.94/94.81 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) = 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). [resolve(5919,a,5913,a)]. 94.94/94.81 5920 -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(1357)]. 94.94/94.81 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(5920,a,5913,a)]. 94.94/94.81 5921 -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(1932)]. 94.94/94.81 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(5921,a,5913,a)]. 94.94/94.81 5922 -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(1978)]. 95.03/94.91 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(5922,a,5913,a)]. 95.03/94.91 5923 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 95.03/94.91 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__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(5923,a,5920,a)]. 95.03/94.91 5924 -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),f264(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(2286)]. 95.03/94.91 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),f264(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(5924,a,5913,a)]. 95.03/94.91 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),f264(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(5924,a,5923,a)]. 95.03/94.91 5925 -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,f264(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(2286)]. 95.03/94.91 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,f264(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(5925,a,5913,a)]. 95.45/95.31 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,f264(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(5925,a,5923,a)]. 95.45/95.31 5926 -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(2796)]. 95.45/95.31 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(5926,a,5913,a)]. 95.45/95.31 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(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_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5926,a,5923,a)]. 95.45/95.31 5927 -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(2853)]. 95.45/95.31 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(5927,a,5913,a)]. 95.45/95.31 5928 -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(2865)]. 95.45/95.31 5929 -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(3135)]. 95.45/95.31 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(5929,a,5913,a)]. 95.45/95.31 5930 -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(3534)]. 95.84/95.76 5931 -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(4141)]. 95.84/95.76 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(5931,a,5913,a)]. 95.84/95.76 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(5931,a,5923,a)]. 95.84/95.76 5932 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 95.84/95.76 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),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(5932,a,5914,a)]. 95.84/95.76 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(5932,a,5915,a)]. 95.84/95.76 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) != A | 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). [resolve(5932,a,5917,a)]. 95.84/95.76 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) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B). [resolve(5932,a,5918,a)]. 95.84/95.76 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) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B). [resolve(5932,a,5919,a)]. 95.84/95.76 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(5932,a,5920,a)]. 97.94/97.81 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(5932,a,5921,a)]. 97.94/97.81 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(5932,a,5922,a)]. 97.94/97.81 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),f264(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(5932,a,5924,a)]. 97.94/97.81 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,f264(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(5932,a,5925,a)]. 97.94/97.81 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(5932,a,5926,a)]. 97.94/97.81 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(5932,a,5927,a)]. 97.94/97.81 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(5932,a,5929,a)]. 97.94/97.81 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(5932,a,5931,a)]. 97.94/97.81 5933 class_Divides_Osemiring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Osemiring__div) # label(axiom). [assumption]. 97.94/97.81 5934 -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_Groups_Otimes__class_Otimes(A),D),C) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(170)]. 97.94/97.81 5935 -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_Groups_Otimes__class_Otimes(A),D),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(170)]. 97.94/97.81 5936 -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(239)]. 97.94/97.81 5937 -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(315)]. 97.94/97.81 5938 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | c_Divides_Odiv__class_Omod(A,C,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_dvd__eq__mod__eq__0) # label(axiom). [clausify(360)]. 97.94/97.81 5939 -class_Divides_Osemiring__div(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | c_Divides_Odiv__class_Omod(A,C,B) != c_Groups_Ozero__class_Ozero(A) # label(fact_dvd__eq__mod__eq__0) # label(axiom). [clausify(360)]. 97.94/97.81 5940 -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(374)]. 97.94/97.81 5941 -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(393)]. 97.94/97.81 5942 -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(729)]. 97.94/97.81 5943 -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(774)]. 97.94/97.81 5944 -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(804)]. 97.94/97.81 5945 -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(935)]. 97.94/97.81 5946 -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(998)]. 97.94/97.81 5947 -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(1179)]. 97.94/97.81 5948 -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(1204)]. 97.94/97.82 5949 -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(1226)]. 97.94/97.82 5950 -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(1526)]. 97.94/97.82 5951 -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(1527)]. 97.94/97.82 5952 -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(1542)]. 97.94/97.82 5953 -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(1606)]. 97.94/97.82 5954 -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(1673)]. 97.94/97.82 5955 -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(1768)]. 97.94/97.82 5956 -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(1795)]. 97.94/97.82 5957 -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),B),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),c_Divides_Odiv__class_Omod(A,C,D))) # label(fact_dvd__mod) # label(axiom). [clausify(1872)]. 97.94/97.82 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_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B). [resolve(5933,a,5934,a)]. 97.94/97.82 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_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B). [resolve(5933,a,5935,a)]. 97.94/97.82 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(5933,a,5937,a)]. 97.94/97.82 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5933,a,5938,a)]. 97.94/97.85 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5933,a,5939,a)]. 97.94/97.85 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(5933,a,5940,a)]. 97.94/97.85 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(5933,a,5941,a)]. 97.94/97.85 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(5933,a,5942,a)]. 97.94/97.85 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(5933,a,5943,a)]. 97.94/97.85 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(5933,a,5944,a)]. 97.94/97.85 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),A). [resolve(5933,a,5946,a)]. 97.94/97.85 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(5933,a,5947,a)]. 97.94/97.85 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(5933,a,5949,a)]. 97.94/97.85 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(5933,a,5950,a)]. 97.94/97.85 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(5933,a,5951,a)]. 97.94/97.85 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(5933,a,5953,a)]. 97.94/97.85 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(5933,a,5954,a)]. 97.94/97.85 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(5933,a,5955,a)]. 97.94/97.85 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(5933,a,5956,a)]. 97.94/97.85 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C))). [resolve(5933,a,5957,a)]. 98.24/98.10 5958 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | c_Divides_Odiv__class_Omod(A,c_Divides_Odiv__class_Omod(A,D,C),B) = c_Divides_Odiv__class_Omod(A,D,B) # label(fact_mod__mod__cancel) # label(axiom). [clausify(2056)]. 98.24/98.10 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B),A) = c_Divides_Odiv__class_Omod(tc_Int_Oint,C,A). [resolve(5958,a,5933,a)]. 98.24/98.10 5959 -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(2133)]. 98.24/98.10 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(5959,a,5933,a)]. 98.24/98.10 5960 -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(2193)]. 98.24/98.10 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(5960,a,5933,a)]. 98.24/98.10 5961 -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)]. 98.24/98.10 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(5961,a,5933,a)]. 98.24/98.10 5962 -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(2365)]. 98.24/98.10 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(5962,a,5933,a)]. 98.24/98.10 5963 -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),B),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),c_Divides_Odiv__class_Omod(A,D,C))) # label(fact_dvd__mod__iff) # label(axiom). [clausify(2481)]. 98.24/98.10 5964 -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),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),c_Divides_Odiv__class_Omod(A,D,C))) # label(fact_dvd__mod__iff) # label(axiom). [clausify(2481)]. 98.24/98.10 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B))). [resolve(5964,a,5933,a)]. 98.24/98.10 5965 -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(2749)]. 98.25/98.17 5966 class_Divides_Osemiring__div(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Divides_Osemiring__div) # label(axiom). [assumption]. 98.25/98.17 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_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5966,a,5934,a)]. 98.25/98.17 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_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5966,a,5935,a)]. 98.25/98.17 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(5966,a,5936,a)]. 98.25/98.17 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(5966,a,5937,a)]. 98.25/98.17 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5966,a,5938,a)]. 98.25/98.17 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5966,a,5939,a)]. 98.25/98.17 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(5966,a,5940,a)]. 98.25/98.17 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(5966,a,5941,a)]. 98.25/98.17 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(5966,a,5942,a)]. 98.25/98.17 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(5966,a,5943,a)]. 98.25/98.17 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(5966,a,5944,a)]. 98.25/98.18 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(5966,a,5945,a)]. 98.25/98.18 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(5966,a,5946,a)]. 98.25/98.18 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(5966,a,5947,a)]. 98.25/98.18 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(5966,a,5948,a)]. 98.25/98.18 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(5966,a,5949,a)]. 98.25/98.18 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(5966,a,5950,a)]. 98.25/98.18 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(5966,a,5951,a)]. 98.25/98.18 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(5966,a,5953,a)]. 98.25/98.18 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(5966,a,5954,a)]. 98.25/98.18 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(5966,a,5955,a)]. 98.25/98.18 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(5966,a,5956,a)]. 98.35/98.25 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,C))). [resolve(5966,a,5957,a)]. 98.35/98.25 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B),A) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,A). [resolve(5966,a,5958,a)]. 98.35/98.25 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(5966,a,5959,a)]. 98.35/98.25 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(5966,a,5960,a)]. 98.35/98.25 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(5966,a,5961,a)]. 98.35/98.25 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(5966,a,5962,a)]. 98.35/98.25 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B))). [resolve(5966,a,5964,a)]. 98.35/98.25 5967 -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(2958)]. 98.35/98.25 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(5967,a,5933,a)]. 98.35/98.25 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(5967,a,5966,a)]. 98.35/98.25 5968 -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(3032)]. 98.35/98.25 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(5968,a,5933,a)]. 98.35/98.25 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(5968,a,5966,a)]. 98.56/98.42 5969 -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(3276)]. 98.56/98.42 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(5969,a,5933,a)]. 98.56/98.42 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(5969,a,5966,a)]. 98.56/98.42 5970 -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(3290)]. 98.56/98.42 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(5970,a,5933,a)]. 98.56/98.42 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(5970,a,5966,a)]. 98.56/98.42 5971 -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(3342)]. 98.56/98.42 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(5971,a,5933,a)]. 98.56/98.42 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(5971,a,5966,a)]. 98.56/98.42 5972 -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(3424)]. 98.56/98.42 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(5972,a,5933,a)]. 98.56/98.42 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(5972,a,5966,a)]. 98.56/98.46 5973 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),c_Divides_Odiv__class_Omod(A,C,D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_dvd__mod__imp__dvd) # label(axiom). [clausify(3431)]. 98.56/98.46 5974 -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(3449)]. 98.56/98.46 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(5974,a,5933,a)]. 98.56/98.46 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(5974,a,5966,a)]. 98.56/98.46 5975 -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(3486)]. 98.56/98.46 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(5975,a,5933,a)]. 98.56/98.46 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(5975,a,5966,a)]. 98.56/98.46 5976 -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(3493)]. 98.56/98.46 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(5976,a,5933,a)]. 98.56/98.46 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(5976,a,5966,a)]. 98.56/98.46 5977 class_Divides_Osemiring__div(tc_Nat_Onat) # label(arity_Nat__Onat__Divides_Osemiring__div) # label(axiom). [assumption]. 98.56/98.46 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B). [resolve(5977,a,5934,a)]. 98.56/98.48 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B). [resolve(5977,a,5935,a)]. 98.56/98.48 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(5977,a,5936,a)]. 98.56/98.48 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(5977,a,5937,a)]. 98.56/98.48 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5977,a,5938,a)]. 98.56/98.48 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,A) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5977,a,5939,a)]. 98.56/98.48 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(5977,a,5940,a)]. 98.56/98.48 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(5977,a,5941,a)]. 98.56/98.48 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(5977,a,5942,a)]. 98.56/98.48 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(5977,a,5943,a)]. 98.56/98.48 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(5977,a,5944,a)]. 98.56/98.48 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(5977,a,5945,a)]. 98.56/98.48 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(5977,a,5946,a)]. 98.56/98.48 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(5977,a,5947,a)]. 98.56/98.48 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(5977,a,5948,a)]. 98.56/98.48 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(5977,a,5949,a)]. 98.56/98.50 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(5977,a,5950,a)]. 98.56/98.50 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(5977,a,5951,a)]. 98.56/98.50 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(5977,a,5953,a)]. 98.56/98.50 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(5977,a,5954,a)]. 98.56/98.50 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(5977,a,5955,a)]. 98.56/98.50 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(5977,a,5956,a)]. 98.56/98.50 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C))). [resolve(5977,a,5957,a)]. 98.56/98.50 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B),A) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,A). [resolve(5977,a,5958,a)]. 98.56/98.50 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(5977,a,5959,a)]. 98.56/98.50 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(5977,a,5960,a)]. 98.56/98.50 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(5977,a,5961,a)]. 98.56/98.50 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B))). [resolve(5977,a,5964,a)]. 98.56/98.50 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5977,a,5967,a)]. 98.56/98.50 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),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,C)),B). [resolve(5977,a,5968,a)]. 98.56/98.50 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(5977,a,5969,a)]. 98.65/98.57 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(5977,a,5970,a)]. 98.65/98.57 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(5977,a,5971,a)]. 98.65/98.57 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(5977,a,5972,a)]. 98.65/98.57 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(5977,a,5974,a)]. 98.65/98.57 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(5977,a,5975,a)]. 98.65/98.57 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(5977,a,5976,a)]. 98.65/98.57 5978 -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(3535)]. 98.65/98.57 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(5978,a,5933,a)]. 98.65/98.57 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(5978,a,5966,a)]. 98.65/98.57 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(5978,a,5977,a)]. 98.65/98.57 5979 -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(3592)]. 98.65/98.57 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(5979,a,5933,a)]. 98.65/98.57 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(5979,a,5966,a)]. 98.65/98.57 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(5979,a,5977,a)]. 98.65/98.57 5980 -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(3645)]. 98.86/98.79 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(5980,a,5933,a)]. 98.86/98.79 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_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(5980,a,5966,a)]. 98.86/98.79 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(5980,a,5977,a)]. 98.86/98.79 5981 -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(3741)]. 98.86/98.79 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(5981,a,5933,a)]. 98.86/98.79 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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A)),A) = B. [resolve(5981,a,5966,a)]. 98.86/98.79 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),A) = B. [resolve(5981,a,5977,a)]. 98.86/98.79 5982 -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(3815)]. 