0.09/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.09/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.33 % Computer : n024.cluster.edu 0.13/0.33 % Model : x86_64 x86_64 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.33 % Memory : 8042.1875MB 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.33 % CPULimit : 1200 0.13/0.33 % DateTime : Wed Jul 14 15:56:17 EDT 2021 0.13/0.34 % CPUTime : 6.13/6.21 ============================== Prover9 =============================== 6.13/6.21 Prover9 (32) version 2009-11A, November 2009. 6.13/6.21 Process 6299 was started by sandbox on n024.cluster.edu, 6.13/6.21 Wed Jul 14 15:56:23 2021 6.13/6.21 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_6145_n024.cluster.edu". 6.13/6.21 ============================== end of head =========================== 6.13/6.21 6.13/6.21 ============================== INPUT ================================= 6.13/6.21 6.13/6.21 % Reading from file /tmp/Prover9_6145_n024.cluster.edu 6.13/6.21 6.13/6.21 set(prolog_style_variables). 6.13/6.21 set(auto2). 6.13/6.21 % set(auto2) -> set(auto). 6.13/6.21 % set(auto) -> set(auto_inference). 6.13/6.21 % set(auto) -> set(auto_setup). 6.13/6.21 % set(auto_setup) -> set(predicate_elim). 6.13/6.21 % set(auto_setup) -> assign(eq_defs, unfold). 6.13/6.21 % set(auto) -> set(auto_limits). 6.13/6.21 % set(auto_limits) -> assign(max_weight, "100.000"). 6.13/6.21 % set(auto_limits) -> assign(sos_limit, 20000). 6.13/6.21 % set(auto) -> set(auto_denials). 6.13/6.21 % set(auto) -> set(auto_process). 6.13/6.21 % set(auto2) -> assign(new_constants, 1). 6.13/6.21 % set(auto2) -> assign(fold_denial_max, 3). 6.13/6.21 % set(auto2) -> assign(max_weight, "200.000"). 6.13/6.21 % set(auto2) -> assign(max_hours, 1). 6.13/6.21 % assign(max_hours, 1) -> assign(max_seconds, 3600). 6.13/6.21 % set(auto2) -> assign(max_seconds, 0). 6.13/6.21 % set(auto2) -> assign(max_minutes, 5). 6.13/6.21 % assign(max_minutes, 5) -> assign(max_seconds, 300). 6.13/6.21 % set(auto2) -> set(sort_initial_sos). 6.13/6.21 % set(auto2) -> assign(sos_limit, -1). 6.13/6.21 % set(auto2) -> assign(lrs_ticks, 3000). 6.13/6.21 % set(auto2) -> assign(max_megs, 400). 6.13/6.21 % set(auto2) -> assign(stats, some). 6.13/6.21 % set(auto2) -> clear(echo_input). 6.13/6.21 % set(auto2) -> set(quiet). 6.13/6.21 % set(auto2) -> clear(print_initial_clauses). 6.13/6.21 % set(auto2) -> clear(print_given). 6.13/6.21 assign(lrs_ticks,-1). 6.13/6.21 assign(sos_limit,10000). 6.13/6.21 assign(order,kbo). 6.13/6.21 set(lex_order_vars). 6.13/6.21 clear(print_given). 6.13/6.21 6.13/6.21 % formulas(sos). % not echoed (5234 formulas) 6.13/6.21 6.13/6.21 ============================== end of input ========================== 6.13/6.21 6.13/6.21 % From the command line: assign(max_seconds, 1200). 6.13/6.21 6.13/6.21 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 6.13/6.21 6.13/6.21 % Formulas that are not ordinary clauses: 6.13/6.21 1 (all V_n all V_m ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) & (V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = 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]. 6.13/6.21 2 (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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_folding_Ounion) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 3 (all V_list_H all V_a_H all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H) != c_List_Olist_ONil(T_a)) # label(fact_list_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 4 (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(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__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))))) # label(fact_setsum__Sigma) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 5 (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]. 6.13/6.21 6 (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)))) -> 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)) = 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_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)))))))))) # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 7 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n)),hAPP(c_Code__Numeral_Oof__nat,V_m))) # label(fact_plus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 8 (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) -> (V_A_2 = hAPP(c_Relation_ODomain(T_c,T_c),V_r_2) -> c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2)))) # label(fact_comp__equivI) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 9 (all V_m hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m))) # label(fact_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 10 (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]. 6.13/6.21 11 (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]. 6.13/6.21 12 (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]. 6.13/6.21 13 (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_Oimage(T_c,T_d,V_f_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)) = 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))))) # label(fact_bij__image__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 14 (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]. 6.13/6.21 15 (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]. 6.13/6.21 16 (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]. 6.13/6.21 17 (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]. 6.13/6.21 18 (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_Nat_Onat,T_c),V_f_2),V_A_2) = 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)))) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 19 (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]. 6.13/6.21 20 (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]. 6.13/6.21 21 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_R_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__is__UN__rel__pow) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 22 (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_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))) & (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))))) # label(fact_UN__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 23 (all V_x all V_xs all T_a V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))))) # label(fact_last__snoc) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 24 (all V_B_2 all V_A_2 all V_C_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_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__extend__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 25 (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]. 6.13/6.21 26 (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]. 6.13/6.21 27 (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]. 6.13/6.21 28 (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__raw) # label(axiom) # label(non_clause). [assumption]. 6.13/6.21 29 (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]. 6.13/6.21 30 (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]. 6.13/6.21 31 (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]. 6.13/6.21 32 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_t_2)))))))) # label(fact_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 33 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> 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))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omin(T_c),V_x_2),V_xs_2))) # label(fact_min__max_OInf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 34 (all V_n_2 all V_m_2 all V_Pa_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) | (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,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]. 6.22/6.21 35 (all V_Pa_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(V_Pa_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_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)))))) # label(fact_Int__Collect) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 36 (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]. 6.22/6.21 37 (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]. 6.22/6.21 38 (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]. 6.22/6.21 39 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),V_k_2)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_k_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))))) # label(fact_atLeast__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 40 (all V_m_2 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)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2)) # label(fact_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 41 (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]. 6.22/6.21 42 (all V_n_2 all T_c all V_x_2 all V_Pa_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))) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Oreplicate(T_c,V_n_2,V_x_2) = 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]. 6.22/6.21 43 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2) <-> V_m_2 = V_n_2)) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 44 (all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_d,T_c)),V_r_2)))) # label(fact_finite__converse) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 45 (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]. 6.22/6.21 46 (all V_x all V_i 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(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x))) # label(fact_length__list__update) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 47 (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]. 6.22/6.21 48 (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]. 6.22/6.21 49 (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]. 6.22/6.21 50 (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))) -> (V_i_2 = V_j_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)))))) # label(fact_nth__eq__iff__index__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 51 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Nat_OSuc,V_m_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 52 (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]. 6.22/6.21 53 (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]. 6.22/6.21 54 (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]. 6.22/6.21 55 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (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) = V_a | c_Groups_Ozero__class_Ozero(T_a) = V_b))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 56 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_2) & class_Finite__Set_Ofinite(T_1) -> class_Finite__Set_Ofinite(tc_sum(T_2,T_1)))) # label(arity_sum__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 57 (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]. 6.22/6.21 58 (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]. 6.22/6.21 59 (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,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))) = 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__inter) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 60 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2))))) # label(fact_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 61 (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]. 6.22/6.21 62 (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)),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)))))) = 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)))))) # label(fact_sublist__shift__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 63 (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) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_diff__minus__eq__add) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 64 (all T_a all V_xs (c_List_Olist_ONil(T_a) != 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))) = V_xs)) # label(fact_append__butlast__last__id) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 65 (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_d),V_B_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_bij__betw__finite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 66 (all V_a all T_a (class_Int_Onumber__ring(T_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))) = V_a)) # label(fact_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 67 (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),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)))) # label(fact_split__eta) # label(axiom) # label(non_clause). [assumption]. 6.22/6.21 68 (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]. 6.22/6.21 69 (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]. 6.22/6.22 70 (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]. 6.22/6.22 71 (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))) -> 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))) & (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)))) # label(fact_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 72 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> (exists B_k hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k)) = V_n_2))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 73 (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_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))) = 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)))))) # label(fact_setsum__Un) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 74 (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]. 6.22/6.22 75 (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]. 6.22/6.22 76 (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]. 6.22/6.22 77 (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]. 6.22/6.22 78 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),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),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))) # label(fact_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 79 (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]. 6.22/6.22 80 (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]. 6.22/6.22 81 (all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 82 (all V_B_2 all T_c 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) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Un__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 83 (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]. 6.22/6.22 84 (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]. 6.22/6.22 85 (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]. 6.22/6.22 86 (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]. 6.22/6.22 87 (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]. 6.22/6.22 88 (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]. 6.22/6.22 89 (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]. 6.22/6.22 90 (all V_g_2 all V_f_2 all T_d all T_c 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))) = V_f_2) # label(fact_override__on__emptyset) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 91 (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]. 6.22/6.22 92 (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]. 6.22/6.22 93 (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),V_x_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),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2)))))) # label(fact_less__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 94 (all V_A_2 all V_x_2 all T_d all T_c 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))) = hAPP(c_Finite__Set_Ocard(T_d),V_A_2)) # label(fact_card__cartesian__product__singleton) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 95 (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]. 6.22/6.22 96 (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)),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,hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_setsum__shift__bounds__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 97 (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]. 6.22/6.22 98 (all V_zs all V_ys all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Oappend(T_a),V_ys),V_zs)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_zs)) # label(fact_append__assoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 99 (all V_ys all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(c_List_Ohd(T_a),V_xs))) # label(fact_hd__append2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 100 (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]. 6.22/6.22 101 (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) = V_y_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) = V_x_2))) # label(fact_inv__f__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 102 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa))))) -> (B_x != V_t_2 -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2) != V_t_2)))))) # label(fact_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 103 (all V_l all V_k c_Int_OBit1(V_k) != c_Int_OBit0(V_l)) # label(fact_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 104 (all 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)) & (-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))))) # label(fact_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 105 (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]. 6.22/6.22 106 (all V_z all V_y all V_x all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_z)))) # label(fact_sup__inf__distrib1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 107 (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),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)))) = hAPP(hAPP(c_List_Onth(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)),V_j_2))) # label(fact_dropWhile__nth) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 108 (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))) -> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_listrel__Nil1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 109 (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_Int_Oint),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(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),V_m_2))))) # label(fact_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 110 (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]. 6.22/6.22 111 (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]. 6.22/6.22 112 (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__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]. 6.22/6.22 113 (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]. 6.22/6.22 114 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_i_2,V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) # label(fact_set__upt) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 115 (all V_r_2 all T_c (c_Relation_Otrans(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_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)))))))) # label(fact_trans__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 116 (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)) -> c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2) = V_x_2))))) # label(fact_Min__eqI) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 117 (all V_m V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 118 (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]. 6.22/6.22 119 (all V_n hAPP(c_Int_Onat,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = V_n) # label(fact_nat__int) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 120 (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]. 6.22/6.22 121 (all V_a2_2 all V_r_2 all T_d all T_c ((exists B_a hBOOL(hAPP(hAPP(V_r_2,B_a),V_a2_2))) <-> hBOOL(hAPP(c_Predicate_ORangeP(T_c,T_d,V_r_2),V_a2_2)))) # label(fact_RangeP_Oequations) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 122 (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))) & 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__last) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 123 (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) <-> (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)))))) # label(fact_congruent__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 124 (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]. 6.22/6.22 125 (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]. 6.22/6.22 126 (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]. 6.22/6.22 127 (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]. 6.22/6.22 128 (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]. 6.22/6.22 129 (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_Olistrel1(T_c,hAPP(c_Relation_Oconverse(T_c,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)),hAPP(c_Relation_Oconverse(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olistrel1(T_c,V_r_2)))))) # label(fact_in__listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 130 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (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(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)))))) # label(fact_of__int__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 131 (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]. 6.22/6.22 132 (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]. 6.22/6.22 133 (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]. 6.22/6.22 134 (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_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),V_A_2))))))) # label(fact_min__max_OInf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 135 (all V_Pa_2 all V_xs_2 all T_c ((exists B_x (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__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_Bex__set__list__ex) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 136 (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]. 6.22/6.22 137 (all V_A_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_Relation_OId__on(T_c,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) & V_x_2 = V_y_2)) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 138 (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]. 6.22/6.22 139 (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]. 6.22/6.22 140 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Ocancel__semigroup__add(T_c) -> (V_b_2 = V_c_2 <-> 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_a_2),V_b_2)))) # label(fact_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 141 (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_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_fdisj),V_Pa_2)),V_Qa_2))) # label(fact_Collect__disj__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 142 (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]. 6.22/6.22 143 (all V_xs_2 all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> 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))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_xs_2))) # label(fact_Inf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 144 (all V_com_H_2 all V_fun_H_2 all V_loc_H_2 c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 145 (all V_R_2 all V_a2_2 all V_a1_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_a1_2),V_a2_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a2_2)) & (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(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)))))) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_a2_2)) # label(fact_max__ext_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 146 (all V_zs_2 all V_ys_2 all V_xs_2 all T_c (V_zs_2 = V_ys_2 <-> 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))) # label(fact_same__append__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 147 (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]. 6.22/6.22 148 (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]. 6.22/6.22 149 (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_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))) & (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)) = V_A_2))) # label(fact_UN__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 150 (all T_c (class_Orderings_Olinorder(T_c) -> c_Orderings_Oord__class_Omin(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))))) # label(fact_dual__max) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 151 (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]. 6.22/6.22 152 (all V_x_2 all V_n_2 all T_c 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,hAPP(c_Nat_OSuc,V_n_2),V_x_2))) # label(fact_set__replicate__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 153 (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]. 