98.86/98.79 5983 -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(3815)]. 98.86/98.79 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(5983,a,5933,a)]. 98.86/98.79 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(5983,a,5966,a)]. 98.86/98.79 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(5983,a,5977,a)]. 98.86/98.79 5984 -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(3978)]. 98.86/98.79 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(5984,a,5933,a)]. 99.13/99.07 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(5984,a,5966,a)]. 99.13/99.07 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(5984,a,5977,a)]. 99.13/99.07 5985 -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(4243)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5985,a,5933,a)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5985,a,5966,a)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(5985,a,5977,a)]. 99.13/99.07 5986 -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(4272)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5986,a,5933,a)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5986,a,5966,a)]. 99.13/99.07 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5986,a,5977,a)]. 99.13/99.07 5987 -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_Groups_Otimes__class_Otimes(A),D),B) != C | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B) = D # label(fact_dvd__div__eq__mult) # label(axiom). [clausify(4322)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A) != B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A) = C. [resolve(5987,a,5933,a)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A) != B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A) = C. [resolve(5987,a,5966,a)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A) != B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A) = C. [resolve(5987,a,5977,a)]. 99.13/99.07 5988 -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_Groups_Otimes__class_Otimes(A),D),B) = C | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B) != D # label(fact_dvd__div__eq__mult) # label(axiom). [clausify(4322)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A) != C. [resolve(5988,a,5933,a)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),B),A) != C. [resolve(5988,a,5966,a)]. 99.13/99.07 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A) != C. [resolve(5988,a,5977,a)]. 99.13/99.07 5989 -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(4358)]. 99.26/99.20 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(5989,a,5933,a)]. 99.26/99.20 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(5989,a,5966,a)]. 99.26/99.20 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(5989,a,5977,a)]. 99.26/99.20 5990 -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(4404)]. 99.26/99.20 5991 -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(4426)]. 99.26/99.20 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(5991,a,5933,a)]. 99.26/99.20 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(5991,a,5966,a)]. 99.26/99.20 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(5991,a,5977,a)]. 99.26/99.20 5992 -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(4549)]. 99.26/99.20 5993 -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(4561)]. 99.26/99.20 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(5993,a,5933,a)]. 99.26/99.20 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(5993,a,5966,a)]. 99.26/99.20 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(5993,a,5977,a)]. 99.26/99.20 5994 -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(4568)]. 99.26/99.20 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(5994,a,5933,a)]. 99.26/99.20 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(5994,a,5966,a)]. 99.45/99.36 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(5994,a,5977,a)]. 99.45/99.36 5995 -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),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B))) # label(fact_div__dvd__div) # label(axiom). [clausify(4697)]. 99.45/99.36 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5995,a,5933,a)]. 99.45/99.36 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5995,a,5966,a)]. 99.45/99.36 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),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5995,a,5977,a)]. 99.45/99.36 5996 -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),B),D)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B))) # label(fact_div__dvd__div) # label(axiom). [clausify(4697)]. 99.45/99.36 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5996,a,5933,a)]. 99.45/99.36 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5996,a,5966,a)]. 99.45/99.36 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),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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),C),A))). [resolve(5996,a,5977,a)]. 99.45/99.36 5997 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | c_Divides_Odiv__class_Omod(A,C,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_dvd__imp__mod__0) # label(axiom). [clausify(4715)]. 99.45/99.36 5998 -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(4803)]. 102.34/102.28 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(5998,a,5933,a)]. 102.34/102.28 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(5998,a,5966,a)]. 102.34/102.28 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(5998,a,5977,a)]. 102.34/102.28 5999 -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(4836)]. 102.34/102.28 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(5999,a,5933,a)]. 102.34/102.28 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(5999,a,5966,a)]. 102.34/102.28 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(5999,a,5977,a)]. 102.34/102.28 6000 class_Rings_Oordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring) # label(axiom). [assumption]. 102.34/102.28 6001 -class_Rings_Oordered__ring(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_Otimes__class_Otimes(A),B),C))) # label(fact_split__mult__pos__le) # label(axiom). [clausify(184)]. 102.34/102.28 6002 -class_Rings_Oordered__ring(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),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_split__mult__pos__le) # label(axiom). [clausify(184)]. 102.34/102.28 6003 -class_Rings_Oordered__ring(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),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__nonpos__nonpos) # label(axiom). [clausify(226)]. 102.34/102.28 6004 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),E),B)),C)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(482)]. 102.34/102.28 6005 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),E),B)),C)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(482)]. 102.34/102.28 6006 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_le__add__iff2) # label(axiom). [clausify(1721)]. 102.34/102.28 6007 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)),E)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E)),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)),F))) # label(fact_le__add__iff2) # label(axiom). [clausify(1721)]. 102.34/102.28 6008 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__right__mono__neg) # label(axiom). [clausify(2758)]. 102.34/102.28 6009 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E)),C)),D)),F)) # label(fact_le__add__iff1) # label(axiom). [clausify(2910)]. 102.34/102.28 6010 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E)),C)),D)),F)) # label(fact_le__add__iff1) # label(axiom). [clausify(2910)]. 102.34/102.28 6011 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E)),C)),D)),F)) # label(fact_less__add__iff1) # label(axiom). [clausify(3024)]. 102.34/102.28 6012 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E)),C)),D)),F)) # label(fact_less__add__iff1) # label(axiom). [clausify(3024)]. 102.34/102.28 6013 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__left__mono__neg) # label(axiom). [clausify(3563)]. 102.34/102.28 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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))). [resolve(6000,a,6001,a)]. 102.34/102.28 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),D),A)),B)),E))). [resolve(6000,a,6004,a)]. 102.34/102.28 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),D),A)),B)),E))). [resolve(6000,a,6005,a)]. 102.34/102.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | 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),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(6000,a,6006,a)]. 102.34/102.28 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)),D)),E))) | -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),C),D)),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)),E))). [resolve(6000,a,6007,a)]. 104.55/104.40 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),c_Groups_Ozero__class_Ozero(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),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(6000,a,6008,a)]. 104.55/104.40 Derived: -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),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D)),B)),C)),E)). [resolve(6000,a,6009,a)]. 104.55/104.40 Derived: 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),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D)),B)),C)),E)). [resolve(6000,a,6010,a)]. 104.55/104.40 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D)),B)),C)),E)). [resolve(6000,a,6011,a)]. 104.55/104.40 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D)),B)),C)),E)). [resolve(6000,a,6012,a)]. 104.55/104.40 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),c_Groups_Ozero__class_Ozero(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),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(6000,a,6013,a)]. 104.55/104.40 6014 -class_Rings_Olinordered__comm__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_comm__mult__strict__left__mono) # label(axiom). [clausify(1686)]. 104.55/104.40 6015 class_Rings_Olinordered__comm__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 104.55/104.40 6016 class_Rings_Olinordered__comm__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 104.55/104.40 6017 class_Rings_Olinordered__comm__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 107.65/107.54 6018 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]. 107.65/107.54 6019 -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(205)]. 107.65/107.54 6020 -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(466)]. 107.65/107.54 Derived: -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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(6018,a,6019,a)]. 107.65/107.54 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(6018,a,6020,a)]. 107.65/107.54 6021 -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(1412)]. 107.65/107.54 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(6021,a,6018,a)]. 107.65/107.54 6022 -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(1412)]. 107.65/107.54 6023 -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(1818)]. 107.65/107.54 Derived: -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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(6023,a,6018,a)]. 107.65/107.54 6024 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_max__add__distrib__left) # label(axiom). [clausify(1975)]. 107.65/107.54 Derived: 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),B)),C) = 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),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(6024,a,6018,a)]. 107.86/107.79 6025 -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(2249)]. 107.86/107.79 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(6025,a,6018,a)]. 107.86/107.79 6026 -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(2249)]. 107.86/107.79 6027 -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(2719)]. 107.86/107.79 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(6027,a,6018,a)]. 107.86/107.79 6028 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]. 107.86/107.79 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Nat_Onat),A),C)). [resolve(6028,a,6019,a)]. 107.86/107.79 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(6028,a,6020,a)]. 107.86/107.79 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(6028,a,6021,a)]. 107.86/107.79 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(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(6028,a,6024,a)]. 107.86/107.79 6029 -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(2861)]. 107.86/107.79 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(6029,a,6018,a)]. 107.86/107.79 6030 -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(2861)]. 108.85/108.71 6031 -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(3269)]. 108.85/108.71 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(6031,a,6018,a)]. 108.85/108.71 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(6031,a,6028,a)]. 108.85/108.71 6032 -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(4867)]. 108.85/108.71 6033 -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(4867)]. 108.85/108.71 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(6033,a,6018,a)]. 108.85/108.71 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(6033,a,6028,a)]. 108.85/108.71 6034 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]. 108.85/108.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Int_Oint),A),C)). [resolve(6034,a,6019,a)]. 108.85/108.71 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(6034,a,6020,a)]. 108.85/108.71 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(6034,a,6021,a)]. 108.85/108.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Int_Oint),B),C)). [resolve(6034,a,6023,a)]. 108.85/108.71 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(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(6034,a,6024,a)]. 111.96/111.82 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(6034,a,6025,a)]. 111.96/111.82 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(6034,a,6027,a)]. 111.96/111.82 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(6034,a,6029,a)]. 111.96/111.82 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(6034,a,6031,a)]. 111.96/111.82 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(6034,a,6033,a)]. 111.96/111.82 6035 -class_Rings_Ozero__neq__one(A) | c_Groups_Ozero__class_Ozero(A) != c_Groups_Oone__class_Oone(A) # label(fact_zero__neq__one) # label(axiom). [clausify(410)]. 111.96/111.82 6036 class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ozero__neq__one) # label(axiom). [assumption]. 111.96/111.82 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral). [resolve(6035,a,6036,a)]. 111.96/111.82 6037 -class_Rings_Ozero__neq__one(A) | c_Groups_Ozero__class_Ozero(A) != c_Groups_Oone__class_Oone(A) # label(fact_one__neq__zero) # label(axiom). [clausify(3210)]. 111.96/111.82 6038 class_Rings_Ozero__neq__one(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ozero__neq__one) # label(axiom). [assumption]. 111.96/111.82 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != c_Groups_Oone__class_Oone(tc_Nat_Onat). [resolve(6038,a,6035,a)]. 111.96/111.82 6039 class_Rings_Ozero__neq__one(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ozero__neq__one) # label(axiom). [assumption]. 111.96/111.82 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Groups_Oone__class_Oone(tc_Int_Oint). [resolve(6039,a,6035,a)]. 111.96/111.82 6040 -class_Rings_Omult__zero(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(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(4673)]. 111.96/111.82 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Power_Opower(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(6040,b,6036,a)]. 111.96/111.82 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Power_Opower(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(6040,b,6038,a)]. 111.96/111.82 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Power_Opower(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(6040,b,6039,a)]. 113.84/113.71 6041 -class_Rings_Omult__zero(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(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(4673)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Power_Opower(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(6041,b,6036,a)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Power_Opower(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(6041,b,6038,a)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Power_Opower(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(6041,b,6039,a)]. 113.84/113.71 6042 -class_Rings_Omult__zero(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Ono__zero__divisors(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 | c_Groups_Ozero__class_Ozero(A) != B # label(fact_power__eq__0__iff) # label(axiom). [clausify(4673)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) | -class_Power_Opower(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 | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A. [resolve(6042,b,6036,a)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ono__zero__divisors(tc_Nat_Onat) | -class_Power_Opower(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 | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(6042,b,6038,a)]. 113.84/113.71 Derived: -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ono__zero__divisors(tc_Int_Oint) | -class_Power_Opower(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 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(6042,b,6039,a)]. 113.84/113.71 6043 class_Groups_Ocancel__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 113.84/113.71 6044 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C = D # label(fact_add__left__cancel) # label(axiom). [clausify(245)]. 113.84/113.71 6045 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C != D # label(fact_add__left__cancel) # label(axiom). [clausify(245)]. 113.84/113.71 6046 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B = D # label(fact_add__right__cancel) # label(axiom). [clausify(914)]. 113.84/113.71 6047 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B != D # label(fact_add__right__cancel) # label(axiom). [clausify(914)]. 113.84/113.71 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),A),C) | B != C. [resolve(6043,a,6045,a)]. 120.04/119.91 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),C),B) | A = C. [resolve(6043,a,6046,a)]. 120.04/119.91 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),C),B) | A != C. [resolve(6043,a,6047,a)]. 120.04/119.91 6048 class_Groups_Ocancel__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C) | B != C. [resolve(6048,a,6045,a)]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B) | A = C. [resolve(6048,a,6046,a)]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B) | A != C. [resolve(6048,a,6047,a)]. 120.04/119.91 6049 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C) | B != C. [resolve(6049,a,6045,a)]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B) | A = C. [resolve(6049,a,6046,a)]. 120.04/119.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B) | A != C. [resolve(6049,a,6047,a)]. 120.04/119.91 6050 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D) | C = D # label(fact_add__left__imp__eq) # label(axiom). [clausify(3362)]. 120.04/119.91 6051 -class_Groups_Ocancel__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) | B = D # label(fact_add__right__imp__eq) # label(axiom). [clausify(3715)]. 120.04/119.91 6052 -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(2179)]. 120.04/119.91 6053 class_Groups_Oab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 120.04/119.91 6054 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 120.04/119.91 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(6052,a,6053,a)]. 120.04/119.91 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(6052,a,6054,a)]. 120.04/119.91 6055 class_Groups_Oab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 120.04/119.91 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(6055,a,6052,a)]. 120.04/119.91 6056 -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(4289)]. 120.04/119.91 6057 class_Groups_Oab__semigroup__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 120.26/120.13 6058 -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(261)]. 120.26/120.13 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(6057,a,6058,a)]. 120.26/120.13 6059 -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(1844)]. 120.26/120.13 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(6059,a,6057,a)]. 120.26/120.13 6060 -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(1965)]. 120.26/120.13 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(6060,a,6057,a)]. 120.26/120.13 6061 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 120.26/120.13 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_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,E),D)),F),B). [resolve(6061,a,6058,a)]. 120.54/120.48 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(6061,a,6059,a)]. 120.54/120.48 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(6061,a,6060,a)]. 120.54/120.48 6062 -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(3237)]. 120.54/120.48 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(6062,a,6057,a)]. 120.54/120.48 Derived: c_Finite__Set_Ofun__left__comm(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)). [resolve(6062,a,6061,a)]. 120.54/120.48 6063 -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),f378(B,A)),f379(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f378(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f379(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(3291)]. 120.54/120.48 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),f378(A,tc_Code__Numeral_Ocode__numeral)),f379(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f378(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f379(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(6063,a,6057,a)]. 120.54/120.48 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),f378(A,tc_Int_Oint)),f379(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f378(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f379(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(6063,a,6061,a)]. 121.05/120.92 6064 -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(3666)]. 121.05/120.92 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(6064,a,6057,a)]. 121.05/120.92 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(6064,a,6061,a)]. 121.05/120.92 6065 -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(3868)]. 121.05/120.92 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(6065,a,6057,a)]. 121.05/120.92 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(6065,a,6061,a)]. 121.05/120.92 6066 -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(3896)]. 121.05/120.92 6067 -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(4217)]. 121.05/120.98 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(6067,a,6057,a)]. 121.05/120.98 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(6067,a,6061,a)]. 121.05/120.98 6068 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 121.05/120.98 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_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,E),D)),F),B). [resolve(6068,a,6058,a)]. 121.05/120.98 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(6068,a,6059,a)]. 121.05/120.98 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(6068,a,6060,a)]. 121.05/120.98 Derived: c_Finite__Set_Ofun__left__comm(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)). [resolve(6068,a,6062,a)]. 121.05/120.98 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),f378(A,tc_Nat_Onat)),f379(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f378(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f379(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(6068,a,6063,a)]. 121.05/120.98 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(6068,a,6064,a)]. 121.14/121.03 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(6068,a,6065,a)]. 121.14/121.03 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(6068,a,6067,a)]. 121.14/121.03 6069 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f512(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),F),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),D),C) # label(fact_fold__image__cong) # label(axiom). [clausify(4394)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f512(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)),E),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)),D),C),B). [resolve(6069,a,6057,a)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f512(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)),E),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),C),B). [resolve(6069,a,6061,a)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f512(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)),E),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),C),B). [resolve(6069,a,6068,a)]. 121.14/121.03 6070 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f512(E,F,D,C,B,A)) != hAPP(F,f512(E,F,D,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(4394)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f512(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(E,f512(D,E,C,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(6070,a,6057,a)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f512(D,E,C,B,A,tc_Int_Oint)) != hAPP(E,f512(D,E,C,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(6070,a,6061,a)]. 121.14/121.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f512(D,E,C,B,A,tc_Nat_Onat)) != hAPP(E,f512(D,E,C,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(6070,a,6068,a)]. 123.24/123.19 6071 -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(4670)]. 123.24/123.19 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(6071,a,6057,a)]. 123.24/123.19 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(6071,a,6061,a)]. 123.24/123.19 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(6071,a,6068,a)]. 123.24/123.19 6072 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != B | c_Orderings_Otop__class_Otop(A) != C | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(2019)]. 123.24/123.19 6073 class_Lattices_Obounded__lattice__top(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__top) # label(axiom). [assumption]. 123.24/123.19 6074 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__top(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom). [clausify(363)]. 123.24/123.19 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != B | c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B). [resolve(6072,a,6073,a)]. 123.24/123.19 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) != C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != D | c_Orderings_Otop__class_Otop(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),D) | -class_Lattices_Obounded__lattice(B). [resolve(6072,a,6074,b)]. 123.24/123.19 6075 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(2019)]. 123.24/123.19 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B). [resolve(6075,a,6073,a)]. 123.24/123.19 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),D) | -class_Lattices_Obounded__lattice(B). [resolve(6075,a,6074,b)]. 123.24/123.19 6076 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),C),B) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(2019)]. 123.24/123.19 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),B),A). [resolve(6076,a,6073,a)]. 129.75/129.62 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),D),C) | -class_Lattices_Obounded__lattice(B). [resolve(6076,a,6074,b)]. 129.75/129.62 6077 -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(2085)]. 129.75/129.62 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(6077,a,6073,a)]. 129.75/129.62 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),c_Orderings_Otop__class_Otop(tc_fun(A,B))) | -class_Lattices_Obounded__lattice(B). [resolve(6077,a,6074,b)]. 129.75/129.62 6078 -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(2397)]. 129.75/129.62 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(6078,a,6073,a)]. 129.75/129.62 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),c_Orderings_Otop__class_Otop(tc_fun(A,B))),C) | -class_Lattices_Obounded__lattice(B). [resolve(6078,a,6074,b)]. 129.75/129.62 6079 -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(4064)]. 129.75/129.62 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_HOL_Obool)),A) = A. [resolve(6079,a,6073,a)]. 129.75/129.62 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),c_Orderings_Otop__class_Otop(tc_fun(A,B))),C) = C | -class_Lattices_Obounded__lattice(B). [resolve(6079,a,6074,b)]. 129.75/129.62 6080 -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(4326)]. 129.75/129.62 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = A. [resolve(6080,a,6073,a)]. 129.75/129.62 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),c_Orderings_Otop__class_Otop(tc_fun(A,B))) = C | -class_Lattices_Obounded__lattice(B). [resolve(6080,a,6074,b)]. 129.75/129.62 6081 class_Int_Onumber(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Int_Onumber) # label(axiom). [assumption]. 129.75/129.62 6082 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6083 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6084 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6085 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6086 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6087 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6088 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -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(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6089 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -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(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6090 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6091 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(304)]. 129.75/129.62 6092 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6093 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -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_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6094 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6095 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -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_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6096 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6097 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6098 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6099 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6100 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6101 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(704)]. 129.75/129.62 6102 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)),C)) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6103 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | 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_Ozero__class_Ozero(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))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6104 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | 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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6105 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | 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_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6106 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | -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(A),D),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6107 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | -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(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6108 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6109 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6110 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)),C)) | -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(A),D),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6111 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)),C)) | -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(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(913)]. 129.75/129.63 6112 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) | c_Groups_Ozero__class_Ozero(A) = D # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1131)]. 129.75/129.63 6113 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1131)]. 129.75/129.63 6114 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != B # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1131)]. 129.75/129.63 6115 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) != D | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1131)]. 129.75/129.63 6116 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) != D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != B # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1131)]. 129.75/129.63 6117 -class_Int_Onumber(A) | -class_Rings_Osemiring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)) # label(fact_right__distrib__number__of) # label(axiom). [clausify(1307)]. 129.75/129.63 6118 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B) != D | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(1453)]. 129.75/129.63 6119 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Groups_Ozero__class_Ozero(A) != C | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(1453)]. 129.75/129.63 6120 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C) != D | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(1453)]. 129.75/129.63 6121 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(1453)]. 129.75/129.63 6122 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B) = D | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(1453)]. 129.75/129.63 6123 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) | c_Groups_Ozero__class_Ozero(A) = D # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(1875)]. 129.75/129.63 6124 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(1875)]. 129.75/129.63 6125 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != B # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(1875)]. 129.75/129.63 6126 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) != D | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(1875)]. 129.75/129.63 6127 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = D | c_Groups_Ozero__class_Ozero(A) != D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != B # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(1875)]. 129.75/129.63 6128 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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_Ozero__class_Ozero(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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6129 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6130 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6131 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6132 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6133 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.63 6134 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.64 6135 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.64 6136 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.64 6137 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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_Ozero__class_Ozero(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),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_divide__le__eq__number__of) # label(axiom). [clausify(2493)]. 129.75/129.64 6138 -class_Int_Onumber(A) | -class_Rings_Osemiring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_left__distrib__number__of) # label(axiom). [clausify(2573)]. 129.75/129.64 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) != C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = C. [resolve(6081,a,6112,b)]. 129.75/129.64 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) != C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) = A. [resolve(6081,a,6113,b)]. 129.75/129.64 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) = C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) != A. [resolve(6081,a,6114,b)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) = C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B). [resolve(6081,a,6115,b)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) = C | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)) != A. [resolve(6081,a,6116,b)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),A) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A). [resolve(6081,a,6118,a)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != B | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),B). [resolve(6081,a,6119,a)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),B) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),B). [resolve(6081,a,6120,a)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(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) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A). [resolve(6081,a,6121,a)]. 129.85/129.78 Derived: -class_Fields_Ofield__inverse__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),A) = C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A). [resolve(6081,a,6122,a)]. 129.85/129.78 6139 -class_Orderings_Oord(A) | -class_Int_Onumber(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))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_max__number__of) # label(axiom). [clausify(2942)]. 130.06/129.94 Derived: -class_Orderings_Oord(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(6139,b,6081,a)]. 130.06/129.94 6140 -class_Orderings_Oord(A) | -class_Int_Onumber(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))) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_max__number__of) # label(axiom). [clausify(2942)]. 130.06/129.94 Derived: -class_Orderings_Oord(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(6140,b,6081,a)]. 130.06/129.94 6141 -class_Int_Onumber(A) | -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_left__diff__distrib__number__of) # label(axiom). [clausify(3238)]. 130.06/129.94 Derived: -class_Rings_Oring(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))). [resolve(6141,a,6081,a)]. 130.06/129.94 6142 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6143 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6144 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(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),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6145 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6146 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6147 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6148 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.06/129.94 6149 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.25/130.12 6150 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(A),hAPP(c_Int_Onumber__class_Onumber__of(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_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.25/130.12 6151 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(3368)]. 130.25/130.12 6152 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B) != D | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_eq__divide__eq__number__of) # label(axiom). [clausify(3731)]. 130.25/130.12 6153 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Groups_Ozero__class_Ozero(A) != C | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C) # label(fact_eq__divide__eq__number__of) # label(axiom). [clausify(3731)]. 130.25/130.12 6154 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C) != D | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C) # label(fact_eq__divide__eq__number__of) # label(axiom). [clausify(3731)]. 130.25/130.12 6155 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_eq__divide__eq__number__of) # label(axiom). [clausify(3731)]. 130.25/130.12 6156 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B) = D | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B) # label(fact_eq__divide__eq__number__of) # label(axiom). [clausify(3731)]. 130.25/130.12 6157 -class_Orderings_Oord(A) | -class_Int_Onumber(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))) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_min__number__of) # label(axiom). [clausify(3758)]. 130.25/130.12 Derived: -class_Orderings_Oord(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(6157,b,6081,a)]. 130.46/130.39 6158 -class_Orderings_Oord(A) | -class_Int_Onumber(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))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_min__number__of) # label(axiom). [clausify(3758)]. 130.46/130.39 Derived: -class_Orderings_Oord(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(6158,b,6081,a)]. 130.46/130.39 6159 -class_Int_Onumber(A) | -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_power2__eq__square__number__of) # label(axiom). [clausify(3798)]. 