6.22/6.22 154 (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]. 6.22/6.22 155 (all V_v_H all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(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),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))))) # label(fact_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 156 (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_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_bij__image__Compl__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 157 (all V_xs all V_n all V_m all T_a hAPP(c_List_Orotate(T_a,V_m),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_List_Orotate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_xs)) # label(fact_rotate__rotate) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 158 (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]. 6.22/6.22 159 (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]. 6.22/6.22 160 (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]. 6.22/6.22 161 (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]. 6.22/6.22 162 (all V_B_2 all V_A_2 all T_d 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_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)))) = c_Nitpick_Oprod(T_c,T_d,V_A_2,V_B_2)) # label(fact_Nitpick_Oprod__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 163 (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_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_OlessThan(T_c),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2))))) # label(fact_ivl__disj__un_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 164 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c 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)) = 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))) # label(fact_vimage__INT) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 165 (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_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)) = V_A_2)) # label(fact_image__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 166 (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)) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) <-> hAPP(c_Finite__Set_Ocard(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))))) # label(fact_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 167 (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]. 6.22/6.22 168 (all V_k c_Int_OBit1(c_Int_Opred(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(V_k))) # label(fact_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 169 (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]. 6.22/6.22 170 (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]. 6.22/6.22 171 (all T_d all T_c 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_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]. 6.22/6.22 172 (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]. 6.22/6.22 173 (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_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),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_I19_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 174 (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]. 6.22/6.22 175 (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),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_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__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]. 6.22/6.22 176 (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]. 6.22/6.22 177 (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]. 6.22/6.22 178 (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]. 6.22/6.22 179 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(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(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_power__minus1__odd) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 180 (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]. 6.22/6.22 181 (all V_r_2 all T_d all T_c (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)) = V_r_2 <-> hAPP(c_Relation_ODomain(T_c,T_d),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 182 (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]. 6.22/6.22 183 (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)),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]. 6.22/6.22 184 (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]. 6.22/6.22 185 (all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> V_a = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_a))) # label(fact_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 186 (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_x_2) = 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)))) # label(fact_inv__into__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 187 (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_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2)))))) # label(fact_setprod__mono__one__left) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 188 (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]. 6.22/6.22 189 (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]. 6.22/6.22 190 (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]. 6.22/6.22 191 (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]. 6.22/6.22 192 (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]. 6.22/6.22 193 (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]. 6.22/6.22 194 (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]. 6.22/6.22 195 (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)) -> (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_xa),B_x)),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_xa)),V_r_2))))))) <-> c_Relation_Ototal__on(T_c,V_A_2,V_r_2))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 196 (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(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(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2))))) # label(fact_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 197 (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_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_OgreaterThanAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2))))) # label(fact_ivl__disj__un_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 198 (all V_n all V_m all V_k 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)) = 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))) # label(fact_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 199 (all V_n all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))) # label(fact_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 200 (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]. 6.22/6.22 201 (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]. 6.22/6.22 202 (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_y_2 != V_x_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 203 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) # label(fact_minus__mult__minus) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 204 (all V_k c_Int_OBit0(c_Int_Osucc(V_k)) = c_Int_Osucc(c_Int_OBit1(V_k))) # label(fact_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 205 (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_y_2)) <-> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2) = V_y_2))) # label(fact_le__iff__sup) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 206 (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]. 6.22/6.22 207 (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_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)) = V_A_2)) # label(fact_the__inv__into__onto) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 208 (all V_i_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,hAPP(c_Int_Onat,V_i_2))) <-> (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)))) & (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)))))) # label(fact_split__nat) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 209 (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]. 6.22/6.22 210 (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_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))) = 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))))))) # label(fact_setprod__Un) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 211 (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]. 6.22/6.22 212 (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]. 6.22/6.22 213 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),c_Transitive__Closure_Otrancl(T_c,V_r_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)))) # label(fact_finite__trancl) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 214 (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]. 6.22/6.22 215 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)))) # label(fact_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 216 (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]. 6.22/6.22 217 (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) -> (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) <-> (c_Groups_Ozero__class_Ozero(T_c) = V_c_2 -> 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))))) # label(fact_divide__eq__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 218 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 219 (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]. 6.22/6.22 220 (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]. 6.22/6.22 221 (all V_t_2 all V_A_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)))))))) # label(fact_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 222 (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_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 223 (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]. 6.22/6.22 224 (all T_1 class_Nat_Osize(tc_Option_Ooption(T_1))) # label(arity_Option__Ooption__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 225 (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]. 6.22/6.22 226 (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]. 6.22/6.22 227 (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]. 6.22/6.22 228 (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]. 6.22/6.22 229 (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]. 6.22/6.22 230 (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]. 6.22/6.22 231 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_Osublists(T_a,V_xs)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) # label(fact_length__sublists) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 232 (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]. 6.22/6.22 233 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_z))) # label(fact_mult__minus1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 234 (all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hAPP(c_Int_Onat,V_i_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 235 (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]. 6.22/6.22 236 (all V_n all V_m (V_m = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) -> V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_n = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 237 (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]. 6.22/6.22 238 (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]. 6.22/6.22 239 (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]. 6.22/6.22 240 (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]. 6.22/6.22 241 (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_Ouminus__class_Ouminus(T_a),V_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__of__neg) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 242 (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]. 6.22/6.22 243 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 244 (all V_r all V_q all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_a),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_a -> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_q))) # label(fact_self__quotient) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 245 (all V_l all V_k c_Int_OBit0(V_k) != c_Int_OBit1(V_l)) # label(fact_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.22 246 (all V_f_2 all V_A_2 all V_B_2 all T_c all T_d (class_Rings_Odvd(T_d) & class_Groups_Ocomm__monoid__mult(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]. 6.22/6.22 247 (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)) & V_B_2 = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),B_AA))) <-> 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]. 6.22/6.22 248 (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]. 6.22/6.22 249 (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]. 6.22/6.22 250 (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]. 6.22/6.22 251 (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]. 6.22/6.22 252 (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__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)) = 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)) # label(fact_New__Random__Sequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 253 (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_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)) & hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),B_n),B_y) = V_ys_2 & 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(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]. 6.22/6.23 254 (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_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)) = 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)))))))))) # label(fact_setsum__mult__setsum__if__inj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 255 (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_OgreaterThan(T_c),V_k_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2))) # label(fact_Compl__greaterThan) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 256 (all T_c 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))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Inter__empty) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 257 (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]. 6.22/6.23 258 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 259 (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__sup__class_Osup(T_d)),V_f_2)),V_B_2),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))))) # label(fact_sup__SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 260 (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]. 6.22/6.23 261 (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]. 6.22/6.23 262 (all V_ys_2 all V_xs_2 all V_x_2 all T_d all T_c 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) = 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)) # label(fact_zip__Cons1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 263 (all V_y_2 all V_x_2 (V_y_2 = V_x_2 <-> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))) # label(fact_transfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 264 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 265 (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]. 6.22/6.23 266 (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) -> V_A_2 = hAPP(c_Set_Oimage(T_c,T_c,V_f_2),V_A_2))))) # label(fact_endo__inj__surj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 267 (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]. 6.22/6.23 268 (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]. 6.22/6.23 269 (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]. 6.22/6.23 270 (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))) -> V_a = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_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]. 6.22/6.23 271 (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_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),V_A_2)))))) # label(fact_min__max_OInf__fin_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 272 (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]. 6.22/6.23 273 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H))))))))) # label(fact_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 274 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_fequal,V_x_2),V_y_2)) | V_y_2 != V_x_2)) # label(help_c__fequal__2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 275 (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]. 6.22/6.23 276 (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]. 6.22/6.23 277 (all V_n all V_m 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)) = 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))))) # label(fact_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 278 (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]. 6.22/6.23 279 (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]. 6.22/6.23 280 (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]. 6.22/6.23 281 (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]. 6.22/6.23 282 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (V_y != c_Groups_Ozero__class_Ozero(T_a) -> (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]. 6.22/6.23 283 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = V_y))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 284 (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_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)) = 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)))))) # label(fact_setsum__diff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 285 (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]. 6.22/6.23 286 (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]. 6.22/6.23 287 (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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_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)))))) # label(fact_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 288 (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]. 6.22/6.23 289 (all V_n_2 all V_f_2 all T_c all V_A_2 (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))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_nat__seg__image__imp__finite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 290 (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(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__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 291 (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]. 6.22/6.23 292 (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]. 6.22/6.23 293 (all V_ys all V_x all V_xs all T_a V_x = 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))) # label(fact_nth__append__length) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 294 (all V_xs_2 all V_n_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))),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))) = c_Enum_On__lists(T_c,hAPP(c_Nat_OSuc,V_n_2),V_xs_2)) # label(fact_n__lists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 295 (all V_i all V_b ((-(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)))) -> c_Random_Olog(V_b,V_i) = 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)))) & (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))) -> c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = c_Random_Olog(V_b,V_i)))) # label(fact_log_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 296 (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]. 6.22/6.23 297 (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(tc_Nat_Onat),V_x_2),V_y_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))))))) # label(fact_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 298 (all V_k c_Int_OBit1(V_k) = c_Int_Osucc(c_Int_OBit0(V_k))) # label(fact_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 299 (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]. 6.22/6.23 300 (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]. 6.22/6.23 301 (all V_A_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(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)))) = c_Nitpick_Ocard_H(T_c,V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Nitpick_Ocard_H(T_c,V_A_2)))) # label(fact_Nitpick_Ocard_H__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 302 (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]. 6.22/6.23 303 (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]. 6.22/6.23 304 (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]. 6.22/6.23 305 (all V_B_2 all V_A_2 all T_d all T_c (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)) -> hAPP(V_B_2,B_x) = 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]. 6.22/6.23 306 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 307 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = c_Groups_Osgn__class_Osgn(T_c,V_a_2) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sgn__0__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 308 (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]. 6.22/6.23 309 (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]. 6.22/6.23 310 (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]. 6.22/6.23 311 (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]. 6.22/6.23 312 (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]. 6.22/6.23 313 (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]. 6.22/6.23 314 (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]. 6.22/6.23 315 (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]. 6.22/6.23 316 (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]. 6.22/6.23 317 (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]. 6.22/6.23 318 (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]. 6.22/6.23 319 (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)) -> (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_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)) = 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)))))))) # label(fact_fold1__Un2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 320 (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),hAPP(hAPP(c_List_Oappend(T_c),V_xs_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),V_ys_2)))) # label(fact_append__in__listsp__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 321 (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]. 6.22/6.23 322 (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]. 6.22/6.23 323 (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]. 6.22/6.23 324 (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]. 6.22/6.23 325 (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]. 6.22/6.23 326 (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]. 6.22/6.23 327 (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)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,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,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)))) # label(fact_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 328 (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),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]. 6.22/6.23 329 (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))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(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,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))),V_m_2)))) # label(fact_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 330 (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]. 6.22/6.23 331 (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),V_a),V_b) = 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))))) # label(fact_mult__divide__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 332 (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_y_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_y_2)),V_g_2)))) # label(fact_folding_Ocommute__left__comp) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 333 (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]. 6.22/6.23 334 (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]. 6.22/6.23 335 (all V_A_2 all T_c 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) = 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))))) # label(fact_set__Cons__sing__Nil) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 336 (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]. 6.22/6.23 337 (all V_xs_2 all V_n_2 all T_c (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) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2))) # label(fact_take__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 338 (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]. 