130.46/130.39 6160 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.46/130.39 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | -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),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6160,a,6081,a)]. 130.46/130.39 6161 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | 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(A),D),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.46/130.39 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6161,a,6081,a)]. 130.56/130.39 6162 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.39 6163 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.39 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -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),hAPP(c_Int_Onumber__class_Onumber__of(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),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)))). [resolve(6163,a,6081,a)]. 130.56/130.39 6164 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.39 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -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),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)))). [resolve(6164,a,6081,a)]. 130.56/130.40 6165 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.40 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))). [resolve(6165,a,6081,a)]. 130.56/130.40 6166 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.40 6167 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(A),hAPP(c_Int_Onumber__class_Onumber__of(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),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.40 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6167,a,6081,a)]. 130.56/130.42 6168 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.42 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6168,a,6081,a)]. 130.56/130.42 6169 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_less__divide__eq__number__of1) # label(axiom). [clausify(4182)]. 130.56/130.42 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | 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),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6169,a,6081,a)]. 130.56/130.42 6170 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.42 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | 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),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(6170,a,6081,a)]. 130.56/130.42 6171 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | -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(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.42 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),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)),A))) | -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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),A))). [resolve(6171,a,6081,a)]. 130.56/130.42 6172 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.42 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),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)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),A))). [resolve(6172,a,6081,a)]. 130.56/130.43 6173 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),D)) | -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(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 Derived: -class_Fields_Olinordered__field__inverse__zero(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),B)),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(6173,a,6081,a)]. 130.56/130.43 6174 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),D)) | 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(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 Derived: -class_Fields_Olinordered__field__inverse__zero(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),B)),C)) | 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(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(6174,a,6081,a)]. 130.56/130.43 6175 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C))) | -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(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 6176 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 6177 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | 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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(6177,a,6081,a)]. 130.56/130.43 6178 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | -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),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)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),A))). [resolve(6178,a,6081,a)]. 130.56/130.43 6179 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4214)]. 130.56/130.43 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),A)) | 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),hAPP(c_Int_Onumber__class_Onumber__of(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),A))). [resolve(6179,a,6081,a)]. 130.56/130.48 6180 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.56/130.48 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | -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),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6180,a,6081,a)]. 130.56/130.48 6181 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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))) | -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),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.56/130.48 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6181,a,6081,a)]. 130.56/130.48 6182 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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))) | -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_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.56/130.49 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6182,a,6081,a)]. 130.56/130.49 6183 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.56/130.49 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))). [resolve(6183,a,6081,a)]. 130.56/130.49 6184 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.56/130.49 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)),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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))). [resolve(6184,a,6081,a)]. 130.66/130.49 6185 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.66/130.49 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))) | -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),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))). [resolve(6185,a,6081,a)]. 130.66/130.49 6186 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.66/130.49 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))) | -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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)))). [resolve(6186,a,6081,a)]. 130.66/130.50 6187 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(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(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.66/130.50 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6187,a,6081,a)]. 130.66/130.50 6188 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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(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(A),C),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.66/130.50 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(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),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6188,a,6081,a)]. 130.66/130.50 6189 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(4291)]. 130.66/130.50 Derived: -class_Fields_Olinordered__field__inverse__zero(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(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))) | 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),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)))). [resolve(6189,a,6081,a)]. 130.76/130.69 6190 -class_Int_Onumber(A) | -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)) # label(fact_right__diff__distrib__number__of) # label(axiom). [clausify(4504)]. 130.76/130.69 Derived: -class_Rings_Oring(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),C)). [resolve(6190,a,6081,a)]. 130.76/130.69 6191 class_Int_Onumber(tc_Int_Oint) # label(arity_Int__Oint__Int_Onumber) # label(axiom). [assumption]. 130.76/130.69 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | 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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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)),C)). [resolve(6191,a,6082,a)]. 130.76/130.69 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | 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),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)). [resolve(6191,a,6083,a)]. 130.76/130.69 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6084,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6085,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)) | 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),B))). [resolve(6191,a,6086,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6087,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6088,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6089,a)]. 130.76/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)) | 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),B))). [resolve(6191,a,6090,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6091,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6092,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -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_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6093,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6094,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | -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_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6095,a)]. 130.86/130.70 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6096,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))). [resolve(6191,a,6099,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6100,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6101,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | -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__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)),B)). [resolve(6191,a,6102,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | 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_Ozero__class_Ozero(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))). [resolve(6191,a,6103,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | 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_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C))). [resolve(6191,a,6104,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | 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_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C))). [resolve(6191,a,6105,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | 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__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(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6191,a,6106,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | 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__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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C))). [resolve(6191,a,6107,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)). [resolve(6191,a,6108,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C))). [resolve(6191,a,6109,a)]. 130.86/130.71 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)),B)) | -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(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6191,a,6110,a)]. 130.86/130.72 Derived: -class_Fields_Olinordered__field__inverse__zero(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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)),B)) | -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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C))). [resolve(6191,a,6111,a)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) != C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = C. [resolve(6191,a,6112,b)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) != C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = A. [resolve(6191,a,6113,b)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) != A. [resolve(6191,a,6114,b)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B). [resolve(6191,a,6115,b)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = C | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) != A. [resolve(6191,a,6116,b)]. 130.86/130.72 Derived: -class_Rings_Osemiring(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = 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),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)). [resolve(6191,a,6117,a)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A). [resolve(6191,a,6118,a)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B). [resolve(6191,a,6119,a)]. 130.86/130.72 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),B) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B). [resolve(6191,a,6120,a)]. 130.86/130.73 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A). [resolve(6191,a,6121,a)]. 130.86/130.73 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A) = C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A). [resolve(6191,a,6122,a)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_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_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6128,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6129,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6130,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | -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__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6131,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),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_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6132,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6133,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)),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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6134,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))). [resolve(6191,a,6135,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_Ozero__class_Ozero(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6136,b)]. 130.86/130.73 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_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_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(6191,a,6137,b)]. 130.86/130.73 Derived: -class_Orderings_Oord(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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(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(6191,a,6139,b)]. 130.86/130.74 Derived: -class_Orderings_Oord(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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(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(6191,a,6140,b)]. 130.86/130.74 Derived: -class_Rings_Oring(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))). [resolve(6191,a,6141,a)]. 130.86/130.74 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(6191,a,6142,b)]. 130.86/130.74 Derived: -class_Fields_Olinordered__field__inverse__zero(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_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | 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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(6191,a,6143,b)]. 130.86/130.74 Derived: -class_Fields_Olinordered__field__inverse__zero(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_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),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(6191,a,6144,b)]. 130.86/130.74 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),B)). [resolve(6191,a,6145,b)]. 130.86/130.74 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),B)). [resolve(6191,a,6147,b)]. 130.86/130.75 Derived: -class_Fields_Olinordered__field__inverse__zero(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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)). [resolve(6191,a,6149,b)]. 130.86/130.75 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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(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_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(6191,a,6150,b)]. 130.86/130.75 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(6191,a,6151,b)]. 130.86/130.75 Derived: -class_Orderings_Oord(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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(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(6191,a,6157,b)]. 130.86/130.75 Derived: -class_Orderings_Oord(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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(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(6191,a,6158,b)]. 130.86/130.75 Derived: -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6160,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(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_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6161,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6163,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)))). [resolve(6191,a,6164,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | 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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(6191,a,6165,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6167,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6168,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6169,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(6191,a,6170,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),A))) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(6191,a,6171,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(6191,a,6172,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),B)),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B))). [resolve(6191,a,6173,a)]. 130.86/130.76 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),B)),C)) | 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(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(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B))). [resolve(6191,a,6174,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(6191,a,6177,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(6191,a,6178,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(6191,a,6179,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(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__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6180,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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))) | -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),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),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6181,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | 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))) | -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_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6182,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(6191,a,6183,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),c_Groups_Ozero__class_Ozero(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(6191,a,6184,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(6191,a,6185,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))) | -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_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(6191,a,6186,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6187,a)]. 130.86/130.77 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6188,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(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__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)))). [resolve(6191,a,6189,a)]. 131.07/130.95 Derived: -class_Rings_Oring(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = 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),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)). [resolve(6191,a,6190,a)]. 131.07/130.95 6192 class_Int_Onumber(tc_Nat_Onat) # label(arity_Nat__Onat__Int_Onumber) # label(axiom). [assumption]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | 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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)). [resolve(6192,a,6082,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | 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),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),A)). [resolve(6192,a,6083,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6084,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6085,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),A)) | 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),B))). [resolve(6192,a,6086,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6087,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6088,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6089,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),A)) | 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),B))). [resolve(6192,a,6090,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6091,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6092,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6093,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -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),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6094,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | -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_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6095,a)]. 131.07/130.95 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6096,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))). [resolve(6192,a,6099,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6100,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6101,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | -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__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)),B)). [resolve(6192,a,6102,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(6192,a,6103,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C))). [resolve(6192,a,6104,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C))). [resolve(6192,a,6105,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | 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__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(6192,a,6106,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | 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__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C))). [resolve(6192,a,6107,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)). [resolve(6192,a,6108,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C))). [resolve(6192,a,6109,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(6192,a,6110,a)]. 131.07/130.96 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C))). [resolve(6192,a,6111,a)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) != C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C. [resolve(6192,a,6112,b)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) != C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = A. [resolve(6192,a,6113,b)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) != A. [resolve(6192,a,6114,b)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B). [resolve(6192,a,6115,b)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) = C | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) != A. [resolve(6192,a,6116,b)]. 131.07/130.97 Derived: -class_Rings_Osemiring(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)). [resolve(6192,a,6117,a)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A). [resolve(6192,a,6118,a)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),B). [resolve(6192,a,6119,a)]. 131.07/130.97 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B) != C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),B). [resolve(6192,a,6120,a)]. 131.07/130.98 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A). [resolve(6192,a,6121,a)]. 131.07/130.98 Derived: -class_Fields_Ofield__inverse__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A) = C | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A). [resolve(6192,a,6122,a)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6128,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6129,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6130,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | -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__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6131,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6132,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6133,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),C)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6134,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))). [resolve(6192,a,6135,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6136,b)]. 131.07/130.98 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))). [resolve(6192,a,6137,b)]. 131.07/130.98 Derived: -class_Orderings_Oord(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)). [resolve(6192,a,6139,b)]. 131.07/130.99 Derived: -class_Orderings_Oord(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)). [resolve(6192,a,6140,b)]. 131.07/130.99 Derived: -class_Rings_Oring(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))). [resolve(6192,a,6141,a)]. 131.07/130.99 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),B)). [resolve(6192,a,6142,b)]. 131.07/130.99 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),B)). [resolve(6192,a,6143,b)]. 131.07/130.99 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),B)). [resolve(6192,a,6144,b)]. 131.07/130.99 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),B)). [resolve(6192,a,6145,b)]. 131.07/130.99 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),B)). [resolve(6192,a,6147,b)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)). [resolve(6192,a,6149,b)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),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)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),B)). [resolve(6192,a,6150,b)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),B)). [resolve(6192,a,6151,b)]. 131.16/131.00 Derived: -class_Orderings_Oord(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)). [resolve(6192,a,6157,b)]. 131.16/131.00 Derived: -class_Orderings_Oord(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)). [resolve(6192,a,6158,b)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6160,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(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_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6161,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6163,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)))). [resolve(6192,a,6164,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | 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),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(6192,a,6165,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | 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),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6167,a)]. 131.16/131.00 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6168,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6169,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A))). [resolve(6192,a,6170,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),A))). [resolve(6192,a,6171,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),A))). [resolve(6192,a,6172,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B)),C)) | -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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),B))). [resolve(6192,a,6173,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B)),C)) | 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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),B))). [resolve(6192,a,6174,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A))). [resolve(6192,a,6177,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),A))). [resolve(6192,a,6178,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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_Ozero__class_Ozero(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),A))). [resolve(6192,a,6179,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6180,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6181,a)]. 131.16/131.01 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6182,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(6192,a,6183,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(6192,a,6184,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -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(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(6192,a,6185,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))) | -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_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(6192,a,6186,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6187,a)]. 131.16/131.02 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | 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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(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),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6188,a)]. 133.37/133.28 Derived: -class_Fields_Olinordered__field__inverse__zero(tc_Nat_Onat) | -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),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)))). [resolve(6192,a,6189,a)]. 133.37/133.28 Derived: -class_Rings_Oring(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)). [resolve(6192,a,6190,a)]. 133.37/133.28 6193 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_mult__1__left) # label(axiom). [clausify(706)]. 133.37/133.28 6194 class_Groups_Omonoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__mult) # label(axiom). [assumption]. 133.37/133.28 6195 -class_Groups_Omonoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oone__class_Oone(A)),B) # label(fact_power__one) # label(axiom). [clausify(835)]. 133.37/133.28 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)),A). [resolve(6195,a,6194,a)]. 133.37/133.28 6196 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_mult__1__right) # label(axiom). [clausify(1001)]. 133.37/133.28 6197 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__commutes) # label(axiom). [clausify(1122)]. 133.37/133.28 6198 -class_Groups_Omonoid__mult(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_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),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). [clausify(1169)]. 133.37/133.28 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_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(6198,a,6194,a)]. 133.37/133.28 6199 -class_Groups_Omonoid__mult(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_power__Suc2) # label(axiom). [clausify(2769)]. 134.18/134.05 6200 class_Groups_Omonoid__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Omonoid__mult) # label(axiom). [assumption]. 134.18/134.05 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)),A). [resolve(6200,a,6195,a)]. 134.18/134.05 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_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(6200,a,6198,a)]. 134.18/134.05 6201 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),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(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__even__eq) # label(axiom). [clausify(2968)]. 134.18/134.05 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),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(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(6201,a,6194,a)]. 134.18/134.05 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)),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),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(6201,a,6200,a)]. 134.18/134.05 6202 -class_Groups_Omonoid__mult(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_power__add) # label(axiom). [clausify(3646)]. 134.18/134.05 6203 -class_Groups_Omonoid__mult(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_power2__eq__square) # label(axiom). [clausify(4252)]. 134.18/134.05 6204 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),B) # label(fact_power3__eq__cube) # label(axiom). [clausify(4706)]. 134.18/134.05 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),A). [resolve(6204,a,6194,a)]. 134.18/134.05 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_OBit1(c_Int_OBit1(c_Int_OPls)))) = 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),A)),A). [resolve(6204,a,6200,a)]. 145.78/145.66 6205 class_Groups_Omonoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__mult) # label(axiom). [assumption]. 145.78/145.66 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)),A). [resolve(6205,a,6195,a)]. 145.78/145.66 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_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(6205,a,6198,a)]. 145.78/145.66 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),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(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)). [resolve(6205,a,6201,a)]. 145.78/145.66 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),A)),A). [resolve(6205,a,6204,a)]. 145.78/145.66 6206 -class_Groups_Omonoid__mult(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_power__mult) # label(axiom). [clausify(4822)]. 145.78/145.66 6207 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B # label(fact_power__one__right) # label(axiom). [clausify(4882)]. 145.78/145.66 6208 c_Relation_Oirrefl(A,B) | hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),f36(B,A)),f36(B,A))),B)) # label(fact_irrefl__def) # label(axiom). [clausify(331)]. 145.78/145.66 6209 -c_Relation_Oirrefl(A,B) | -hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),C),C)),B)) # label(fact_irrefl__def) # label(axiom). [clausify(331)]. 145.78/145.66 Derived: hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),f36(B,A)),f36(B,A))),B)) | -hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),C),C)),B)). [resolve(6208,a,6209,a)]. 145.78/145.66 6210 c_Relation_Oirrefl(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool)),B),c_Relation_OId(A))) # label(fact_irrefl__diff__Id) # label(axiom). [clausify(441)]. 145.78/145.66 Derived: -hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),B),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(A,A),tc_HOL_Obool)),C),c_Relation_OId(A)))). [resolve(6210,a,6209,a)]. 145.78/145.66 6211 -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(1655)]. 146.28/146.15 6212 class_Groups_Olinordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 146.28/146.15 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(6211,a,6212,a)]. 146.28/146.15 6213 class_Groups_Olinordered__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 146.28/146.15 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 | 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(6213,a,6211,a)]. 146.28/146.15 6214 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]. 146.28/146.15 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(6214,a,6211,a)]. 146.28/146.19 6215 -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(3673)]. 146.28/146.19 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(6215,a,6212,a)]. 146.28/146.19 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(6215,a,6213,a)]. 152.37/152.19 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(6215,a,6214,a)]. 152.37/152.19 6216 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Oreflp(A,B) # label(fact_equivpE) # label(axiom). [clausify(1298)]. 152.37/152.19 6217 -c_Predicate_Oreflp(A,B) | hBOOL(hAPP(hAPP(B,C),C)) # label(fact_reflpE) # label(axiom). [clausify(462)]. 152.37/152.19 Derived: -c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(B,C),C)). [resolve(6216,b,6217,a)]. 152.37/152.19 6218 c_Predicate_Oreflp(A,B) | -c_Equiv__Relations_Oequivp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(1598)]. 152.37/152.19 6219 -c_Predicate_Oreflp(A,B) | c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_reflp__def) # label(axiom). [clausify(2583)]. 158.48/158.35 Derived: c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6219,a,6216,b)]. 158.48/158.35 6220 c_Predicate_Oreflp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_reflp__def) # label(axiom). [clausify(2583)]. 158.48/158.35 Derived: -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,C),C)). [resolve(6220,a,6217,a)]. 158.48/158.35 6221 -c_Predicate_Osymp(A,B) | -c_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5666,a,5662,a)]. 158.48/158.35 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6221,b,6220,a)]. 158.48/158.35 6222 -c_Equiv__Relations_Opart__equivp(A,B) | -c_Predicate_Osymp(A,B) | -c_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(5669,a,5666,a)]. 158.48/158.35 Derived: -c_Equiv__Relations_Opart__equivp(A,B) | -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6222,c,6220,a)]. 158.48/158.35 6223 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 158.48/158.35 6224 -class_Groups_Oordered__cancel__ab__semigroup__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_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__strict__left__mono) # label(axiom). [clausify(526)]. 158.48/158.35 6225 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 158.48/158.35 6226 -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),f242(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(2157)]. 158.48/158.35 Derived: -class_Groups_Ocomm__monoid__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),f242(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(6226,b,6223,a)]. 158.48/158.35 Derived: -class_Groups_Ocomm__monoid__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),f242(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(6226,b,6225,a)]. 158.57/158.43 6227 -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,f242(E,D,C,B,A))),hAPP(E,f242(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(2157)]. 158.57/158.43 Derived: -class_Groups_Ocomm__monoid__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,f242(D,C,B,A,tc_Int_Oint))),hAPP(D,f242(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(6227,b,6223,a)]. 158.57/158.43 Derived: -class_Groups_Ocomm__monoid__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,f242(D,C,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f242(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(6227,b,6225,a)]. 158.57/158.43 6228 class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 158.57/158.43 Derived: -class_Groups_Ocomm__monoid__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),f242(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(6228,a,6226,b)]. 158.57/158.43 Derived: -class_Groups_Ocomm__monoid__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,f242(D,C,B,A,tc_Nat_Onat))),hAPP(D,f242(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(6228,a,6227,b)]. 158.57/158.43 6229 -class_Groups_Oordered__cancel__ab__semigroup__add(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__strict__mono) # label(axiom). [clausify(2454)]. 158.57/158.43 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),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(6229,a,6223,a)]. 158.57/158.43 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),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(6229,a,6225,a)]. 159.47/159.34 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),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(6229,a,6228,a)]. 159.47/159.34 6230 -class_Groups_Oordered__cancel__ab__semigroup__add(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__le__less__mono) # label(axiom). [clausify(4186)]. 159.47/159.34 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),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(6230,a,6223,a)]. 159.47/159.34 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),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(6230,a,6225,a)]. 159.47/159.34 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),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(6230,a,6228,a)]. 159.47/159.34 6231 -class_Groups_Oordered__cancel__ab__semigroup__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_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__strict__right__mono) # label(axiom). [clausify(4382)]. 159.47/159.34 6232 -class_Groups_Oordered__cancel__ab__semigroup__add(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(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),E))) # label(fact_add__less__le__mono) # label(axiom). [clausify(4851)]. 