6.22/6.23 339 (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]. 6.22/6.23 340 (all V_n all V_m ((V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) & (V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_n = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 341 (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_Olist__update(T_c,V_xs_2),V_i_2),V_x_2) = V_xs_2 <-> hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_i_2) = V_x_2))) # label(fact_list__update__same__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 342 (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_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_b_2),V_a_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_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))))) # label(fact_converse__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 343 (all V_l all V_k (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_k),V_l) -> 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]. 6.22/6.23 344 (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_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__assoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 345 (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_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)) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 346 (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))) -> 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_0621_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) = V_ys_092_060_094isub_0622_2) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_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))) -> 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_0622_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) = V_xs_092_060_094isub_0621_2) <-> hAPP(hAPP(c_List_Oappend(T_c),V_ys_092_060_094isub_0621_2),V_ys_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))) # label(fact_append__eq__append__conv__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 347 (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]. 6.22/6.23 348 (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) <-> V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)))) # label(fact_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 349 (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)))) -> 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))) = 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))) & (-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]. 6.22/6.23 350 (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]. 6.22/6.23 351 (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)),c_Groups_Oone__class_Oone(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls)))))) # label(fact_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 352 (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]. 6.22/6.23 353 (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(tc_Int_Oint),V_x_2),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_Ozero__class_Ozero(T_c)))))) # label(fact_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 354 (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_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)) = 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)))) # label(fact_setprod_Odistrib) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 355 (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]. 6.22/6.23 356 (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_equivp__reflp__symp__transp) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 357 (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_B_2),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_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_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 358 (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]. 6.22/6.23 359 (all V_a all V_y all V_x 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_x),V_y)) = V_a -> V_a = V_y)) # label(fact_snd__eqD) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 360 (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]. 6.22/6.23 361 (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_Divides_Odiv__class_Odiv(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_Divides_Odiv__class_Odiv(T_a),V_b),V_c))))) # label(fact_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 362 (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]. 6.22/6.23 363 (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_x_2 = V_y_2)) # label(fact_rel__pow__0__E) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 364 (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]. 6.22/6.23 365 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b))))) # label(fact_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 366 (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]. 6.22/6.23 367 (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)) -> V_k_2 = hAPP(c_Finite__Set_Ocard(T_c),B_c))) -> ((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_c2 != B_c1 -> 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_c1),B_c2)))))) -> 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]. 6.22/6.23 368 (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]. 6.22/6.23 369 (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]. 6.22/6.23 370 (all V_w all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 371 (all V_x_2 all V_A_2 all T_c (class_Lattices_Oab__semigroup__idem__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)) -> 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]. 6.22/6.23 372 (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]. 6.22/6.23 373 (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]. 6.22/6.23 374 (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]. 6.22/6.23 375 (all V_B_2 all V_A_2 all T_c all T_d 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)) = 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))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 376 (all V_xs all V_i all T_a 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)) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs))) # label(fact_rev__drop) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 377 (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]. 6.22/6.23 378 (all V_l_2 all V_k_2 (c_Int_OBit1(V_k_2) = c_Int_OBit1(V_l_2) <-> V_l_2 = V_k_2)) # label(fact_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 379 (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_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),V_A_2)))))) # label(fact_min__max_OInf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 380 (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(hAPP(c_member(T_c),B_x),c_Quotient_ORespects(T_c,V_R_2))) & hBOOL(hAPP(V_Pa_2,B_x)))))))) # label(fact_bex__reg__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 381 (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]. 6.22/6.23 382 (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]. 6.22/6.23 383 (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_ys_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_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]. 6.22/6.23 384 (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]. 6.22/6.23 385 (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)),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__Diff1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 386 (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_b_2) = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = V_b_2))) # label(fact_minus__equation__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 387 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a))) # label(fact_minus__diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 388 (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_OgreaterThanLessThan(T_c,V_x_2,V_y_2)) = 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)))) # label(fact_image__uminus__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 389 (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]. 6.22/6.23 390 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_Opred(V_k)),V_l))) # label(fact_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 391 (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]. 6.22/6.23 392 (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]. 6.22/6.23 393 (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(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__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 394 (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]. 6.22/6.23 395 (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_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2))))) # label(fact_min__max_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 396 (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]. 6.22/6.23 397 (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]. 6.22/6.23 398 (all V_t_2 all V_B_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_t_2))))))) # label(fact_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 399 (all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_abs__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 400 (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]. 6.22/6.23 401 (all V_n_2 all V_m_2 all T_c (class_Nat_Osemiring__char__0(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) <-> V_m_2 = V_n_2))) # label(fact_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 402 (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),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(T_c),V_b_2),V_a_2)))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 403 (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]. 6.22/6.23 404 (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))))) -> (V_B_2 = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_s_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]. 6.22/6.23 405 (all V_z_2 all V_w_2 ((exists B_n V_z_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 406 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> V_x_2 = 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)))))) # label(fact_Min_Osingleton) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 407 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),c_Divides_Odivmod__int(V_a,V_b))))) # label(fact_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 408 (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_Omin(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(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 -> 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]. 6.22/6.23 409 (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) -> (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)) <-> (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))))))) # label(fact_SUP__le__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 410 (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]. 6.22/6.23 411 (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]. 6.22/6.23 412 (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]. 6.22/6.23 413 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) | V_n_2 = V_m_2)) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 414 (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)),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),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]. 6.22/6.23 415 (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_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)) = 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))) # label(fact_map__zip__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 416 (all V_i_1 all T_a c_Lazy__Sequence_Oempty(T_a) = hAPP(c_New__DSequence_Opos__empty(T_a),V_i_1)) # label(fact_New__DSequence_Opos__empty__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 417 (all V_xs all V_x all T_a hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = V_x) # label(fact_hd_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 418 (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]. 6.22/6.23 419 (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]. 6.22/6.23 420 (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]. 6.22/6.23 421 (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]. 6.22/6.23 422 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),V_b) = V_a))) # label(fact_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 423 (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]. 6.22/6.23 424 (all 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_a_2)) # label(fact_dvd_OatLeastAtMost__singleton) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 425 (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]. 6.22/6.23 426 (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]. 6.22/6.23 427 (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 (hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_vs)) = V_xs_2 -> V_ys_2 != hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_vs)))))))) # label(fact_listrel1E) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 428 (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]. 6.22/6.23 429 (all V_A_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))))) # label(fact_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 430 (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]. 6.22/6.23 431 (all V_a_2 all V_f_2 all T_c all V_g_2 (V_g_2 = c_Finite__Set_Ofold1(T_c,V_f_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]. 6.22/6.23 432 (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]. 6.22/6.23 433 (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]. 6.22/6.23 434 (all V_xs_2 all T_c 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))))) = hAPP(c_List_Oset(T_c),V_xs_2)) # label(fact_set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.22/6.23 435 (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]. 6.22/6.23 436 (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_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(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)) & (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,V_A_2) = V_x_2))))) # label(fact_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 437 (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]. 6.22/6.24 438 (all V_a_2 all T_c (class_Rings_Oring__1__no__zero__divisors(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)))) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 439 (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))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)) & (-(c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2) -> hAPP(hAPP(c_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))))) # label(fact_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 440 (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]. 6.22/6.24 441 (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]. 6.22/6.24 442 (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]. 6.22/6.24 443 (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]. 6.22/6.24 444 (all V_C_2 all V_B_2 all V_A_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_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),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_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)))) # label(fact_Times__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 445 (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]. 6.22/6.24 446 (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_Olinorder__class_OMin(T_c,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))))))) # label(fact_Min_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 447 (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_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_I35_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 448 (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]. 6.22/6.24 449 (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]. 6.22/6.24 450 (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]. 6.22/6.24 451 (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]. 6.22/6.24 452 (all V_v_2 all V_n_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_OSuc,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n_2)),c_Int_Onat)))) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 453 (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]. 6.22/6.24 454 (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]. 6.22/6.24 455 (all V_q all V_y 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_Groups_Otimes__class_Otimes(T_a),V_x),V_y)),V_q) = 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)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 456 (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 (hAPP(c_Nat_OSuc,B_m) = V_n_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)) -> -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]. 6.22/6.24 457 (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_y_2 = V_x_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]. 6.22/6.24 458 (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]. 6.22/6.24 459 (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]. 6.22/6.24 460 (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]. 6.22/6.24 461 (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_b = V_c))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 462 (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]. 6.22/6.24 463 (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_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)))) # label(fact_Int__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 464 (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]. 6.22/6.24 465 (all V_n all V_m hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n))) # label(fact_int__power) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 466 (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]. 6.22/6.24 467 (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]. 6.22/6.24 468 (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]. 6.22/6.24 469 (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]. 6.22/6.24 470 (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]. 6.22/6.24 471 (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]. 6.22/6.24 472 (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]. 6.22/6.24 473 (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]. 6.22/6.24 474 (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]. 6.22/6.24 475 (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]. 6.22/6.24 476 (all V_xs_2 all T_c 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) = hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_xs_2)) # label(fact_zip__same__conv__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 477 (all V_n_2 all V_v_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))) <-> 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)))))) # label(fact_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 478 (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_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]. 6.22/6.24 479 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = c_Orderings_Obot__class_Obot(T_a) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = c_Orderings_Otop__class_Otop(T_a) -> V_y = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))))) # label(fact_compl__unique) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 480 (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_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),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_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 481 (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,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)),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_Int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 482 (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]. 6.22/6.24 483 (all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Domain__empty) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 484 (all V_r_2 all T_c (-(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))) <-> c_Wellfounded_Owf(T_c,V_r_2))) # label(fact_wf__iff__no__infinite__down__chain) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 485 (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]. 6.22/6.24 486 (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]. 6.22/6.24 487 (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]. 6.22/6.24 488 (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) = hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),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__id__dest) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 489 (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]. 6.22/6.24 490 (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]. 6.22/6.24 491 (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)) -> c_Groups_Oone__class_Oone(T_d) = 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)) & (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) = hAPP(V_b_2,V_a_2))))) # label(fact_setprod__delta_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 492 (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]. 6.22/6.24 493 (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),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)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_neg__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 494 (all V_ys all V_xs all V_x all T_a 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)) = hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_removeAll__append) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 495 (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]. 6.22/6.24 496 (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]. 6.22/6.24 497 (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]. 6.22/6.24 498 (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]. 6.22/6.24 499 (all V_A_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)) = c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)))) # label(fact_Sup__insert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 500 (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]. 6.22/6.24 501 (all V_a all V_b all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c) = V_a)))) # label(fact_divide__eq__imp) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 502 (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]. 6.22/6.24 503 (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)) -> hAPP(V_f_2,V_i_2) = c_Groups_Ozero__class_Ozero(T_d))))))) # label(fact_setsum__nonneg__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 504 (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__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)),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))) # label(fact_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 505 (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]. 6.22/6.24 506 (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]. 6.22/6.24 507 (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]. 6.22/6.24 508 (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_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)))) # label(fact_ord_OgreaterThanLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 509 (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) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = V_m | c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m)) # label(fact_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 510 (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]. 6.22/6.24 511 (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]. 6.22/6.24 512 (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(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))) = 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))))) # label(fact_takeWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 513 (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]. 6.22/6.24 514 (all V_m all V_l hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),V_l),V_m) = c_List_Oreturn__list(hAPP(hAPP(c_List_Oappend(tc_Int_Oint),c_List_Oembed__list(V_l)),c_List_Oembed__list(V_m)))) # label(fact_transfer__nat__int__list__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 515 (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]. 