159.47/159.34 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),D)) | 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),D))). [resolve(6232,a,6223,a)]. 159.47/159.34 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(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(6232,a,6225,a)]. 159.47/159.34 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(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(6232,a,6228,a)]. 161.87/161.75 6233 -class_Rings_Oordered__cancel__semiring(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)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonpos__nonneg) # label(axiom). [clausify(778)]. 161.87/161.75 6234 class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 161.87/161.75 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),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(6233,a,6234,a)]. 161.87/161.75 6235 class_Rings_Oordered__cancel__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 161.87/161.75 6236 -class_Rings_Oordered__cancel__semiring(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),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonneg__nonpos) # label(axiom). [clausify(1173)]. 161.87/161.75 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),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_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(6236,a,6234,a)]. 161.87/161.75 6237 -class_Rings_Oordered__cancel__semiring(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_Otimes__class_Otimes(A),B),C))) # label(fact_mult__nonneg__nonneg) # label(axiom). [clausify(2871)]. 161.87/161.75 6238 -class_Rings_Oordered__cancel__semiring(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),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_split__mult__neg__le) # label(axiom). [clausify(2913)]. 161.87/161.75 6239 -class_Rings_Oordered__cancel__semiring(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)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_split__mult__neg__le) # label(axiom). [clausify(2913)]. 161.87/161.75 6240 class_Rings_Oordered__cancel__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 161.87/161.75 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),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),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(6240,a,6233,a)]. 167.78/167.66 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_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_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(6240,a,6236,a)]. 167.78/167.66 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)) | 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_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(6240,a,6237,a)]. 167.78/167.66 6241 -class_Rings_Oordered__cancel__semiring(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),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__nonneg__nonpos2) # label(axiom). [clausify(4583)]. 167.78/167.66 6242 class_Rings_Olinordered__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 167.78/167.66 6243 -class_Rings_Olinordered__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__le) # label(axiom). [clausify(586)]. 167.78/167.66 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | 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),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(6242,a,6243,a)]. 167.78/167.66 6244 -class_Nat_Osemiring__char__0(A) | B != C | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C) # label(fact_of__nat__eq__iff) # label(axiom). [clausify(1811)]. 167.78/167.66 6245 class_Nat_Osemiring__char__0(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osemiring__char__0) # label(axiom). [assumption]. 167.78/167.66 Derived: A != B | 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(6244,a,6245,a)]. 167.78/167.66 6246 -class_Nat_Osemiring__char__0(A) | B = C | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C) # label(fact_of__nat__eq__iff) # label(axiom). [clausify(1811)]. 167.78/167.66 Derived: A = B | 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(6246,a,6245,a)]. 174.88/174.78 6247 class_Nat_Osemiring__char__0(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osemiring__char__0) # label(axiom). [assumption]. 174.88/174.78 Derived: A != B | 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(6247,a,6244,a)]. 174.88/174.78 Derived: A = B | 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(6247,a,6246,a)]. 174.88/174.78 6248 -class_Nat_Osemiring__char__0(A) | c_Fun_Oinj__on(tc_Nat_Onat,A,c_Nat_Osemiring__1__class_Oof__nat(A),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_inj__of__nat) # label(axiom). [clausify(3619)]. 174.88/174.78 Derived: c_Fun_Oinj__on(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))). [resolve(6248,a,6245,a)]. 174.88/174.78 Derived: c_Fun_Oinj__on(tc_Nat_Onat,tc_Nat_Onat,c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))). [resolve(6248,a,6247,a)]. 174.97/174.81 6249 class_Nat_Osemiring__char__0(tc_Int_Oint) # label(arity_Int__Oint__Nat_Osemiring__char__0) # label(axiom). [assumption]. 174.97/174.81 Derived: A != B | 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(6249,a,6244,a)]. 174.97/174.81 Derived: A = B | 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(6249,a,6246,a)]. 174.97/174.81 Derived: c_Fun_Oinj__on(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))). [resolve(6249,a,6248,a)]. 174.97/174.81 6250 class_Rings_Olinordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring) # label(axiom). [assumption]. 174.97/174.81 6251 -class_Rings_Olinordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__square__less__zero) # label(axiom). [clausify(802)]. 174.97/174.81 6252 -class_Rings_Olinordered__ring(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_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)))) # label(fact_sum__squares__ge__zero) # label(axiom). [clausify(1020)]. 174.97/174.81 6253 -class_Rings_Olinordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__sum__squares__lt__zero) # label(axiom). [clausify(1621)]. 174.97/174.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6250,a,6251,a)]. 174.97/174.81 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)))). [resolve(6250,a,6252,a)]. 174.97/174.81 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_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6250,a,6253,a)]. 174.97/174.81 6254 -class_Rings_Olinordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B))) # label(fact_zero__le__square) # label(axiom). [clausify(4179)]. 174.97/174.81 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_Otimes__class_Otimes(tc_Int_Oint),A),A))). [resolve(6254,a,6250,a)]. 177.97/177.83 6255 class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 177.97/177.83 6256 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) # label(fact_field__power__not__zero) # label(axiom). [clausify(830)]. 177.97/177.83 6257 -class_Rings_Oring__1__no__zero__divisors(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) != B | c_Groups_Oone__class_Oone(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_power2__eq__1__iff) # label(axiom). [clausify(1791)]. 177.97/177.83 6258 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) != B | c_Groups_Oone__class_Oone(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_power2__eq__1__iff) # label(axiom). [clausify(1791)]. 177.97/177.83 6259 -class_Rings_Oring__1__no__zero__divisors(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = B | c_Groups_Oone__class_Oone(A) = B | c_Groups_Oone__class_Oone(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_power2__eq__1__iff) # label(axiom). [clausify(1791)]. 177.97/177.83 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B). [resolve(6255,a,6256,a)]. 177.97/177.83 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) != A | c_Groups_Oone__class_Oone(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(6255,a,6257,a)]. 177.97/177.83 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != A | c_Groups_Oone__class_Oone(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(6255,a,6258,a)]. 177.97/177.83 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = A | c_Groups_Oone__class_Oone(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(6255,a,6259,a)]. 177.97/177.83 6260 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | c_Groups_Oone__class_Oone(A) = B | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = B # label(fact_square__eq__1__iff) # label(axiom). [clausify(3275)]. 177.97/177.83 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | c_Groups_Oone__class_Oone(tc_Int_Oint) = A | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(6260,a,6255,a)]. 177.97/177.83 6261 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | c_Groups_Oone__class_Oone(A) != B # label(fact_square__eq__1__iff) # label(axiom). [clausify(3275)]. 177.97/177.83 6262 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) != B # label(fact_square__eq__1__iff) # label(axiom). [clausify(3275)]. 177.97/177.83 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) != A. [resolve(6262,a,6255,a)]. 177.97/177.83 6263 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != B | 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__eq__power2) # label(axiom). [clausify(4037)]. 183.48/183.35 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | 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(6263,a,6255,a)]. 183.48/183.35 6264 -class_Rings_Oring__1__no__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) = B | 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__eq__power2) # label(axiom). [clausify(4037)]. 183.48/183.35 6265 c_Nitpick_Oless__frac(A,B,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_Nitpick_Oless__frac__def) # label(axiom). [clausify(839)]. 183.48/183.35 6266 -c_Nitpick_Oless__frac(A,B,C) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_Nitpick_Oless__frac__def) # label(axiom). [clausify(839)]. 183.48/183.35 6267 -class_Rings_Oidom(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) # label(fact_square__eq__iff) # label(axiom). [clausify(1488)]. 183.48/183.35 6268 class_Rings_Oidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oidom) # label(axiom). [assumption]. 183.48/183.35 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B). [resolve(6267,a,6268,a)]. 183.48/183.35 6269 -class_Rings_Oidom(A) | B != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) # label(fact_square__eq__iff) # label(axiom). [clausify(1488)]. 183.48/183.35 Derived: A != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B). [resolve(6269,a,6268,a)]. 183.48/183.35 6270 -class_Rings_Oidom(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = C | B = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) # label(fact_square__eq__iff) # label(axiom). [clausify(1488)]. 183.48/183.35 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = B | A = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B). [resolve(6270,a,6268,a)]. 183.48/183.35 6271 -class_Rings_Oidom(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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_Rings_Odvd__class_Odvd(A),B),D)) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(1559)]. 183.48/183.35 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(6271,a,6268,a)]. 183.48/183.35 6272 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(1559)]. 183.48/183.35 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(6272,a,6268,a)]. 183.48/183.35 6273 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | c_Groups_Ozero__class_Ozero(A) != C # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(1559)]. 186.58/186.46 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(6273,a,6268,a)]. 186.58/186.46 6274 -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(A) != B | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2578)]. 186.58/186.46 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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(6274,a,6268,a)]. 186.58/186.46 6275 -class_Rings_Oidom(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2578)]. 186.58/186.46 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),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(6275,a,6268,a)]. 186.58/186.46 6276 -class_Rings_Oidom(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2578)]. 186.58/186.46 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(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(6276,a,6268,a)]. 186.58/186.46 6277 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono_H) # label(axiom). [clausify(1855)]. 186.58/186.46 6278 class_Rings_Oordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__semiring) # label(axiom). [assumption]. 186.58/186.46 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),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__eq(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(6277,a,6278,a)]. 186.58/186.46 6279 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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)),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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono) # label(axiom). [clausify(2008)]. 187.09/186.92 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),D)) | -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)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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(6279,a,6278,a)]. 187.09/186.92 6280 class_Rings_Oordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__semiring) # label(axiom). [assumption]. 187.09/186.92 6281 class_Rings_Oordered__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__semiring) # label(axiom). [assumption]. 187.09/186.92 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),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__eq(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(6281,a,6277,a)]. 187.09/186.92 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),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)),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),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(6281,a,6279,a)]. 187.09/186.92 6282 -class_Rings_Oordered__semiring(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)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__left__mono) # label(axiom). [clausify(3459)]. 187.09/186.92 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)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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(6282,a,6278,a)]. 187.09/186.92 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B))). [resolve(6282,a,6280,a)]. 187.09/186.92 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)),C)) | 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),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(6282,a,6281,a)]. 192.78/192.69 6283 -class_Rings_Oordered__semiring(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)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_mult__right__mono) # label(axiom). [clausify(4191)]. 192.78/192.69 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)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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(6283,a,6278,a)]. 192.78/192.69 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))). [resolve(6283,a,6280,a)]. 192.78/192.69 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)),C)) | 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),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(6283,a,6281,a)]. 192.78/192.69 6284 -class_Rings_Olinordered__semiring(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__left__less__imp__less) # label(axiom). [clausify(4263)]. 192.78/192.69 6285 class_Rings_Olinordered__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring) # label(axiom). [assumption]. 192.78/192.69 6286 class_Rings_Olinordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring) # label(axiom). [assumption]. 192.78/192.69 6287 class_Rings_Olinordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring) # label(axiom). [assumption]. 192.78/192.69 6288 -class_Rings_Olinordered__semiring(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__right__less__imp__less) # label(axiom). [clausify(4502)]. 192.78/192.69 6289 -c_Equiv__Relations_Oequivp(A,B) | c_Equiv__Relations_Opart__equivp(A,B) # label(fact_equivp__implies__part__equivp) # label(axiom). [clausify(1762)]. 192.78/192.69 6290 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,D),C)) # label(fact_part__equivp__symp) # label(axiom). [clausify(1121)]. 192.78/192.69 6291 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f137(B,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_fun(A,tc_HOL_Obool)),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),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_COMBS(A,A,tc_HOL_Obool),B),c_COMBI(A))))),hAPP(hAPP(c_COMBC(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A)),c_fequal)),B))))) # label(fact_part__equivp__typedef) # label(axiom). [clausify(1292)]. 193.18/193.04 Derived: -c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f137(B,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_fun(A,tc_HOL_Obool)),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),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_COMBS(A,A,tc_HOL_Obool),B),c_COMBI(A))))),hAPP(hAPP(c_COMBC(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A)),c_fequal)),B))))). [resolve(6289,b,6291,a)]. 193.18/193.04 6292 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)) # label(fact_part__equivp__transp) # label(axiom). [clausify(1943)]. 193.18/193.04 6293 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 6294 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hAPP(B,D) = hAPP(B,C) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6294,a,6289,b)]. 193.18/193.04 6295 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,D),D)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 6296 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,C),C)) | hAPP(B,D) != hAPP(B,C) | -hBOOL(hAPP(hAPP(B,D),D)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6296,a,6289,b)]. 193.18/193.04 6297 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f351(B,A)),f351(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 6298 c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.04 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6298,a,6290,a)]. 193.18/193.04 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f137(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),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_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6298,a,6291,a)]. 193.18/193.04 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6298,a,6292,a)]. 193.18/193.04 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(6298,a,6293,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(6298,a,6294,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(6298,a,6295,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6298,a,6296,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f351(A,B)),f351(A,B))). [resolve(6298,a,6297,a)]. 193.18/193.06 6299 c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hAPP(B,f353(B,A)) = hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6299,a,6290,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f137(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),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_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6299,a,6291,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6299,a,6292,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(6299,a,6293,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(6299,a,6294,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(6299,a,6295,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6299,a,6296,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f351(A,B)),f351(A,B))). [resolve(6299,a,6297,a)]. 193.18/193.06 6300 c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6300,a,6290,a)]. 193.18/193.06 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f137(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),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_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6300,a,6291,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6300,a,6292,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(6300,a,6293,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(6300,a,6294,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(6300,a,6295,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6300,a,6296,a)]. 193.18/193.08 Derived: hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f351(A,B)),f351(A,B))). [resolve(6300,a,6297,a)]. 193.18/193.08 6301 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | hAPP(B,f353(B,A)) != hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(3067)]. 193.18/193.08 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6301,a,6290,a)]. 193.18/193.08 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f137(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),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_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6301,a,6291,a)]. 193.18/193.08 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6301,a,6292,a)]. 193.18/193.08 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(6301,a,6293,a)]. 193.58/193.45 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(6301,a,6294,a)]. 193.58/193.45 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(6301,a,6295,a)]. 193.58/193.45 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6301,a,6296,a)]. 193.58/193.45 Derived: -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f351(A,B)),f351(A,B))). [resolve(6301,a,6297,a)]. 193.58/193.45 6302 c_Predicate_Osymp(A,B) | -c_Equiv__Relations_Opart__equivp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(3150)]. 193.58/193.45 Derived: c_Predicate_Osymp(A,B) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6302,b,6289,b)]. 193.58/193.45 Derived: c_Predicate_Osymp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6302,b,6298,a)]. 193.58/193.45 Derived: c_Predicate_Osymp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hAPP(B,f353(B,A)) = hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6302,b,6299,a)]. 193.58/193.45 Derived: c_Predicate_Osymp(A,B) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6302,b,6300,a)]. 193.58/193.45 Derived: c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | hAPP(B,f353(B,A)) != hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6302,b,6301,a)]. 193.58/193.45 6303 hBOOL(hAPP(hAPP(A,f364(A,B)),f364(A,B))) | -c_Equiv__Relations_Opart__equivp(B,A) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(3150)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f364(A,B)),f364(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6303,b,6298,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f364(A,B)),f364(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6303,b,6299,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f364(A,B)),f364(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6303,b,6300,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f364(A,B)),f364(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6303,b,6301,a)]. 193.58/193.45 6304 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f438(B,A)),f438(B,A))) # label(fact_part__equivpE) # label(axiom). [clausify(3919)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f438(A,B)),f438(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6304,a,6298,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f438(A,B)),f438(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hAPP(A,f353(A,B)) = hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6304,a,6299,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f438(A,B)),f438(A,B))) | hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6304,a,6300,a)]. 193.58/193.45 Derived: hBOOL(hAPP(hAPP(A,f438(A,B)),f438(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,f352(A,B)),f352(A,B))) | hAPP(A,f353(A,B)) != hAPP(A,f352(A,B)) | -hBOOL(hAPP(hAPP(A,f353(A,B)),f353(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(6304,a,6301,a)]. 195.06/194.90 6305 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(3919)]. 195.06/194.90 6306 -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | c_Equiv__Relations_Opart__equivp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5668,b,5662,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,E),D)). [resolve(6306,c,6290,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f137(B,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_fun(A,tc_HOL_Obool)),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),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_COMBS(A,A,tc_HOL_Obool),B),c_COMBI(A))))),hAPP(hAPP(c_COMBC(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A)),c_fequal)),B))))). [resolve(6306,c,6291,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,D),E)) | -hBOOL(hAPP(hAPP(B,E),F)) | hBOOL(hAPP(hAPP(B,D),F)). [resolve(6306,c,6292,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,D),D)). [resolve(6306,c,6293,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,D),E)) | hAPP(B,E) = hAPP(B,D). [resolve(6306,c,6294,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,E),E)). [resolve(6306,c,6295,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,D),E)) | -hBOOL(hAPP(hAPP(B,D),D)) | hAPP(B,E) != hAPP(B,D) | -hBOOL(hAPP(hAPP(B,E),E)). [resolve(6306,c,6296,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f351(B,A)),f351(B,A))). [resolve(6306,c,6297,a)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f364(B,A)),f364(B,A))). [resolve(6306,c,6303,b)]. 195.06/194.90 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f438(B,A)),f438(B,A))). [resolve(6306,c,6304,a)]. 195.06/194.90 6307 -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | c_Equiv__Relations_Opart__equivp(A,B) | -c_Equiv__Relations_Oequivp(A,B). [resolve(5668,b,5665,b)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,E),D)). [resolve(6307,c,6290,a)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,D),E)) | -hBOOL(hAPP(hAPP(B,E),F)) | hBOOL(hAPP(hAPP(B,D),F)). [resolve(6307,c,6292,a)]. 195.58/195.44 6308 -c_Equiv__Relations_Opart__equivp(A,B) | c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(5669,a,5663,a)]. 195.58/195.44 Derived: c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6308,a,6289,b)]. 195.58/195.44 Derived: c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6308,a,6298,a)]. 195.58/195.44 Derived: c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hAPP(B,f353(B,A)) = hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6308,a,6299,a)]. 195.58/195.44 Derived: c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6308,a,6300,a)]. 195.58/195.44 Derived: c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | hAPP(B,f353(B,A)) != hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6308,a,6301,a)]. 195.58/195.44 6309 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)). [resolve(5669,a,5664,a)]. 195.58/195.44 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),D)) | hBOOL(hAPP(hAPP(A,B),D)) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6309,a,6289,b)]. 195.58/195.44 6310 -c_Equiv__Relations_Opart__equivp(A,B) | -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6222,c,6220,a)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6310,a,6298,a)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hAPP(B,f353(B,A)) = hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6310,a,6299,a)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6310,a,6300,a)]. 195.58/195.44 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,f352(B,A)),f352(B,A))) | hAPP(B,f353(B,A)) != hAPP(B,f352(B,A)) | -hBOOL(hAPP(hAPP(B,f353(B,A)),f353(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6310,a,6301,a)]. 212.17/212.04 Derived: -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Orefl__on(A,c_Orderings_Otop__class_Otop(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6310,a,6306,c)]. 212.17/212.04 6311 hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -c_Nitpick_Oless__eq__frac(A,B,C) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom). [clausify(1322)]. 212.17/212.04 6312 -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Nitpick_Oless__eq__frac(A,B,C) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom). [clausify(1322)]. 212.17/212.04 6313 class_Nat_Osize(tc_HOL_Obool) # label(arity_HOL__Obool__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 6314 -class_Nat_Osize(A) | c_FunDef_Ois__measure(A,c_Nat_Osize__class_Osize(A)) # label(fact_measure__size) # label(axiom). [clausify(1429)]. 212.17/212.04 6315 class_Nat_Osize(tc_Option_Ooption(A)) # label(arity_Option__Ooption__Nat_Osize) # label(axiom). [clausify(1794)]. 212.17/212.04 6316 class_Nat_Osize(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 6317 class_Nat_Osize(tc_prod(A,B)) # label(arity_prod__Nat_Osize) # label(axiom). [clausify(2203)]. 212.17/212.04 6318 class_Nat_Osize(tc_Com_Ocom) # label(arity_Com__Ocom__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 6319 class_Nat_Osize(tc_sum(A,B)) # label(arity_sum__Nat_Osize) # label(axiom). [clausify(2988)]. 212.17/212.04 6320 class_Nat_Osize(tc_Code__Evaluation_Oterm) # label(arity_Code____Evaluation__Oterm__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 6321 class_Nat_Osize(tc_Hoare__Mirabelle_Otriple(A)) # label(arity_Hoare____Mirabelle__Otriple__Nat_Osize) # label(axiom). [clausify(3204)]. 212.17/212.04 6322 class_Nat_Osize(tc_Lazy__Sequence_Olazy__sequence(A)) # label(arity_Lazy____Sequence__Olazy____sequence__Nat_Osize) # label(axiom). [clausify(3730)]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_Lazy__Sequence_Olazy__sequence(A),c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(A))). [resolve(6322,a,6314,a)]. 212.17/212.04 6323 class_Nat_Osize(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_Code__Numeral_Ocode__numeral,c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral)). [resolve(6323,a,6314,a)]. 212.17/212.04 6324 class_Nat_Osize(tc_List_Olist(A)) # label(arity_List__Olist__Nat_Osize) # label(axiom). [clausify(4002)]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_List_Olist(A),c_Nat_Osize__class_Osize(tc_List_Olist(A))). [resolve(6324,a,6314,a)]. 212.17/212.04 6325 class_Nat_Osize(tc_Com_Ostate) # label(arity_Com__Ostate__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_Com_Ostate,c_Nat_Osize__class_Osize(tc_Com_Ostate)). [resolve(6325,a,6314,a)]. 212.17/212.04 6326 class_Nat_Osize(tc_Nitpick_Opair__box(A,B)) # label(arity_Nitpick__Opair____box__Nat_Osize) # label(axiom). [clausify(4286)]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_Nitpick_Opair__box(A,B),c_Nat_Osize__class_Osize(tc_Nitpick_Opair__box(A,B))). [resolve(6326,a,6314,a)]. 212.17/212.04 6327 class_Nat_Osize(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osize) # label(axiom). [assumption]. 212.17/212.04 Derived: c_FunDef_Ois__measure(tc_Nat_Onat,c_Nat_Osize__class_Osize(tc_Nat_Onat)). [resolve(6327,a,6314,a)]. 212.17/212.04 6328 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 212.17/212.04 6329 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),c_Orderings_Obot__class_Obot(A)) # label(fact_inf__bot__right) # label(axiom). [clausify(1432)]. 213.07/212.93 6330 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),c_Orderings_Obot__class_Obot(A)),B) # label(fact_inf__bot__left) # label(axiom). [clausify(1549)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),c_Orderings_Obot__class_Obot(tc_HOL_Obool)). [resolve(6328,a,6329,a)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_HOL_Obool)),A). [resolve(6328,a,6330,a)]. 213.07/212.93 6331 -class_Lattices_Obounded__lattice__bot(A) | hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),c_Orderings_Obot__class_Obot(A)) = B # label(fact_sup__bot__right) # label(axiom). [clausify(3248)]. 213.07/212.93 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(6331,a,6328,a)]. 213.07/212.93 6332 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(4166)]. 213.07/212.93 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Obot__class_Obot(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),c_Orderings_Obot__class_Obot(tc_fun(B,A))). [resolve(6332,b,6329,a)]. 213.07/212.93 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Obot__class_Obot(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),c_Orderings_Obot__class_Obot(tc_fun(B,A))),C). [resolve(6332,b,6330,a)]. 213.07/212.93 Derived: -class_Lattices_Obounded__lattice(A) | hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),C),c_Orderings_Obot__class_Obot(tc_fun(B,A))) = C. [resolve(6332,b,6331,a)]. 213.07/212.93 6333 -class_Lattices_Obounded__lattice__bot(A) | hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),c_Orderings_Obot__class_Obot(A)),B) = B # label(fact_sup__bot__left) # label(axiom). [clausify(4738)]. 213.07/212.93 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_HOL_Obool)),A) = A. [resolve(6333,a,6328,a)]. 213.07/212.93 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),c_Orderings_Obot__class_Obot(tc_fun(A,B))),C) = C | -class_Lattices_Obounded__lattice(B). [resolve(6333,a,6332,b)]. 213.07/212.93 6334 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) = B # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4806)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A. [resolve(6334,a,6328,a)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | -class_Lattices_Obounded__lattice(B). [resolve(6334,a,6332,b)]. 213.07/212.93 6335 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) = C # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4806)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = B. [resolve(6335,a,6328,a)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = D | -class_Lattices_Obounded__lattice(B). [resolve(6335,a,6332,b)]. 213.07/212.93 6336 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) != B | c_Orderings_Obot__class_Obot(A) != C # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4806)]. 213.07/212.93 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != B. [resolve(6336,a,6328,a)]. 221.68/221.56 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != D | -class_Lattices_Obounded__lattice(B). [resolve(6336,a,6332,b)]. 221.68/221.56 6337 -class_Rings_Ocomm__semiring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_comm__semiring__class_Odistrib) # label(axiom). [clausify(4548)]. 221.68/221.56 6338 class_Rings_Ocomm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring) # label(axiom). [assumption]. 221.68/221.56 6339 class_Rings_Ocomm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring) # label(axiom). [assumption]. 221.68/221.56 6340 class_Rings_Ocomm__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ocomm__semiring) # label(axiom). [assumption]. 221.68/221.56 6341 -class_Rings_Oordered__comm__semiring(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)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_comm__mult__left__mono) # label(axiom). [clausify(3516)]. 221.68/221.56 6342 class_Rings_Oordered__comm__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 221.68/221.56 6343 class_Rings_Oordered__comm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 221.68/221.56 6344 class_Rings_Oordered__comm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 221.68/221.56 6345 class_Groups_Osgn__if(tc_Int_Oint) # label(arity_Int__Oint__Groups_Osgn__if) # label(axiom). [assumption]. 221.68/221.56 6346 -class_Groups_Osgn__if(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Osgn__class_Osgn(A,B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) # label(fact_sgn__if) # label(axiom). [clausify(1676)]. 221.68/221.56 6347 -class_Groups_Osgn__if(A) | c_Groups_Ozero__class_Ozero(A) = B | -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__if) # label(axiom). [clausify(1676)]. 221.68/221.56 6348 -class_Groups_Osgn__if(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Ozero__class_Ozero(A) # label(fact_sgn__if) # label(axiom). [clausify(1676)]. 221.68/221.56 6349 -class_Groups_Osgn__if(A) | c_Groups_Osgn__class_Osgn(A,c_Groups_Ozero__class_Ozero(A)) = c_Groups_Ozero__class_Ozero(A) # label(fact_sgn0) # label(axiom). [clausify(2753)]. 221.68/221.56 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)),A)) | 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)). [resolve(6345,a,6346,a)]. 221.68/221.56 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)),A)) | c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = c_Groups_Oone__class_Oone(tc_Int_Oint). [resolve(6345,a,6347,a)]. 221.68/221.56 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(6345,a,6348,a)]. 221.68/221.56 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(6345,a,6349,a)]. 225.58/225.43 6350 class_Divides_Oring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Oring__div) # label(axiom). [assumption]. 225.58/225.43 6351 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),C) = c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),D),C) # label(fact_mod__minus__cong) # label(axiom). [clausify(1723)]. 225.58/225.43 6352 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Divides_Odiv__class_Omod(A,B,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),C) # label(fact_mod__minus__eq) # label(axiom). [clausify(2515)]. 225.58/225.43 6353 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C),D) # label(fact_mod__diff__right__eq) # label(axiom). [clausify(3088)]. 225.58/225.43 6354 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__left__eq) # label(axiom). [clausify(3289)]. 225.58/225.43 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,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C),B). [resolve(6350,a,6351,a)]. 225.58/225.43 Derived: 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,A,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),B). [resolve(6350,a,6352,a)]. 225.58/225.43 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(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_Ominus__class_Ominus(tc_Int_Oint),A),B),C). [resolve(6350,a,6353,a)]. 225.58/225.43 6355 -class_Divides_Oring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B) # label(fact_dvd__neg__div) # label(axiom). [clausify(3975)]. 225.58/225.43 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(c_Groups_Ouminus__class_Ouminus(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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),A). [resolve(6355,a,6350,a)]. 225.58/225.43 6356 -class_Divides_Oring__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_Ominus__class_Ominus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),F),C) # label(fact_mod__diff__cong) # label(axiom). [clausify(4398)]. 225.58/225.43 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_Ominus__class_Ominus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),E),B). [resolve(6356,a,6350,a)]. 225.58/225.43 6357 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(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_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__eq) # label(axiom). [clausify(4622)]. 241.28/241.13 Derived: 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,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_Ominus__class_Ominus(tc_Int_Oint),A),C),B). [resolve(6357,a,6350,a)]. 241.28/241.13 6358 -class_Divides_Oring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_dvd__div__neg) # label(axiom). [clausify(4839)]. 241.28/241.13 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(c_Groups_Ouminus__class_Ouminus(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),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(6358,a,6350,a)]. 241.28/241.13 6359 -class_Groups_Osemigroup__add(A) | hAPP(c_List_Ofoldl(A,A,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(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_foldl__assoc) # label(axiom). [clausify(3163)]. 241.28/241.13 6360 class_Groups_Osemigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Osemigroup__add) # label(axiom). [assumption]. 241.28/241.13 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,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(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(6359,a,6360,a)]. 241.28/241.13 6361 class_Groups_Osemigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Osemigroup__add) # label(axiom). [assumption]. 241.28/241.13 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,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(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(6361,a,6359,a)]. 241.28/241.13 6362 class_Groups_Osemigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Osemigroup__add) # label(axiom). [assumption]. 241.28/241.13 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,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(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(6362,a,6359,a)]. 241.28/241.13 6363 class_Orderings_Owellorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Owellorder) # label(axiom). [assumption]. 