6.22/6.24 516 (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_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)) -> hAPP(V_f_2,V_y_2) != hAPP(V_f_2,V_x_2)))))) # label(fact_inj__on__contraD) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 517 (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]. 6.22/6.24 518 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z)) # label(fact_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 519 (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,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b)))) # label(fact_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 520 (all V_y_2 all V_n_2 all V_x_2 all T_c (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 & V_x_2 = V_y_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]. 6.22/6.24 521 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Rings_Oring__1(T_c) -> 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)) = 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))) # label(fact_of__int__setprod) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 522 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b)) # label(fact_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 523 (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)))) # label(fact_zle__linear) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 524 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)))) # label(fact_of__int__m1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 525 (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]. 6.22/6.24 526 (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_B_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_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__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 527 (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]. 6.22/6.24 528 (all V_l_2 c_List_Oreturn__list(V_l_2) = hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),c_Int_Onat),V_l_2)) # label(fact_return__list__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 529 (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]. 6.22/6.24 530 (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]. 6.22/6.24 531 (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__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 532 (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)) -> (V_x_2 = V_y_2 <-> hAPP(V_Abs_2,V_y_2) = hAPP(V_Abs_2,V_x_2)))))) # label(fact_type__definition_OAbs__inject) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 533 (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_Omin(T_c)),V_a_2),V_A_2) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_Min_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 534 (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]. 6.22/6.24 535 (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]. 6.22/6.24 536 (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]. 6.22/6.24 537 (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]. 6.22/6.24 538 (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]. 6.22/6.24 539 (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]. 6.22/6.24 540 (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]. 6.22/6.24 541 (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]. 6.22/6.24 542 (all V_w all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)) = 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))) # label(fact_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 543 (all V_uu all T_b all T_a c_List_Olist_ONil(tc_prod(T_a,T_b)) = c_Enum_Oproduct(T_a,T_b,c_List_Olist_ONil(T_a),V_uu)) # label(fact_product_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 544 (all V_xs all T_a all V_y all V_x ((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))) & (V_y != V_x -> c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Oremove1(T_a,V_x,V_xs))))) # label(fact_remove1_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 545 (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]. 6.22/6.24 546 (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]. 6.22/6.24 547 (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]. 6.22/6.24 548 (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),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b)),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_Oassoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 549 (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]. 6.22/6.24 550 (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]. 6.22/6.24 551 (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]. 6.22/6.24 552 (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]. 6.22/6.24 553 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k_2),V_l_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 554 (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),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))))) <-> 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))))) # label(fact_divide__le__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 555 (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)) -> hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x) != V_c_2)))) # label(fact_Id__onE) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 556 (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]. 6.22/6.24 557 (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]. 6.22/6.24 558 (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]. 6.22/6.24 559 (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_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))))))) # label(fact_equiv__class__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 560 (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]. 6.22/6.24 561 (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]. 6.22/6.24 562 (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_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_xs_2) -> (c_List_Oconcat(T_c,V_ys_2) = c_List_Oconcat(T_c,V_xs_2) <-> V_ys_2 = V_xs_2)))) # label(fact_concat__eq__concat__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 563 (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]. 6.22/6.24 564 (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_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__minus__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 565 (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(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)) = hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_e,T_f),V_x_2))) # label(fact_snd__prod__fun) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 566 (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_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_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_x_2),V_A_2))))) # label(fact_min__max_OInf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 567 (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]. 6.22/6.24 568 (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]. 6.22/6.24 569 (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]. 6.22/6.24 570 (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]. 6.22/6.24 571 (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]. 6.22/6.24 572 (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]. 6.22/6.24 573 (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_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) = 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)))))) # label(fact_fold__image__reindex) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 574 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d 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) & (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))))) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2))) # label(fact_list__all2__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 575 (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 (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)))) = V_X_2 -> -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_quotientE) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 576 (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]. 6.22/6.24 577 (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]. 6.22/6.24 578 (all V_R_2 all T_c c_Transitive__Closure_Ortrancl(T_c,c_Transitive__Closure_Otrancl(T_c,V_R_2)) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)) # label(fact_trancl__rtrancl__absorb) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 579 (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)) <-> 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_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 580 (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]. 6.22/6.24 581 (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]. 6.22/6.24 582 (all V_b_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (V_b_2 = V_a_2 | V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_b_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_a_2)))) # label(fact_square__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 583 (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)) -> (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]. 6.22/6.24 584 (all T_c all V_A_2 all T_d 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_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]. 6.22/6.24 585 (all V_w_2 all V_z_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))) = 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)))) # label(fact_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 586 (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_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)) = 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)))))) # label(fact_inj__on__image__Int) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 587 (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]. 6.22/6.24 588 (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_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_S_2)))))) # label(fact_setsum__mono__zero__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 589 (all V_c all V_b all V_a all T_a (class_Groups_Oab__semigroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 590 (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]. 6.22/6.24 591 (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]. 6.22/6.24 592 (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]. 6.22/6.24 593 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Ofield__inverse__zero(T_c) -> ((c_Groups_Ozero__class_Ozero(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 -> 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) <-> 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_eq__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 594 (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_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]. 6.22/6.24 595 (all V_xs_2 all T_c (V_xs_2 != c_List_Olist_ONil(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))))) # label(fact_length__greater__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 596 (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))) -> 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)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)))) # label(fact_atLeastAtMostSuc__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 597 (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]. 6.22/6.24 598 (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]. 6.22/6.24 599 (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]. 6.22/6.24 600 (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]. 6.22/6.24 601 (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_OInf__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__inf__class_Oinf(T_c)),V_x_2),V_A_2))))) # label(fact_Inf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 602 (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]. 6.22/6.24 603 (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]. 6.22/6.24 604 (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]. 6.22/6.24 605 (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]. 6.22/6.24 606 (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]. 6.22/6.24 607 (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_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2)))) # label(fact_greaterThanAtMost__empty) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 608 (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]. 6.22/6.24 609 (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(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 610 (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)) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)))) # label(fact_setsum_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 611 (all V_zs all V_y all V_x all T_a c_List_Oremove1(T_a,V_y,c_List_Oremove1(T_a,V_x,V_zs)) = c_List_Oremove1(T_a,V_x,c_List_Oremove1(T_a,V_y,V_zs))) # label(fact_remove1__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 612 (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]. 6.22/6.24 613 (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]. 6.22/6.24 614 (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]. 6.22/6.24 615 (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]. 6.22/6.24 616 (all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)) = c_Divides_OnegDivAlg(V_k,V_l)))) # label(fact_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 617 (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_Omax(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)))))) # label(fact_min__max_OSup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 618 (all V_x_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oremdups(T_c,V_x_2) <-> c_List_Olist_ONil(T_c) = V_x_2)) # label(fact_remdups__eq__nil__right__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 619 (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]. 6.22/6.24 620 (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]. 6.22/6.24 621 (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]. 6.22/6.24 622 (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]. 6.22/6.24 623 (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,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)))))) # label(fact_derivs__insertD) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 624 (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_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)) = 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)))) # label(fact_map__zip__map2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 625 (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]. 6.22/6.24 626 (all 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))) = hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_code__numeral))) # label(fact_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.24 627 (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]. 6.22/6.24 628 (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]. 6.22/6.25 629 (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]. 6.22/6.25 630 (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]. 6.22/6.25 631 (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]. 6.22/6.25 632 (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]. 6.22/6.25 633 (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]. 6.22/6.25 634 (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]. 6.22/6.25 635 (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]. 6.22/6.25 636 (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]. 6.22/6.25 637 (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]. 6.22/6.25 638 (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]. 6.22/6.25 639 (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_R_2) = c_Transitive__Closure_Ortrancl(T_c,V_S_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 640 (all V_x_2 all V_f_2 all T_e all T_d all T_c hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)) = 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))) # label(fact_fst__apfst) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 641 (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_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2))))) # label(fact_eqset__imp__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 642 (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]. 6.22/6.25 643 (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]. 6.22/6.25 644 (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]. 6.22/6.25 645 (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]. 6.22/6.25 646 (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]. 6.22/6.25 647 (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_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)) = 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]. 6.22/6.25 648 (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_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 649 (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]. 6.22/6.25 650 (all V_r_1 all V_y all V_b_1 all V_a_1 (V_a_1 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_1),V_y)),V_r_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))) -> (V_b_1 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> 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]. 6.22/6.25 651 (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]. 6.22/6.25 652 (all V_w_2 all V_v_2 all T_c (class_Int_Onumber__ring(T_c) -> 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))) = 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))))) # label(fact_power__number__of__even__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 653 (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]. 6.22/6.25 654 (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_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 655 (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]. 6.22/6.25 656 (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]. 6.22/6.25 657 (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]. 6.22/6.25 658 (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(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))))) # label(fact_folding__one_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 659 (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))) -> 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))) & (-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)))))) # label(fact_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 660 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w)) = 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)))) # label(fact_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 661 (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]. 6.22/6.25 662 (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]. 6.22/6.25 663 (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]. 6.22/6.25 664 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) = c_Groups_Oone__class_Oone(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 665 (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]. 6.22/6.25 666 (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]. 6.22/6.25 667 (all V_n_2 all V_m_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 | V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 668 (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]. 6.22/6.25 669 (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]. 6.22/6.25 670 (all V_Qa_2 all V_Pa_2 (-hBOOL(V_Pa_2) | hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)))) # label(help_c__fdisj__1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 671 (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]. 6.22/6.25 672 (all V_Pa_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),V_Pa_2) = 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))))) # label(fact_split__paired__The) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 673 (all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (c_Divides_Odiv__class_Omod(T_c,V_b_2,V_a_2) = c_Groups_Ozero__class_Ozero(T_c) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2))))) # label(fact_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 674 (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]. 6.22/6.25 675 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 676 (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]. 6.22/6.25 677 (all V_n all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),hAPP(c_List_Orotate(T_a,V_n),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))))) # label(fact_hd__rotate__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 678 (all V_r_2 all T_c all T_d V_r_2 = c_Predicate_Oconversep(T_d,T_c,c_Predicate_Oconversep(T_c,T_d,V_r_2))) # label(fact_conversep__conversep) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 679 (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]. 6.22/6.25 680 (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]. 6.22/6.25 681 (all V_r_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) | c_Nitpick_Ounknown(tc_HOL_Obool)) & 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]. 6.22/6.25 682 (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]. 6.22/6.25 683 (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(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))) = 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)))))) # label(fact_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 684 (all V_n all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_x),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),V_n))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 685 (all V_xs_2 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),hAPP(c_List_Oset(T_c),V_xs_2))))) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__iff__raw) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 686 (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]. 6.22/6.25 687 (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_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)) = 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)))) # label(fact_length__filter__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 688 (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]. 6.22/6.25 689 (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]. 6.22/6.25 690 (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]. 6.22/6.25 691 (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]. 6.22/6.25 692 (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]. 6.22/6.25 693 (all V_r_2 all V_ys_2 all V_xs_2 all T_c (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)))) <-> 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__nth) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 694 (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]. 6.22/6.25 695 (all V_u_2 all V_l_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))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2)) # label(fact_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 696 (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]. 6.22/6.25 697 (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]. 6.22/6.25 698 (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]. 6.22/6.25 699 (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]. 6.22/6.25 700 (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]. 6.22/6.25 701 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 702 (all V_n_2 all V_m_2 (V_m_2 = 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)) <-> 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_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 703 (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]. 6.22/6.25 704 (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))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_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_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_H_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 705 (all V_u_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(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)))) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2))) # label(fact_ivl__disj__un_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 706 (all T_d all T_c c_Fun_Oid(tc_prod(T_c,T_d)) = c_Product__Type_Oapfst(T_c,T_c,T_d,c_Fun_Oid(T_c))) # label(fact_apfst__id) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 707 (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),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(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),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),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]. 