241.28/241.13 6364 -class_Orderings_Owellorder(A) | c_Wellfounded_Owf(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),c_Orderings_Oord__class_Oless(A)))) # label(fact_wf) # label(axiom). [clausify(2116)]. 241.28/241.13 Derived: c_Wellfounded_Owf(tc_Nat_Onat,hAPP(c_Set_OCollect(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)))). [resolve(6363,a,6364,a)]. 241.28/241.13 6365 class_Rings_Oordered__ring__abs(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring__abs) # label(axiom). [assumption]. 241.28/241.13 6366 -class_Rings_Oordered__ring__abs(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),c_Groups_Ozero__class_Ozero(A))) | 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)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) # label(fact_abs__eq__mult) # label(axiom). [clausify(2208)]. 243.78/243.68 6367 -class_Rings_Oordered__ring__abs(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)) | 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)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) # label(fact_abs__eq__mult) # label(axiom). [clausify(2208)]. 243.78/243.68 6368 -class_Rings_Oordered__ring__abs(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))) | 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)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) # label(fact_abs__eq__mult) # label(axiom). [clausify(2208)]. 243.78/243.68 6369 -class_Rings_Oordered__ring__abs(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)),C)) | 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)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) # label(fact_abs__eq__mult) # label(axiom). [clausify(2208)]. 243.78/243.68 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_Groups_Ozero__class_Ozero(tc_Int_Oint))) | 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)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)). [resolve(6365,a,6366,a)]. 243.78/243.68 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)) | 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)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)). [resolve(6365,a,6367,a)]. 243.78/243.68 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))) | 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)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)). [resolve(6365,a,6368,a)]. 243.78/243.68 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)),B)) | 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)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)). [resolve(6365,a,6369,a)]. 243.78/243.68 6370 class_Rings_Olinordered__semiring__1__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 243.78/243.68 6371 -class_Rings_Olinordered__semiring__1__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__lt) # label(axiom). [clausify(2676)]. 256.65/256.47 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),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | 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),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(6370,a,6371,a)]. 256.65/256.47 6372 -c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | c_List_Onat__list(A) # label(fact_nat__list__def) # label(axiom). [clausify(4356)]. 256.65/256.47 6373 -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(2831)]. 256.65/256.47 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(6372,b,6373,a)]. 256.65/256.47 6374 c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | -c_List_Onat__list(A) # label(fact_nat__list__def) # label(axiom). [clausify(4356)]. 256.65/256.47 6375 c_Nitpick_Orefl_H(A,B) | -hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),f325(B,A)),f325(B,A))),B)) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(2860)]. 256.65/256.47 6376 -c_Nitpick_Orefl_H(A,B) | hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),C),C)),B)) # label(fact_Nitpick_Orefl_H__def) # label(axiom). [clausify(2860)]. 256.65/256.47 Derived: -hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),f325(B,A)),f325(B,A))),B)) | hBOOL(hAPP(hAPP(c_member(tc_prod(A,A)),hAPP(hAPP(c_Product__Type_OPair(A,A),C),C)),B)). [resolve(6375,a,6376,a)]. 256.65/256.47 6377 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C))),F))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),F))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(3984)]. 256.65/256.47 6378 class_Rings_Ocomm__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__ring) # label(axiom). [assumption]. 256.65/256.47 Derived: -class_Rings_Odvd(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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B))),E))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),E))). [resolve(6377,b,6378,a)]. 256.65/256.47 6379 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C))),F))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),F))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(3984)]. 256.65/256.47 Derived: -class_Rings_Odvd(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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B))),E))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),E))). [resolve(6379,b,6378,a)]. 278.30/278.14 6380 -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_I4_J) # label(axiom). [clausify(4245)]. 278.30/278.14 6381 -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_I4_J) # label(axiom). [clausify(4245)]. 278.30/278.14 6382 c_Nitpick_Ounknown(tc_HOL_Obool) | hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(A,A)),B)) | -c_Nitpick_Owf_H(A,B) # label(fact_Nitpick_Owf_H__def) # label(axiom). [clausify(3255)]. 278.30/278.14 6383 -c_Nitpick_Ounknown(tc_HOL_Obool) | -c_Wellfounded_Oacyclic(A,B) | c_Nitpick_Owf_H(A,B) # label(fact_Nitpick_Owf_H__def) # label(axiom). [clausify(3255)]. 278.30/278.14 6384 -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(A,A)),B)) | -c_Wellfounded_Oacyclic(A,B) | c_Nitpick_Owf_H(A,B) # label(fact_Nitpick_Owf_H__def) # label(axiom). [clausify(3255)]. 278.30/278.14 6385 c_Wellfounded_Oacyclic(A,B) | -c_Nitpick_Owf_H(A,B) # label(fact_Nitpick_Owf_H__def) # label(axiom). [clausify(3255)]. 278.30/278.14 278.30/278.14 ============================== end predicate elimination ============= 278.30/278.14 278.30/278.14 Auto_denials: (non-Horn, no changes). 278.30/278.14 278.30/278.14 Term ordering decisions: 278.30/278.14 Function symbol KB weights: tc_Int_Oint=1. tc_Nat_Onat=1. tc_HOL_Obool=1. tc_Code__Numeral_Ocode__numeral=1. c_Nat_OSuc=1. c_Int_OPls=1. c_Int_Onat=1. c_fequal=1. c_fconj=1. c_Int_OMin=1. c_Nat__Numeral_Oneg=1. c_fNot=1. c_Code__Numeral_Oof__nat=1. c_Code__Numeral_Onat__of=1. t_b=1. c_Int_Ointrel=1. c_Int_OAbs__Integ=1. tc_Com_Ostate=1. c_fimplies=1. c_Int_ORep__Integ=1. tc_Com_Ocom=1. c_fdisj=1. c_Int_OInteg=1. t_a=1. c_Divides_OnegateSnd=1. c_Wellfounded_Oless__than=1. tc_Product__Type_Ounit=1. c_FunDef_Opair__less=1. c_Nitpick_Onorm__frac__rel=1. c_Divides_OnegDivAlg__rel=1. c_Divides_OposDivAlg__rel=1. c_FunDef_Opair__leq=1. c_Wellfounded_Opred__nat=1. c_Code__Numeral_Oint__of=1. c_Lazy__Sequence_Osmall__lazy_H__rel=1. c_List_Oupto__rel=1. c_Nitpick_Onat__gcd__rel=1. c_fFalse=1. tc_Code__Evaluation_Oterm=1. v_c0=1. c_FunDef_Omax__strict=1. c_FunDef_Omax__weak=1. c_FunDef_Omin__strict=1. c_FunDef_Omin__weak=1. v_G=1. c_Com_Ocom_OSKIP=1. c_Nitpick_Oint__gcd=1. c_fTrue=1. v_F=1. v_P_H=1. v_Q_H=1. c_Code__Numeral_Osubtract__code__numeral=1. v_x=1. v_P=1. v_Q=1. c_Nitpick_OFrac=1. hAPP=1. tc_fun=1. tc_prod=1. c_Product__Type_OPair=1. c_Big__Operators_Ocomm__monoid__add__class_Osetsum=1. c_List_Omap=1. c_Big__Operators_Ocomm__monoid__mult__class_Osetprod=1. c_Complete__Lattice_Ocomplete__lattice__class_OSUPR=1. c_Complete__Lattice_Ocomplete__lattice__class_OINFI=1. c_SetInterval_Oord__class_OatLeastLessThan=1. c_Transitive__Closure_Ortrancl=1. c_Product__Type_OSigma=1. c_List_Ofilter=1. c_COMBK=1. c_Big__Operators_Olinorder__class_OMax=1. c_Big__Operators_Olinorder__class_OMin=1. c_Complete__Lattice_OSup__class_OSup=1. c_Product__Type_Osnd=1. c_Product__Type_Ofst=1. c_Finite__Set_Ofold1=1. c_List_Ozip=1. c_Relation_Oconverse=1. c_List_Oupt=1. c_Transitive__Closure_Otrancl=1. c_Nat_Ocompow=1. c_Wellfounded_Oaccp=1. c_Complete__Lattice_OInf__class_OInf=1. c_List_Olist__update=1. c_List_Olistrel1=1. c_List_Oconcat=1. c_List_Olistrel=1. c_Big__Operators_Olattice__class_OInf__fin=1. c_Big__Operators_Olattice__class_OSup__fin=1. c_List_Olists=1. c_List_Oremdups=1. c_Relation_ODomain=1. c_List_Obutlast=1. c_List_Olistsp=1. c_List_Olexord=1. c_Relation_ORange=1. c_HOL_OLet=1. c_List_Olast=1. c_List_Otranspose=1. c_Groups_Osgn__class_Osgn=1. c_Wellfounded_Oacc=1. c_Divides_OposDivAlg=1. c_Divides_Odivmod__int__rel=1. c_Divides_Odivmod__nat__rel=1. c_Divides_OnegDivAlg=1. c_List_Olinorder__class_Osorted__list__of__set=1. c_List_Orotate=1. c_List_Oupto=1. tc_sum=1. c_Relation_OId__on=1. c_Wellfounded_Omax__ext=1. c_List_OremoveAll=1. c_HOL_OThe=1. c_Hilbert__Choice_OEps=1. c_List_Olex=1. c_Divides_Odivmod__int=1. c_Relation_Oinv__image=1. c_Divides_Odivmod__nat=1. c_List_Omeasures=1. c_Nitpick_Onorm__frac=1. c_Com_Ocom_OWhile=1. c_Com_Ocom_OSemi=1. c_Quotient_ORespects=1. c_Nitpick_Onat__gcd=1. c_Random_Opick=1. c_Divides_Opdivmod=1. c_Lazy__Sequence_Osmall__lazy_H=1. c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy=1. c_List_Olexn=1. c_Nat__Transfer_Otsub=1. c_Predicate_OPowp=1. c_Random_Oselect__weight=1. c_Enum_Osublists=1. c_Wellfounded_Omin__ext=1. c_Code__Numeral_Onat__of__aux=1. c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size=1. c_Nitpick_OAbs__Frac=1. c_Lazy__Sequence_Oflat=1. c_Orderings_Oord_Omax=1. c_Orderings_Oord_Omin=1. c_Random_Olog=1. c_List_Olenlex=1. c_List_Olistset=1. c_Quickcheck_Obeyond=1. c_Random_Oselect=1. c_Set_Othe__elem=1. tc_Nitpick_Opair__box=1. c_FunDef_Orp__inv__image=1. c_New__DSequence_Oneg__single=1. c_New__DSequence_Opos__single=1. c_Nitpick_Ocard_H=1. c_Nitpick_Oint__lcm=1. c_Nitpick_Onat__lcm=1. c_Random_Oinc__shift=1. c_SMT_Oz3div=1. c_SMT_Oz3mod=1. c_Code__Numeral_Odiv__mod__code__numeral=1. c_Enum_Oenum__the=1. c_Int_Onat__aux=1. c_Lazy__Sequence_Ohb__single=1. c_Nitpick_Oinverse__frac=1. c_Nitpick_Onumber__of__frac=1. c_Nitpick_Ouminus__frac=1. f5=1. f8=1. f9=1. f15=1. f17=1. f19=1. f20=1. f23=1. f24=1. f25=1. f26=1. f35=1. f36=1. f40=1. f41=1. f45=1. f61=1. f69=1. f72=1. f77=1. f78=1. f79=1. f100=1. f101=1. f117=1. f126=1. f128=1. f137=1. f148=1. f149=1. f162=1. f165=1. f169=1. f181=1. f182=1. f200=1. f201=1. f202=1. f203=1. f204=1. f212=1. f213=1. f217=1. f218=1. f230=1. f235=1. f248=1. f284=1. f289=1. f291=1. f293=1. f294=1. f295=1. f296=1. f297=1. f300=1. f311=1. f312=1. f318=1. f325=1. f326=1. f327=1. f330=1. f331=1. f332=1. f344=1. f345=1. f351=1. f352=1. f353=1. f361=1. f364=1. f370=1. f373=1. f378=1. f379=1. f381=1. f382=1. f387=1. f392=1. f400=1. f401=1. f402=1. f409=1. f410=1. f411=1. f436=1. f437=1. f438=1. f444=1. f452=1. f453=1. f459=1. f460=1. f461=1. f474=1. f493=1. f494=1. f495=1. f504=1. f514=1. f515=1. f522=1. f523=1. f532=1. f567=1. c_Groups_Ozero__class_Ozero=1. c_Orderings_Oord__class_Oless__eq=1. c_Orderings_Oord__class_Oless=1. c_member=1. tc_List_Olist=1. c_Groups_Otimes__class_Otimes=1. c_Int_Onumber__class_Onumber__of=1. c_Groups_Oplus__class_Oplus=1. c_Finite__Set_Ofinite=1. c_Orderings_Obot__class_Obot=1. c_Groups_Ominus__class_Ominus=1. c_Groups_Oone__class_Oone=1. c_Set_Oinsert=1. c_Lattices_Osemilattice__sup__class_Osup=1. c_Lattices_Osemilattice__inf__class_Oinf=1. c_Orderings_Oord__class_Omin=1. c_Nat_Osize__class_Osize=1. c_List_Olist_OCons=1. c_Orderings_Oord__class_Omax=1. c_List_Oset=1. c_Orderings_Otop__class_Otop=1. c_List_Olist_ONil=1. c_Divides_Odiv__class_Odiv=1. c_Groups_Ouminus__class_Ouminus=1. c_Power_Opower__class_Opower=1. c_Nat_Osemiring__1__class_Oof__nat=1. c_Rings_Oinverse__class_Odivide=1. c_Int_OBit1=1. c_List_Oappend=1. c_Rings_Odvd__class_Odvd=1. c_Int_OBit0=1. c_Set_OCollect=1. c_Finite__Set_Ocard=1. c_List_Onth=1. c_List_Odistinct=1. c_Groups_Oabs__class_Oabs=1. c_List_Otake=1. c_List_Omonoid__add__class_Olistsum=1. c_COMBI=1. c_List_Orev=1. c_SetInterval_Oord__class_OlessThan=1. tc_Lazy__Sequence_Olazy__sequence=1. c_List_Odrop=1. tc_Hoare__Mirabelle_Otriple=1. c_HOL_OEx=1. c_SetInterval_Oord__class_OatMost=1. c_Int_Oring__1__class_Oof__int=1. c_SetInterval_Oord__class_OgreaterThan=1. c_SetInterval_Oord__class_OatLeast=1. c_If=1. c_Equiv__Relations_Oquotient=1. c_Fun_Oid=1. c_Hoare__Mirabelle_Otriple_Otriple=1. c_Int_Opred=1. c_Set_OPow=1. tc_Option_Ooption=1. c_Relation_OId=1. c_List_Otl=1. c_List_Ohd=1. c_Int_Oring__1__class_OInts=1. c_Relation_OField=1. c_List_Orotate1=1. c_Int_Osucc=1. c_Divides_Oadjust=1. c_Enum_Oenum__class_Oenum__all=1. c_Enum_Oenum__class_Oenum__ex=1. c_Com_Ocom_Ocom__size=1. c_Code__Numeral_OSuc__code__numeral=1. c_List_Omember=1. c_Nat_Osemiring__1__class_ONats=1. c_Lazy__Sequence_Olazy__sequence_OEmpty=1. c_Nitpick_Odenom=1. c_Nitpick_Onum=1. c_Lazy__Sequence_Osingle=1. c_Wellfounded_Ofinite__psubset=1. c_HOL_OAll=1. c_Lazy__Sequence_Oempty=1. c_Lazy__Sequence_Oyield=1. c_List_Oembed__list=1. c_List_Oreturn__list=1. c_Nitpick_Ofrac=1. c_Wellfounded_Omeasure=1. c_Code__Numeral_Ocode__numeral_Ocode__numeral__size=1. c_List_Otranspose__rel=1. c_Nat_Onat_Onat__size=1. c_Random_Orange=1. c_DSequence_Oempty=1. c_DSequence_Osingle=1. c_HOL_Obool_Obool__size=1. c_Int_Oint__ge__less__than=1. c_Int_Oint__ge__less__than2=1. c_New__DSequence_Opos__empty=1. c_New__DSequence_Opos__not__seq=1. c_New__Random__Sequence_Oneg__single=1. c_New__Random__Sequence_Opos__single=1. c_Nitpick_ORep__Frac=1. c_Random__Sequence_Osingle=1. c_DSequence_Ounion=1. c_Lazy__Sequence_Ohb__not__seq=1. c_Lazy__Sequence_Ohit__bound=1. c_Lazy__Sequence_Oyieldn=1. c_Nat_Ofunpow=1. c_Nitpick_Oone__frac=1. c_Nitpick_Ozero__frac=1. c_Set_OBall=1. c_Set_OBex=1. f71=1. f145=1. f146=1. f171=1. f186=1. f268=1. f269=1. f270=1. f280=1. f281=1. f307=1. f391=1. f403=1. f413=1. f414=1. f445=1. f458=1. f516=1. f517=1. f542=1. f562=1. c_COMBB=1. c_COMBC=1. c_Divides_Odiv__class_Omod=1. c_Set_Oimage=1. c_COMBS=1. c_Finite__Set_Ofold__image=1. c_SetInterval_Oord__class_OatLeastAtMost=1. c_Product__Type_Oprod_Oprod__case=1. c_Big__Operators_Olattice_OInf__fin=1. c_Big__Operators_Olattice_OSup__fin=1. c_List_Olinorder__class_Oinsort__key=1. c_Relation_OImage=1. c_Finite__Set_Ofold=1. c_Relation_Orel__comp=1. c_Set_Ovimage=1. c_List_Oreplicate=1. c_SetInterval_Oord__class_OgreaterThanAtMost=1. c_SetInterval_Oord__class_OgreaterThanLessThan=1. c_List_OtakeWhile=1. c_List_Oremove1=1. c_List_OdropWhile=1. c_Predicate_Oconversep=1. c_List_Osublist=1. c_Lazy__Sequence_Olazy__sequence_OInsert=1. c_List_Olist_Olist__size=1. c_Com_Ocom_OCond=1. c_Com_Ocom_OLocal=1. c_Finite__Set_Ofold1Set=1. c_Enum_On__lists=1. c_List_Oinsert=1. c_Lazy__Sequence_Oappend=1. c_List_Opartition=1. c_Nat_Onat_Onat__rec=1. c_Hoare__Mirabelle_Opeek__and=1. c_Lazy__Sequence_Obind=1. c_List_Osplice=1. c_Predicate_ODomainP=1. c_Predicate_ORangeP=1. c_FunDef_OTHE__default=1. c_Lazy__Sequence_Oanamorph=1. c_List_Oset__Cons=1. c_Power_Opower_Opower=1. c_Wellfounded_Omlex__prod=1. c_FunDef_Oin__rel=1. c_New__DSequence_Opos__union=1. c_Partial__Function_Oflat__lub=1. c_Product__Type_Ointernal__split=1. c_Random_Ominus__shift=1. c_SetInterval_Oord_OatLeast=1. c_SetInterval_Oord_OatMost=1. c_SetInterval_Oord_OgreaterThan=1. c_SetInterval_Oord_OlessThan=1. c_Hoare__Mirabelle_Otriple_Otriple__size=1. c_Nitpick_Oplus__frac=1. c_Nitpick_Otimes__frac=1. c_Random__Sequence_Oempty=1. f1=1. f2=1. f27=1. f28=1. f31=1. f32=1. f33=1. f34=1. f37=1. f42=1. f43=1. f47=1. f62=1. f63=1. f64=1. f65=1. f66=1. f70=1. f80=1. f81=1. f82=1. f84=1. f85=1. f86=1. f87=1. f88=1. f93=1. f102=1. f103=1. f107=1. f108=1. f111=1. f112=1. f113=1. f114=1. f118=1. f119=1. f120=1. f121=1. f122=1. f123=1. f124=1. f127=1. f132=1. f136=1. f138=1. f143=1. f153=1. f154=1. f155=1. f159=1. f160=1. f161=1. f164=1. f166=1. f172=1. f177=1. f183=1. f184=1. f185=1. f187=1. f193=1. f194=1. f197=1. f210=1. f211=1. f214=1. f215=1. f225=1. f226=1. f227=1. f229=1. f232=1. f233=1. f234=1. f237=1. f238=1. f243=1. f245=1. f249=1. f250=1. f255=1. f256=1. f265=1. f266=1. f271=1. f272=1. f275=1. f282=1. f283=1. f285=1. f286=1. f298=1. f308=1. f310=1. f313=1. f319=1. f320=1. f322=1. f323=1. f324=1. f343=1. f346=1. f348=1. f349=1. f350=1. f354=1. f368=1. f369=1. f384=1. f386=1. f394=1. f395=1. f396=1. f399=1. f412=1. f415=1. f416=1. f417=1. f418=1. f419=1. f420=1. f423=1. f424=1. f426=1. f428=1. f429=1. f432=1. f433=1. f434=1. f446=1. f447=1. f456=1. f457=1. f462=1. f464=1. f466=1. f467=1. f468=1. f478=1. f479=1. f480=1. f481=1. f483=1. f484=1. f489=1. f490=1. f501=1. f506=1. f507=1. f508=1. f509=1. f524=1. f525=1. f526=1. f527=1. f528=1. f529=1. f530=1. f539=1. f543=1. f544=1. f552=1. f555=1. f560=1. f563=1. f564=1. f565=1. f566=1. f570=1. f571=1. f574=1. f575=1. f576=1. f577=1. c_Fun_Ocomp=1. c_List_Ofoldl=1. c_Hilbert__Choice_Oinv__into=1. c_List_Olinorder__class_Osort__key=1. c_Product__Type_Oapsnd=1. c_Nat_Onat_Onat__case=1. c_Product__Type_Oapfst=1. c_Product__Type_Oscomp=1. c_Sum__Type_OPlus=1. c_Fun_Othe__inv__into=1. c_List_Olist_Olist__case=1. c_SetInterval_Oord_OatLeastAtMost=1. c_Nitpick_Opair__box_OPairBox=1. c_Wellfounded_Olex__prod=1. c_Product__Type_Ocurry=1. c_Enum_Oproduct=1. c_Lazy__Sequence_Omap=1. c_List_Omaps=1. c_Nat_Osemiring__1__class_Oof__nat__aux=1. c_SetInterval_Oord_OgreaterThanLessThan=1. c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec=1. c_Lazy__Sequence_Oproduct=1. c_New__DSequence_Oneg__decr__bind=1. c_New__DSequence_Opos__decr__bind=1. c_Random_Oiterate=1. c_Random__Sequence_ORandom=1. c_Recdef_Osame__fst=1. c_Code__Numeral_Ocode__numeral_Ocode__numeral__case=1. c_Lazy__Sequence_Ohb__bind=1. c_New__DSequence_Oneg__bind=1. c_New__DSequence_Opos__bind=1. c_New__Random__Sequence_Oneg__bind=1. c_New__Random__Sequence_Opos__bind=1. c_Nitpick_Osetsum_H=1. c_Quotient_OBabs=1. c_Hoare__Mirabelle_Otriple_Otriple__case=1. c_Hoare__Mirabelle_Otriple_Otriple__rec=1. c_New__Random__Sequence_Oneg__map=1. c_New__Random__Sequence_Opos__empty=1. c_New__Random__Sequence_Opos__map=1. c_New__Random__Sequence_Opos__not__random__dseq=1. c_Nitpick_Oprod=1. c_Random__Sequence_Obind=1. c_Random__Sequence_Omap=1. f3=1. f4=1. f6=1. f10=1. f11=1. f12=1. f13=1. f14=1. f21=1. f29=1. f30=1. f44=1. f46=1. f51=1. f52=1. f54=1. f55=1. f56=1. f58=1. f59=1. f60=1. f67=1. f68=1. f83=1. f105=1. f115=1. f116=1. f142=1. f144=1. f147=1. f163=1. f168=1. f170=1. f175=1. f180=1. f195=1. f196=1. f198=1. f199=1. f205=1. f236=1. f240=1. f246=1. f251=1. f252=1. f253=1. f254=1. f267=1. f274=1. f277=1. f278=1. f279=1. f302=1. f303=1. f304=1. f305=1. f306=1. f315=1. f316=1. f333=1. f334=1. f355=1. f356=1. f359=1. f360=1. f362=1. f363=1. f365=1. f366=1. f367=1. f374=1. f375=1. f388=1. f389=1. f397=1. f398=1. f404=1. f408=1. f422=1. f430=1. f431=1. f435=1. f463=1. f470=1. f471=1. f472=1. f473=1. f476=1. f477=1. f482=1. f485=1. f486=1. f487=1. f488=1. f492=1. f497=1. f500=1. f510=1. f513=1. f521=1. f531=1. f535=1. f536=1. f537=1. f538=1. f547=1. f548=1. f549=1. f551=1. f553=1. f556=1. f557=1. f558=1. f559=1. f568=1. f569=1. c_Finite__Set_Ofold__graph=1. c_List_Ofoldr=1. c_List_Olinorder__class_Oinsort__insert__key=1. c_Fun_Ofun__upd=1. c_Predicate_Opred__comp=1. c_SetInterval_Oord_OatLeastLessThan=1. c_SetInterval_Oord_OgreaterThanAtMost=1. c_Fun_Ooverride__on=1. c_Option_Ooption_Ooption__case=1. c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__case=1. c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec=1. c_New__DSequence_Opos__map=1. c_Nitpick_Opair__box_Opair__box__case=1. c_Nitpick_Opair__box_Opair__box__rec=1. c_Nitpick_Opair__box_Opair__box__size=1. c_Product__Type_Oprod_Oprod__rec=1. c_Product__Type_Oprod_Oprod__size=1. f7=1. f16=1. f22=1. f48=1. f49=1. f50=1. f53=1. f57=1. f96=1. f97=1. f98=1. f99=1. f104=1. f109=1. f129=1. f130=1. f131=1. f133=1. f134=1. f135=1. f150=1. f173=1. f174=1. f176=1. f178=1. f179=1. f206=1. f207=1. f208=1. f209=1. f219=1. f220=1. f222=1. f231=1. f239=1. f242=1. f244=1. f247=1. f257=1. f259=1. f260=1. f263=1. f264=1. f276=1. f287=1. f288=1. f292=1. f301=1. f309=1. f328=1. f329=1. f335=1. f338=1. f347=1. f371=1. f372=1. f376=1. f377=1. f380=1. f383=1. f385=1. f390=1. f393=1. f421=1. f425=1. f427=1. f443=1. f448=1. f449=1. f450=1. f454=1. f455=1. f465=1. f469=1. f491=1. f499=1. f518=1. f545=1. f546=1. f550=1. f554=1. f561=1. f572=1. c_Product__Type_Omap__pair=1. c_New__Random__Sequence_Opos__union=1. c_Partial__Function_Ofun__lub=1. f75=1. f76=1. f94=1. f95=1. f151=1. f152=1. f156=1. f157=1. f158=1. f167=1. f188=1. f189=1. f190=1. f191=1. f192=1. f216=1. f241=1. f290=1. f321=1. f357=1. f358=1. f405=1. f406=1. f407=1. f439=1. f440=1. f441=1. f451=1. f496=1Cputime limit exceeded (core dumped) 300.00/299.83 EOF