6.22/6.25 708 (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]. 6.22/6.25 709 (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]. 6.22/6.25 710 (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]. 6.22/6.25 711 (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]. 6.22/6.25 712 (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]. 6.22/6.25 713 (all V_Pa_2 all V_f_2 all T_d all T_c hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(c_Set_OCollect(T_d),V_Pa_2)) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(T_d,tc_HOL_Obool,T_c),V_Pa_2),V_f_2))) # label(fact_vimage__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 714 (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]. 6.22/6.25 715 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = 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)))) # label(fact_sum__length__filter__compl) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 716 (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]. 6.22/6.25 717 (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]. 6.22/6.25 718 (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_OgreaterThan(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__greaterThan) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 719 (all V_B_2 all V_A_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))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 720 (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]. 6.22/6.25 721 (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_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) = 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))) # label(fact_UN__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 722 (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]. 6.22/6.25 723 (all V_n_2 all V_a_2 all T_c (class_Power_Opower(T_c) & class_Rings_Omult__zero(T_c) & class_Rings_Ono__zero__divisors(T_c) & class_Rings_Ozero__neq__one(T_c) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 & 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),V_n_2)))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 724 (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]. 6.22/6.25 725 (all V_z all V_y all V_x 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_x),V_y)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 726 (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]. 6.22/6.25 727 (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]. 6.22/6.25 728 (all V_y all V_x all V_i all V_xs all T_a hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_y) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i),V_y)) # label(fact_list__update__overwrite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 729 (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]. 6.22/6.25 730 (all V_P all T_a V_P = hAPP(c_COMBI(T_a),V_P)) # label(help_c__COMBI__1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 731 (all V_xs_2 all V_r_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> 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) = 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)))) # label(fact_listsum__triv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 732 (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),c_SetInterval_Oord__class_OatLeastAtMost(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)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_setsum__shift__bounds__cl__Suc__ivl) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 733 (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]. 6.22/6.25 734 (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]. 6.22/6.25 735 (all V_xs all T_a c_List_Osplice(T_a,V_xs,c_List_Olist_ONil(T_a)) = V_xs) # label(fact_splice__Nil2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 736 (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]. 6.22/6.25 737 (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 -> 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)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> 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]. 6.22/6.25 738 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_m_2)) -> (c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_m_2),V_n_2) <-> c_Groups_Oone__class_Oone(tc_Int_Oint) = V_n_2 & c_Groups_Oone__class_Oone(tc_Int_Oint) = V_m_2))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 739 (all V_m_2 all V_n_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) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) # label(fact_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 740 (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]. 6.22/6.25 741 (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]. 6.22/6.25 742 (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]. 6.22/6.25 743 (all V_xs_2 all V_f_2 all T_d all T_c (V_xs_2 = c_List_Olist_ONil(T_d) <-> hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) = c_List_Olist_ONil(T_c))) # label(fact_Nil__is__map__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 744 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit1(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 745 (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]. 6.22/6.25 746 (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]. 6.22/6.25 747 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_c -> 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))) & (V_c != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)))) # label(fact_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 748 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c 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)) = hAPP(hAPP(c_List_Ozip(T_c,T_d),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),V_ys_2)) # label(fact_zip__takeWhile__fst) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 749 (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_OgreaterThanAtMost(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_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 750 (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]. 6.22/6.25 751 (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]. 6.22/6.25 752 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),B_x),B_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(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__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]. 6.22/6.25 753 (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]. 6.22/6.25 754 (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]. 6.22/6.25 755 (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]. 6.22/6.25 756 (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]. 6.22/6.25 757 (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]. 6.22/6.25 758 (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]. 6.22/6.25 759 (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]. 6.22/6.25 760 (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]. 6.22/6.25 761 (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_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__less__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 762 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_r_2) = 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)))) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 763 (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_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__Un) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 764 (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]. 6.22/6.25 765 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(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),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 766 (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]. 6.22/6.25 767 (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]. 6.22/6.25 768 (all V_us all V_ys all V_zs all V_xs1 all V_xs all T_a (hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_xs1) = V_zs -> (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]. 6.22/6.25 769 (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]. 6.22/6.25 770 (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]. 6.22/6.25 771 (all V_Pa_2 all T_c (hAPP(c_Set_OCollect(T_c),V_Pa_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> (all B_x -hBOOL(hAPP(V_Pa_2,B_x))))) # label(fact_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 772 (all V_s_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_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) = c_List_Osublist(T_c,V_xs_2,V_s_2))) # label(fact_filter__in__sublist) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 773 (all V_Pa_2 all V_a_2 all V_n_2 all T_c ((exists B_x (hBOOL(hAPP(V_Pa_2,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)))))) <-> V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & hBOOL(hAPP(V_Pa_2,V_a_2)))) # label(fact_Bex__set__replicate) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 774 (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),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_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 775 (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]. 6.22/6.25 776 (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]. 6.22/6.25 777 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Osgn__class_Osgn(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = 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)))) # label(fact_sgn__times) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 778 (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]. 6.22/6.25 779 (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__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_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__setsum) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 780 (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]. 6.22/6.25 781 (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]. 6.22/6.25 782 (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]. 6.22/6.25 783 (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]. 6.22/6.25 784 (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]. 6.22/6.25 785 (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]. 6.22/6.25 786 (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),c_Product__Type_Ocurry(T_c,T_d,T_e,V_f_2)) = V_f_2) # label(fact_split__curry) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 787 (all V_B_2 all V_A_2 all T_d all T_c 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)) = 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))) # label(fact_Domain__Un__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 788 (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 (V_c_2 = hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_a_2),V_b_2) -> hAPP(V_c_2,V_v_2) = hAPP(V_a_2,hAPP(V_b_2,V_v_2)))) # label(fact_o__eq__dest__lhs) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 789 (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]. 6.22/6.25 790 (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]. 6.22/6.25 791 (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]. 6.22/6.25 792 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))) # label(fact_divide__minus1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 793 (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]. 6.22/6.25 794 (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]. 6.22/6.25 795 (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]. 6.22/6.25 796 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_R_2 c_New__DSequence_Opos__not__seq(hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2)) = c_New__Random__Sequence_Opos__not__random__dseq(V_R_2,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__not__random__dseq__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 797 (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]. 6.22/6.25 798 (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]. 6.22/6.25 799 (all V_w_2 all V_b_2 all V_a_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) & class_Int_Onumber(T_c) -> ((hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) != c_Groups_Ozero__class_Ozero(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))) & (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)) <-> 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_eq__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 800 (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))) -> V_f_2 = 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)))) # label(fact_inv__inv__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 801 (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]. 6.22/6.25 802 (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(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))) <-> 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)))) # label(fact_ord_OatLeastLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 803 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 804 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__no__zero__divisors(T_c) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_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) = V_b_2))) # label(fact_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 805 (all V_z_2 all V_m_2 (V_m_2 = hAPP(c_Int_Onat,V_z_2) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = V_z_2)) # label(fact_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 806 (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]. 6.22/6.25 807 (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__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)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)))))) # label(fact_Inf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 808 (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]. 6.22/6.25 809 (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]. 6.22/6.25 810 (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]. 6.22/6.25 811 (all V_x_2 -(all B_y (hAPP(c_Int_OAbs__Integ,B_y) = V_x_2 -> -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]. 6.22/6.25 812 (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,V_xs_2) = V_xs_2)) # label(fact_remove1__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 813 (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]. 6.22/6.25 814 (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),V_A_2) = hAPP(c_Finite__Set_Ofold1(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]. 6.22/6.25 815 (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__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.25 816 (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]. 6.22/6.25 817 (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]. 6.22/6.26 818 (all V_n_2 all V_m_2 (c_Nat__Transfer_Ois__nat(V_m_2) -> (c_Nat__Transfer_Ois__nat(V_n_2) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_m_2,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)))))) # label(fact_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 819 (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_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_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__set__diff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 820 (all V_x all V_n all T_a hAPP(c_List_Orev(T_a),c_List_Oreplicate(T_a,V_n,V_x)) = c_List_Oreplicate(T_a,V_n,V_x)) # label(fact_rev__replicate) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 821 (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]. 6.22/6.26 822 (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]. 6.22/6.26 823 (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__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),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__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__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)))))) # label(fact_le__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 824 (all V_x all T_a all V_xs (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)) = V_x)) # label(fact_last__ConsL) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 825 (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_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_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_bij__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 826 (all V_m 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))))) # label(fact_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 827 (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]. 6.22/6.26 828 (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__fun__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 829 (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]. 6.22/6.26 830 (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]. 6.22/6.26 831 (all V_x all V_n all T_a V_n = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_length__replicate) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 832 (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]. 6.22/6.26 833 (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_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_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))))) # label(fact_neg__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 834 (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]. 6.22/6.26 835 (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_a_2 = V_b_2)) # label(fact_singletonE) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 836 (all V_n_2 all V_m_2 all V_k_2 (V_n_2 = V_m_2 | 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))) # label(fact_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 837 (all V_d_2 c_Int_Oint__ge__less__than(V_d_2) = hAPP(c_Set_OCollect(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_d_2)))),c_Orderings_Oord__class_Oless(tc_Int_Oint))))) # label(fact_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 838 (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]. 6.22/6.26 839 (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]. 6.22/6.26 840 (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),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_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))))) # label(fact_sorted__insort__key) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 841 (all V_c_2 all T_e all T_d all T_c all V_q_2 all V_p_2 (V_q_2 = V_p_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]. 6.22/6.26 842 (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)) | 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)) | B_b = B_a)) -> 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 | 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))))) # label(fact_lexord__linear) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 843 (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]. 6.22/6.26 844 (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),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))) = hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OSemi(V_com1,V_com2))) # label(fact_com_Osize_I12_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 845 (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]. 6.22/6.26 846 (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]. 6.22/6.26 847 (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]. 6.22/6.26 848 (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]. 6.22/6.26 849 (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]. 6.22/6.26 850 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 851 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield__inverse__zero(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]. 6.22/6.26 852 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y) = V_y))) # label(fact_min__max_Osup__absorb2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 853 (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]. 6.22/6.26 854 (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]. 6.22/6.26 855 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_d 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)) = 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))) # label(fact_image__Un) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 856 (all V_xs_2 all V_Pa_2 all T_c 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)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) # label(fact_partition__filter1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 857 (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]. 6.22/6.26 858 (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]. 6.22/6.26 859 (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]. 6.22/6.26 860 (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]. 6.22/6.26 861 (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]. 6.22/6.26 862 (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_Oequations) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 863 (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(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),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_UN__empty2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 864 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_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__mult) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 865 (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]. 6.22/6.26 866 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]. 6.22/6.26 867 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b))) # label(fact_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 868 (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]. 6.22/6.26 869 (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_x_2),hAPP(hAPP(V_f_2,V_y_2),V_z_2)) = hAPP(hAPP(V_f_2,V_y_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)))) # label(fact_fun__left__comm_Ofun__left__comm) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 870 (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]. 6.22/6.26 871 (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]. 6.22/6.26 872 (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,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,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_folding_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 873 (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]. 6.22/6.26 874 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n)) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 875 (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]. 6.22/6.26 876 (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]. 6.22/6.26 877 (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]. 6.22/6.26 878 (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]. 6.22/6.26 879 (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]. 6.22/6.26 880 (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]. 6.22/6.26 881 (all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j),V_i)) -> c_List_Olist_ONil(tc_Int_Oint) = c_List_Oupto(V_i,V_j))) # label(fact_upto__empty) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 882 (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]. 6.22/6.26 883 (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]. 6.22/6.26 884 (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]. 6.22/6.26 885 (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_OBit1(V_k)),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 886 (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]. 6.22/6.26 887 (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]. 6.22/6.26 888 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 889 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_n))) # label(fact_diff__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 890 (all V_xs_2 all V_a_2 all T_c (class_Groups_Omonoid__add(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_xs_2) = 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))))) # label(fact_foldl__foldr1__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 891 (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]. 6.22/6.26 892 (all V_r_2 all T_c (c_Wellfounded_Oacyclic(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) <-> c_Wellfounded_Oacyclic(T_c,V_r_2))) # label(fact_acyclic__converse) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 893 (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]. 6.22/6.26 894 (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]. 6.22/6.26 895 (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]. 6.22/6.26 896 (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]. 6.22/6.26 897 (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]. 6.22/6.26 898 (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]. 6.22/6.26 899 (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),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_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))))))) # label(fact_setprod_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 900 (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)),V_b_2)) & 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_b_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),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__less__divide__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 901 (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) = c_Groups_Oone__class_Oone(T_d)))) # label(fact_setprod_Oinfinite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 902 (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]. 6.22/6.26 903 (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)) <-> (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]. 6.22/6.26 904 (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]. 6.22/6.26 905 (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]. 6.22/6.26 906 (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_Omin(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(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_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),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))))))) # label(fact_min__max_OInf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 907 (all V_k c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = c_Quickcheck_Obeyond(V_k,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) # label(fact_beyond__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 908 (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]. 6.22/6.26 909 (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]. 6.22/6.26 910 (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]. 6.22/6.26 911 (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]. 6.22/6.26 912 (all V_n hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),V_n) = V_n) # label(fact_nat__size) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 913 (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]. 6.22/6.26 914 (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_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)) = 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)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 915 (all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))))) # label(fact_subset__Compl__self__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 916 (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]. 6.22/6.26 917 (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]. 6.22/6.26 918 (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]. 6.22/6.26 919 (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]. 6.22/6.26 920 (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]. 6.22/6.26 921 (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]. 6.22/6.26 922 (all V_B_2 all V_A_2 all V_I_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_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)) = 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))) # label(fact_INT__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 923 (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]. 6.22/6.26 924 (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]. 6.22/6.26 925 (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]. 6.22/6.26 926 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> (c_Wellfounded_Owf(T_c,V_r_2) <-> c_Wellfounded_Oacyclic(T_c,V_r_2)))) # label(fact_wf__iff__acyclic__if__finite) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 927 (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]. 6.22/6.26 928 (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]. 6.22/6.26 929 (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]. 6.22/6.26 930 (all V_xs_2 all 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_Olist_OCons(T_c)),c_List_Olist_ONil(T_c)),V_xs_2) = hAPP(c_List_Orev(T_c),V_xs_2)) # label(fact_rev__foldl__cons) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 931 (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]. 6.22/6.26 932 (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(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]. 6.22/6.26 933 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x))) # label(fact_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 934 (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]. 6.22/6.26 935 (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]. 6.22/6.26 936 (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]. 6.22/6.26 937 (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]. 6.22/6.26 938 (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),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)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),V_B_2))) # label(fact_vimage__insert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 939 (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]. 6.22/6.26 940 (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]. 6.22/6.26 941 (all V_A_2 all V_B_2 all T_d all T_c 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)) = 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__image__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 942 (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_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_INT__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 943 (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]. 6.22/6.26 944 (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_OlessThan(tc_Nat_Onat),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_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_setsum__lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 945 (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(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)) = 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))))) # label(fact_card__Diff__subset) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 946 (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))) -> ((exists B_x (hAPP(V_f_2,B_x) = V_b_2 & (all B_y (hAPP(V_f_2,B_y) = V_b_2 -> B_y = B_x)))) <-> 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)))))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 947 (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]. 6.22/6.26 948 (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 -> ((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_Orderings_Oord__class_Omin(T_c)),V_A_2)))))))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 949 (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]. 6.22/6.26 950 (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]. 6.22/6.26 951 (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]. 6.22/6.26 952 (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]. 6.22/6.26 953 (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_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)) = 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))))) # label(fact_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 954 (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]. 6.22/6.26 955 (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]. 6.22/6.26 956 (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]. 6.22/6.26 957 (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]. 6.22/6.26 958 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield(T_a) -> V_x = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_divide__Numeral1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 959 (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]. 6.22/6.26 960 (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]. 6.22/6.26 961 (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(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_order__le__less) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 962 (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]. 6.22/6.26 963 (all V_A_2 all V_f_2 all T_d all T_c (class_Fields_Olinordered__field(T_c) -> 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)) = 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))) # label(fact_abs__setprod) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 964 (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]. 6.22/6.26 965 (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]. 6.22/6.26 966 (all V_code__numeral c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 967 (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),V_xs_2) = 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)) # label(fact_foldl__foldr) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 968 (all V_xs_2 all V_x_2 all V_zs_2 all V_ys_2 all T_c (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) <-> (exists B_ys_H (hAPP(hAPP(c_List_Oappend(T_c),B_ys_H),V_zs_2) = V_xs_2 & V_ys_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_ys_H))) | 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)) # label(fact_append__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 969 (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]. 6.22/6.26 970 (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]. 6.22/6.26 971 (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]. 6.22/6.26 972 (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)))) & (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)))) <-> c_Relation_Orefl__on(T_c,V_A_2,V_r_2))) # label(fact_refl__on__def_H) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 973 (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]. 6.22/6.26 974 (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]. 6.22/6.26 975 (all V_r_2 all V_x_2 all V_y_2 all T_c (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)) <-> c_Wellfounded_Oacyclic(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_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 976 (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]. 6.22/6.26 977 (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]. 6.22/6.26 978 (all V_a all V_r all V_q all V_c all V_b (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_c -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r),V_c))))))) # label(fact_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 979 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 980 (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_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 981 (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)) & hAPP(V_f_2,B_i) = V_k_2)))))) # label(fact_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 982 (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]. 6.22/6.26 983 (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]. 6.22/6.26 984 (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),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]. 6.22/6.26 985 (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]. 6.22/6.26 986 (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]. 6.22/6.26 987 (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]. 6.22/6.26 988 (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]. 6.22/6.26 989 (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]. 6.22/6.26 990 (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]. 6.22/6.26 991 (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]. 6.22/6.26 992 (all V_c all V_b all V_a 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) = 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))) # label(fact_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 993 (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]. 6.22/6.26 994 (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__sup__class_Osup(T_c),V_x_2),V_xs_2) = 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))))) # label(fact_Sup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 995 (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]. 6.22/6.26 996 (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]. 6.22/6.26 997 (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]. 6.22/6.26 998 (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]. 6.22/6.26 999 (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),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_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)),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]. 6.22/6.26 1000 (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]. 6.22/6.26 1001 (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]. 6.22/6.26 1002 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist_ONil(T_c) = 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__empty__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.26 1003 (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]. 6.22/6.26 1004 (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]. 6.22/6.27 1005 (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]. 6.22/6.27 1006 (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]. 6.22/6.27 1007 (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]. 6.22/6.27 1008 (all V_n all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n))) # label(fact_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1009 (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]. 6.22/6.27 1010 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z)) # label(fact_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1011 (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) <-> c_List_Oremdups(T_c,V_xs_2) = V_xs_2)) # label(fact_length__remdups__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1012 (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)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1013 (all V_xs all T_a hAPP(c_List_Orev(T_a),hAPP(c_List_Orev(T_a),V_xs)) = V_xs) # label(fact_rev__rev__ident) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1014 (all V_xs all T_a all V_ys ((V_ys = c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) & (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))))) # label(fact_last__append) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1015 (all V_xa_2 all V_x_2 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)),V_x_2),V_xa_2)),c_List_Olistrel(T_c,V_r_2))) <-> 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))) # label(fact_listrelp__listrel__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1016 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1017 (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]. 6.22/6.27 1018 (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]. 6.22/6.27 1019 (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]. 6.22/6.27 1020 (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]. 6.22/6.27 1021 (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]. 6.22/6.27 1022 (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]. 6.22/6.27 1023 (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]. 6.22/6.27 1024 (all V_n all V_m ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> c_List_Oupto(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)) = 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)))) & (-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]. 6.22/6.27 1025 (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]. 6.22/6.27 1026 (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_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__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1027 (all T_c hAPP(c_Set_OCollect(T_c),hAPP(c_COMBK(tc_HOL_Obool,T_c),c_fFalse)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_empty__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1028 (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 (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) -> (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]. 6.22/6.27 1029 (all V_y all V_x all T_a (class_Rings_Olinordered__idom(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__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1030 (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]. 6.22/6.27 1031 (all V_q all T_a c_Nitpick_Ouminus__frac(T_a,V_q) = 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)))) # label(fact_Nitpick_Ouminus__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1032 (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]. 6.22/6.27 1033 (all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> c_Enum_On__lists(T_a,V_n,c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(tc_List_Olist(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]. 6.22/6.27 1034 (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))) <-> (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)))))) # label(fact_rtrancl__power) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1035 (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(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_bij__image__INT) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1036 (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]. 6.22/6.27 1037 (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]. 6.22/6.27 1038 (all V_a all V_b (c_Groups_Ozero__class_Ozero(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),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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b)))) # label(fact_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1039 (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]. 6.22/6.27 1040 (all V_R_2 all T_c (c_Wellfounded_Owf(T_c,V_R_2) <-> c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_R_2)))) # label(fact_wf__comp__self) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1041 (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]. 6.22/6.27 1042 (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]. 6.22/6.27 1043 (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]. 6.22/6.27 1044 (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(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)))) # label(fact_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1045 (all T_c all T_d c_Fun_Oid(tc_prod(T_c,T_d)) = c_Product__Type_Oapsnd(T_d,T_d,T_c,c_Fun_Oid(T_d))) # label(fact_apsnd__id) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1046 (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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_folding__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1047 (all T_a c_Nitpick_Ozero__frac(T_a) = c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_Nitpick_Ozero__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1048 (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]. 6.22/6.27 1049 (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]. 6.22/6.27 1050 (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]. 6.22/6.27 1051 (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]. 6.22/6.27 1052 (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_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__iff2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1053 (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]. 6.22/6.27 1054 (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]. 6.22/6.27 1055 (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]. 6.22/6.27 1056 (all V_a (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_a))) # label(fact_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1057 (all V_f2_2 all V_f1_2 all T_c V_f1_2 = 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))) # label(fact_code__numeral_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1058 (all V_k_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_i_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),V_k_2)) | V_k_2 = c_Groups_Ozero__class_Ozero(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_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1059 (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]. 6.22/6.27 1060 (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]. 6.22/6.27 1061 (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]. 6.22/6.27 1062 (all V_r all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) & class_Int_Onumber__ring(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]. 6.22/6.27 1063 (all V_n all V_ys all V_xs all T_a hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) = hAPP(hAPP(c_List_Onth(T_a),V_ys),V_n)) # label(fact_nth__append__length__plus) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1064 (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]. 6.22/6.27 1065 (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]. 6.22/6.27 1066 (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]. 6.22/6.27 1067 (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]. 6.22/6.27 1068 (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]. 6.22/6.27 1069 (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]. 6.22/6.27 1070 (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]. 6.22/6.27 1071 (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]. 6.22/6.27 1072 (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)),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__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1073 (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]. 6.22/6.27 1074 (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]. 6.22/6.27 1075 (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]. 6.22/6.27 1076 (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]. 6.22/6.27 1077 (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]. 6.22/6.27 1078 (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_b_2) = V_a_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]. 6.22/6.27 1079 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))),c_List_Olist_ONil(T_a))))) # label(fact_take__hd__drop) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1080 (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]. 6.22/6.27 1081 (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]. 6.22/6.27 1082 (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]. 6.22/6.27 1083 (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]. 6.22/6.27 1084 (all V_xs_2 all V_Qa_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2)) = 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)) # label(fact_filter__filter) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1085 (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) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_d) = V_xs_2)) # label(fact_map__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1086 (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]. 6.22/6.27 1087 (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)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1088 (all T_c (class_Orderings_Olinorder(T_c) & class_Lattices_Osemilattice__inf(T_c) -> c_Lattices_Osemilattice__inf__class_Oinf(T_c) = c_Orderings_Oord__class_Omin(T_c))) # label(fact_inf__min) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1089 (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_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]. 6.22/6.27 1090 (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__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1091 (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_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)) & (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),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(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)))))) # label(fact_setsum__diff1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1092 (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]. 6.22/6.27 1093 (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]. 6.22/6.27 1094 (all V_n all V_a all V_b all T_a (class_Fields_Ofield(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(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_nonzero__power__divide) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1095 (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_OatMost(T_c,V_less__eq_2,V_k_2))) <-> hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_k_2)))) # label(fact_ord_OatMost__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1096 (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_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_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]. 6.22/6.27 1097 (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]. 6.22/6.27 1098 (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]. 6.22/6.27 1099 (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]. 6.22/6.27 1100 (all V_z3 all V_z2 all V_z1 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)) = 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)) # label(fact_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1101 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y))) # label(fact_min__max_Osup__left__idem) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1102 (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]. 6.22/6.27 1103 (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]. 6.22/6.27 1104 (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]. 6.22/6.27 1105 (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]. 6.22/6.27 1106 (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]. 6.22/6.27 1107 (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)),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)) = 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))))))) # label(fact_fold__image__Un__Int) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1108 (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 V_n_2 = hAPP(c_Nat_OSuc,B_m)))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1109 (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]. 6.22/6.27 1110 (all V_R_2 all T_c all T_d hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_c,c_Product__Type_Osnd(T_d,T_c)),V_R_2) = hAPP(c_Relation_ORange(T_d,T_c),V_R_2)) # label(fact_snd__eq__Range) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1111 (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]. 6.22/6.27 1112 (all V_B_2 all V_C_2 all T_d 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_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__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1113 (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]. 6.22/6.27 1114 (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(c_SetInterval_Oord__class_OatLeastLessThan(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)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)))))) # label(fact_ivl__disj__un_I17_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1115 (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]. 6.22/6.27 1116 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,V_j_2))) # label(fact_set__upto) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1117 (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]. 6.22/6.27 1118 (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]. 6.22/6.27 1119 (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]. 6.22/6.27 1120 (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]. 6.22/6.27 1121 (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]. 6.22/6.27 1122 (all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_a) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__self) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1123 (all V_n_2 all V_m_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) <-> c_Groups_Ozero__class_Ozero(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]. 6.22/6.27 1124 (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]. 6.22/6.27 1125 (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]. 6.22/6.27 1126 (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]. 6.22/6.27 1127 (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]. 6.22/6.27 1128 (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]. 6.22/6.27 1129 (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]. 6.22/6.27 1130 (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]. 6.22/6.27 1131 (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]. 6.22/6.27 1132 (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]. 6.22/6.27 1133 (all V_n all V_i ((V_i = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> c_Code__Numeral_Onat__of__aux(V_i,V_n) = V_n) & (V_i != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> c_Code__Numeral_Onat__of__aux(V_i,V_n) = c_Code__Numeral_Onat__of__aux(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_OSuc,V_n))))) # label(fact_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1134 (all V_n all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> (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]. 6.22/6.27 1135 (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]. 6.22/6.27 1136 (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_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)))) & (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))))) # label(fact_UN__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1137 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OPls)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OPls)))) # label(fact_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1138 (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]. 6.22/6.27 1139 (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]. 6.22/6.27 1140 (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]. 6.22/6.27 1141 (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]. 6.22/6.27 1142 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2))))) # label(fact_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1143 (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]. 6.22/6.27 1144 (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]. 6.22/6.27 1145 (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]. 6.22/6.27 1146 (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]. 6.22/6.27 1147 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1148 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all 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(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))) # label(fact_filter__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1149 (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,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) = 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)))))) # label(fact_fold__image__Sigma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1150 (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]. 6.22/6.27 1151 (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]. 6.22/6.27 1152 (all V_Pa_2 all T_d all T_c c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),V_Pa_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_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))))) # label(fact_split__paired__Eps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1153 (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_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)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_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)) = 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]. 6.22/6.27 1154 (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]. 6.22/6.27 1155 (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]. 6.22/6.27 1156 (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]. 6.22/6.27 1157 (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]. 6.22/6.27 1158 (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]. 6.22/6.27 1159 (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]. 6.22/6.27 1160 (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]. 6.22/6.27 1161 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k))) # label(fact_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1162 (all V_i c_List_Oupt(V_i,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_List_Olist_ONil(tc_Nat_Onat)) # label(fact_upt__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1163 (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]. 6.22/6.27 1164 (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)) | 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_subset__iff__psubset__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1165 (all V_A_2 all V_xs_2 all T_c c_List_Osublist(T_c,V_xs_2,V_A_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(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))))) # label(fact_sublist__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1166 (all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> 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]. 6.22/6.27 1167 (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]. 6.22/6.27 1168 (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]. 6.22/6.27 1169 (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]. 6.22/6.27 1170 (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]. 6.22/6.27 1171 (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]. 6.22/6.27 1172 (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]. 6.22/6.27 1173 (all V_a_2 all V_b_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_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2) <-> V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))) # label(fact_nonzero__divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1174 (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]. 6.22/6.27 1175 (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]. 6.22/6.27 1176 (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]. 6.22/6.27 1177 (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]. 6.22/6.27 1178 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))))) # label(fact_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1179 (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_yes_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_no_2))) -> -hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_partition__P) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1180 (all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_minus__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1181 (all V_A_2 all T_d all T_c all V_B_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_B_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))) = 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]. 6.22/6.27 1182 (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__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)) = 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)))) # label(fact_inf__INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1183 (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]. 6.22/6.27 1184 (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]. 6.22/6.27 1185 (all V_Y_2 all T_c hAPP(c_Set_Oimage(T_c,T_c,c_COMBI(T_c)),V_Y_2) = V_Y_2) # label(fact_image__ident) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1186 (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(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))))) = c_Nitpick_Osetsum_H(T_c,T_d,V_f_2,V_A_2)) & (-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) = c_Groups_Ozero__class_Ozero(T_d)))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1187 (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]. 6.22/6.27 1188 (all V_b_2 all V_a_2 all T_c (class_Groups_Oab__group__add(T_c) -> (V_b_2 = V_a_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1189 (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]. 6.22/6.27 1190 (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]. 6.22/6.27 1191 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_f1_2 all T_c all T_d hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2) = 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))) # label(fact_triple_Orecs) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1192 (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]. 6.22/6.27 1193 (all V_A_H_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)) -> ((exists B_g hAPP(c_Set_Oimage(T_d,T_c,B_g),V_A_H_2) = V_A_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_A_H_2))))))) # label(fact_inj__on__iff__surj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1194 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(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_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__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1195 (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]. 6.22/6.27 1196 (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]. 6.22/6.27 1197 (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]. 6.22/6.27 1198 (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]. 6.22/6.27 1199 (all V_r_2 all T_c c_Transitive__Closure_Otrancl(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_trancl__unfold__right) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1200 (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_A_2 = V_B_2 -> (c_Nat__Transfer_Onat__set(V_B_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_g_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_f_2),V_A_2)))))) # label(fact_transfer__nat__int__sum__prod__cong_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1201 (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]. 6.22/6.27 1202 (all V_m all V_n (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_OSuc,V_m))) -> V_m = V_n))) # label(fact_less__antisym) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1203 (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]. 6.22/6.27 1204 (all V_B_2 all V_A_2 all T_c all T_d 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)) = 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__eq__Inter__image) # label(axiom) # label(non_clause). [assumption]. 6.22/6.27 1205 (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]. 6.22/6.27 1206 (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]. 6.22/6.27 1207 (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]. 6.22/6.27 1208 (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]. 6.22/6.28 1209 (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]. 6.22/6.28 1210 (all V_y_2 all V_x_2 all V_c_2 all T_c all T_e all T_d hAPP(V_c_2,hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2)) = hAPP(hAPP(c_Product__Type_Ocurry(T_d,T_e,T_c,V_c_2),V_x_2),V_y_2)) # label(fact_curry__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1211 (all V_v_H all V_v ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = 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))) & (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]. 6.22/6.28 1212 (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]. 6.22/6.28 1213 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1214 (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]. 6.22/6.28 1215 (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]. 6.22/6.28 1216 (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_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)) = 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))) # label(fact_zip__map__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1217 (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__sup__class_Osup(T_c)),V_B_2),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))))) # label(fact_sup__Sup__fold__sup) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1218 (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]. 6.22/6.28 1219 (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]. 6.22/6.28 1220 (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(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]. 6.22/6.28 1221 (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]. 6.22/6.28 1222 (all V_k all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k)) # label(fact_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1223 (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]. 6.22/6.28 1224 (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]. 6.22/6.28 1225 (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]. 6.22/6.28 1226 (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]. 6.22/6.28 1227 (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]. 6.22/6.28 1228 (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]. 6.22/6.28 1229 (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_Groups_Ocomm__monoid__add(T_d) & class_Orderings_Oord(T_c) -> (V_c_2 = V_a_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_f_2,B_x) = hAPP(V_g_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]. 6.22/6.28 1230 (all V_b all V_a all T_a hAPP(hAPP(c_Nitpick_Ofrac(T_a),V_a),V_b) = c_Nitpick_OAbs__Frac(T_a,c_Nitpick_Onorm__frac(V_a,V_b))) # label(fact_Nitpick_Ofrac__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1231 (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]. 6.22/6.28 1232 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1233 (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]. 6.22/6.28 1234 (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_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_I45_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1235 (all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j),V_i)) -> c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i,V_j))) # label(fact_upt__conv__Nil) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1236 (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]. 6.22/6.28 1237 (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]. 6.22/6.28 1238 (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]. 6.22/6.28 1239 (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_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(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_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1240 (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]. 6.22/6.28 1241 (all V_X_2 (V_X_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),V_X_2),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))),c_FunDef_Omin__strict)))) # label(fact_smin__emptyI) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1242 (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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_a),B_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> hBOOL(hAPP(hAPP(V_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]. 6.22/6.28 1243 (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]. 6.22/6.28 1244 (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]. 6.22/6.28 1245 (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]. 6.22/6.28 1246 (all V_A_2 all T_c hAPP(c_Set_OPow(T_c),V_A_2) = 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_Pow__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1247 (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]. 6.22/6.28 1248 (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)) -> ((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)))))) <-> 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)))))))) # label(fact_finite__Collect__bex) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1249 (all V_z_2 all T_c (class_Int_Oring__char__0(T_c) -> (hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2) = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_z_2))) # label(fact_of__int__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1250 (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]. 6.22/6.28 1251 (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)),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)))))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_r_2))))) # label(fact_half__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1252 (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]. 6.22/6.28 1253 (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]. 6.22/6.28 1254 (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))) = c_Groups_Oone__class_Oone(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)))))) # label(fact_power__number__of__odd__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1255 (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]. 6.22/6.28 1256 (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),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_x_2)) <-> 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)))))) # label(fact_min__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1257 (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]. 6.22/6.28 1258 (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]. 6.22/6.28 1259 (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)) -> (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))) <-> (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_below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1260 (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)) & (V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_p),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))))))) # label(fact_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1261 (all V_y_2 all V_x_2 all T_c (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)) <-> V_x_2 != V_y_2)) # label(fact_conversep__noteq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1262 (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]. 6.22/6.28 1263 (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),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)),V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1264 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) -> hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs) = V_xs)) # label(fact_take__all) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1265 (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]. 6.22/6.28 1266 (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]. 6.22/6.28 1267 (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]. 6.22/6.28 1268 (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(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = c_List_Oinsert(T_c,V_x_2,V_xs_2))) # label(fact_not__in__set__insert) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1269 (all V_xs all T_a all V_y all V_x ((V_y = V_x -> hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(c_List_OremoveAll(T_a,V_x),V_xs)) & (V_y != V_x -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),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]. 6.22/6.28 1270 (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),V_m),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))))) # label(fact_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1271 (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]. 6.22/6.28 1272 (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]. 6.22/6.28 1273 (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)),V_f_2),V_xs_2)) = c_List_Omaps(T_d,T_c,V_f_2,V_xs_2)) # label(fact_concat__map__maps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1274 (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]. 6.22/6.28 1275 (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]. 6.22/6.28 1276 (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]. 6.22/6.28 1277 (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]. 6.22/6.28 1278 (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)) <-> (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)))))) # label(fact_INF1__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1279 (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(c_Quotient_OBabs(T_c,T_d,V_p_2,V_m_2),V_x_2) = hAPP(V_m_2,V_x_2))) # label(fact_Babs__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1280 (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]. 6.22/6.28 1281 (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)),V_b),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__left__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1282 (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]. 6.22/6.28 1283 (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),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_c))) # label(fact_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1284 (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]. 6.22/6.28 1285 (all V_A_2 all T_c all V_y_2 all V_x_2 (V_y_2 = V_x_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))))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1286 (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]. 6.22/6.28 1287 (all V_A_2 all V_xs_2 all V_x_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(T_c),V_x_2),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]. 6.22/6.28 1288 (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]. 6.22/6.28 1289 (all T_c c_Predicate_Oconversep(T_c,T_c,c_fequal) = c_fequal) # label(fact_conversep__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1290 (all V_xs_2 all T_c (c_List_Oremdups(T_c,V_xs_2) = V_xs_2 <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_remdups__id__iff__distinct) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1291 (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]. 6.22/6.28 1292 (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]. 6.22/6.28 1293 (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]. 6.22/6.28 1294 (all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_c (V_x_2 = V_y_2 & V_ys_2 = V_xs_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]. 6.22/6.28 1295 (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_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_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]. 6.22/6.28 1296 (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]. 6.22/6.28 1297 (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]. 6.22/6.28 1298 (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]. 6.22/6.28 1299 (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]. 6.22/6.28 1300 (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]. 6.22/6.28 1301 (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]. 6.22/6.28 1302 (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]. 6.22/6.28 1303 (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]. 6.22/6.28 1304 (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]. 6.22/6.28 1305 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_x_2 = hAPP(hAPP(c_Orderings_Oord__class_Omin(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_min__max_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1306 (all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,V_r_2) <-> 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))))) # label(fact_wfP__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1307 (all V_w_2 all V_v_2 all T_c (class_Orderings_Olinorder(T_c) & class_Int_Onumber(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]. 6.22/6.28 1308 (all V_l all V_k (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != 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_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l))) # label(fact_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1309 (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)))) -> 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) & (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(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)))))) # label(fact_min__max_OSup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1310 (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]. 6.22/6.28 1311 (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]. 6.22/6.28 1312 (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]. 6.22/6.28 1313 (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]. 6.22/6.28 1314 (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)) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_H_2),V_B_H_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> (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]. 6.22/6.28 1315 (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)) -> (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) <-> V_c_2 = V_a_2 & V_d_2 = V_b_2))))) # label(fact_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1316 (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]. 6.22/6.28 1317 (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]. 6.22/6.28 1318 (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_y_2 | c_Groups_Ozero__class_Ozero(T_c) != V_x_2))) # label(fact_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1319 (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]. 6.22/6.28 1320 (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]. 6.22/6.28 1321 (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),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)))) # label(fact_min__max_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1322 (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]. 6.22/6.28 1323 (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]. 6.22/6.28 1324 (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]. 6.22/6.28 1325 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) <-> 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))))))) # label(fact_sorted__Cons) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1326 (all V_r_2 all V_x_2 all T_c all V_y_2 ((exists B_b exists B_z V_y_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_b),B_z)) -> 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]. 6.22/6.28 1327 (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_Odrop(T_d),V_n_2),V_xs_2)) = 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))) # label(fact_drop__map) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1328 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_b))) # label(fact_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1329 (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]. 6.22/6.28 1330 (all T_d all T_c (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(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,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]. 6.22/6.28 1331 (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]. 6.22/6.28 1332 (all V_b_2 all V_a_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_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OatLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1333 (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]. 6.22/6.28 1334 (all V_ys_2 all V_xs_2 all T_c (V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> V_ys_2 = c_List_Olist_ONil(T_c))) # label(fact_append__self__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1335 (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]. 6.22/6.28 1336 (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]. 6.22/6.28 1337 (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(tc_prod(T_c,T_d)),V_Pa_2)) <-> 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)))))))) # label(fact_enum__ex__prod__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1338 (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]. 6.22/6.28 1339 (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]. 6.22/6.28 1340 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n))) # label(fact_of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1341 (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]. 6.22/6.28 1342 (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]. 6.22/6.28 1343 (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__zero__sym) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1344 (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)) -> 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_OnegDivAlg(V_a,V_b))))) # label(fact_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1345 (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]. 6.22/6.28 1346 (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]. 6.22/6.28 1347 (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]. 6.22/6.28 1348 (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]. 6.22/6.28 1349 (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]. 6.22/6.28 1350 (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]. 6.22/6.28 1351 (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,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,V_a,V_c)))) # label(fact_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1352 (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]. 6.22/6.28 1353 (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]. 6.22/6.28 1354 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2)))) # label(fact_Compl__lessThan) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1355 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_c 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)) = 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)))) # label(fact_Times__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1356 (all V_n V_n = hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_n))) # label(fact_nat__of__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1357 (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]. 6.22/6.28 1358 (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_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)))))) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1359 (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]. 6.22/6.28 1360 (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]. 6.22/6.28 1361 (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]. 6.22/6.28 1362 (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]. 6.22/6.28 1363 (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]. 6.22/6.28 1364 (all V_x_2 all T_c (c_List_Olist_ONil(T_c) = V_x_2 <-> c_List_Oremdups(T_c,V_x_2) = c_List_Olist_ONil(T_c))) # label(fact_remdups__eq__nil__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1365 (all V_m 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,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1366 (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]. 6.22/6.28 1367 (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]. 6.22/6.28 1368 (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]. 6.22/6.28 1369 (all V_a_2 all V_w_2 all V_b_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Ofield__inverse__zero(T_c) -> ((c_Groups_Ozero__class_Ozero(T_c) != hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_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)) = V_b_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)) <-> 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_divide__eq__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1370 (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_c_2 = V_d_2 <-> V_b_2 = V_a_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1371 (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]. 6.22/6.28 1372 (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_m_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_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1373 (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]. 6.22/6.28 1374 (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]. 6.22/6.28 1375 (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_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_c,T_d,V_f_2),V_g_2)),V_r_2) = 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))) # label(fact_image__compose) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1376 (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(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) = hAPP(V_f_2,V_a_2)))) # label(fact_inv__into__inv__into__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1377 (all V_xs_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_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)) = hAPP(c_List_Oset(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_set__filter) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1378 (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]. 6.22/6.28 1379 (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]. 6.22/6.28 1380 (all V_y all V_x hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)),c_Nitpick_Onat__gcd(V_x,V_y)) = c_Nitpick_Onat__lcm(V_x,V_y)) # label(fact_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1381 (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]. 6.22/6.28 1382 (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))) <-> (exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_n_2)) & hAPP(c_Nat_OSuc,B_j) = V_m_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2)) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1383 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_diff__self) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1384 (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]. 6.22/6.28 1385 (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]. 6.22/6.28 1386 (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]. 6.22/6.28 1387 (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_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) # label(fact_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1388 (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]. 6.22/6.28 1389 (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) -> ((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)))) <-> 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))))) # label(fact_INF__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1390 (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))) # label(fact_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1391 (all V_zs_2 all V_y_2 all V_x_2 all T_c (class_Groups_Osemigroup__add(T_c) -> 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) = 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)))) # label(fact_foldl__assoc) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1392 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b)) = V_b)) # label(fact_add__minus__cancel) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1393 (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]. 6.22/6.28 1394 (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]. 6.22/6.28 1395 (all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> 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)))) = V_x_2)) # label(fact_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1396 (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]. 6.22/6.28 1397 (all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = V_i) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_i)))) # label(fact_zabs__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1398 (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]. 6.22/6.28 1399 (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]. 6.22/6.28 1400 (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]. 6.22/6.28 1401 (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]. 6.22/6.28 1402 (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) = V_B_2) & (hBOOL(hAPP(hAPP(c_member(T_c),V_d_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(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)) -> 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_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))))) # label(fact_vimage__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1403 (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)))) <-> c_Relation_Osym(T_c,V_r_2))) # label(fact_sym__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1404 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) <-> (all B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Wellfounded_Oacc(T_c,V_r_2)))))) # label(fact_wf__acc__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1405 (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]. 6.22/6.28 1406 (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))) & (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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__conv__if) # label(axiom) # label(non_clause). [assumption]. 6.22/6.28 1407 (all V_ys all V_xs all T_a all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) -> ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_k)) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),B_i) = hAPP(hAPP(c_List_Onth(T_a),V_ys),B_i))) -> hAPP(hAPP(c_List_Otake(T_a),V_k),V_xs) = hAPP(hAPP(c_List_Otake(T_a),V_k),V_ys))))) # label(fact_nth__take__lemma) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1408 (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]. 6.22/6.29 1409 (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]. 6.22/6.29 1410 (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]. 6.22/6.29 1411 (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(c_List_Odistinct(T_c),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_distinct_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1412 (all V_B_2 all V_A_2 all T_d all T_c (c_Orderings_Obot__class_Obot(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool)) = c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2) <-> 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)))) # label(fact_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1413 (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]. 6.22/6.29 1414 (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)) -> c_List_Oupt(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_i),c_List_Oupt(hAPP(c_Nat_OSuc,V_i),V_j))))) # label(fact_upt__rec) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1415 (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]. 6.22/6.29 1416 (all V_d_2 all V_c_2 all V_b_2 all T_c (class_Orderings_Olinorder(T_c) -> 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)) = 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)))) # label(fact_Int__atLeastAtMostR1) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1417 (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),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),V_a_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),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)))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1418 (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]. 6.22/6.29 1419 (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]. 6.22/6.29 1420 (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) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_r))) # label(fact_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1421 (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]. 6.22/6.29 1422 (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]. 6.22/6.29 1423 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__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_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1424 (all T_a all V_ys all V_xs (V_xs = V_ys -> 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]. 6.22/6.29 1425 (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]. 6.22/6.29 1426 (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]. 6.22/6.29 1427 (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]. 6.22/6.29 1428 (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]. 6.22/6.29 1429 (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]. 6.22/6.29 1430 (all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_a_2) <-> (exists B_xs exists B_x (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)))) & hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) = V_a_2)) | V_a_2 = c_List_Olist_ONil(T_c)))) # label(fact_sorted_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1431 (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]. 6.22/6.29 1432 (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]. 6.22/6.29 1433 (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]. 6.22/6.29 1434 (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]. 6.22/6.29 1435 (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(tc_Int_Oint),V_w_2),V_z_2)) | V_w_2 = V_z_2)) # label(fact_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1436 (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]. 6.22/6.29 1437 (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)) -> V_x_2 = c_Big__Operators_Olattice__class_OInf__fin(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,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]. 6.22/6.29 1438 (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]. 6.22/6.29 1439 (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(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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_lessThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1440 (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]. 6.22/6.29 1441 (all V_xs_2 all V_x_2 all T_c 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)) = c_Random_Oselect__weight(T_c,V_xs_2)) # label(fact_select__weight__cons__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1442 (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]. 6.22/6.29 1443 (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),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2) -> 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]. 6.22/6.29 1444 (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]. 6.22/6.29 1445 (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_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)))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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))))) # label(fact_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1446 (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]. 6.22/6.29 1447 (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]. 6.22/6.29 1448 (all V_x_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Product__Type_Osnd(T_e,T_c),V_x_2) = 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))) # label(fact_snd__apfst) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1449 (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]. 6.22/6.29 1450 (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),hAPP(c_Groups_Ouminus__class_Ouminus(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_abs__power__minus) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1451 (all V_m_2 all V_n_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_m_2))) <-> V_m_2 = V_n_2))) # label(fact_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1452 (all V_xs_2 all T_c c_Random_Opick(T_c,V_xs_2) = 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))) # label(fact_pick__drop__zero) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1453 (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),hAPP(c_Nat_OSuc,V_m)),V_n)),hAPP(c_Nat_OSuc,V_k)) = 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)) # label(fact_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1454 (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)) -> 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))) <-> 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)))) # label(fact_Union__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1455 (all V_s_2 all V_r_2 all T_e all T_c all T_d 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)) = 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))) # label(fact_converse__rel__comp) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1456 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1457 (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]. 6.22/6.29 1458 (all V_x_2 all T_c (class_Groups_Ozero(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> c_Groups_Ozero__class_Ozero(T_c) = V_x_2))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1459 (all V_i all V_xs all T_a V_xs = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i))) # label(fact_list__update__id) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1460 (all T_c hAPP(c_Set_OCollect(T_c),hAPP(c_COMBK(tc_HOL_Obool,T_c),c_fTrue)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_UNIV__def) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1461 (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)) | V_m_2 = V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1462 (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_x_2) = hAPP(V_f_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)) -> V_y_2 = V_x_2))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 6.22/6.29 1463 (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_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) = 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))) # label(fact_INT__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1464 (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]. 6.31/6.29 1465 (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) = 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_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1466 (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]. 6.31/6.29 1467 (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]. 6.31/6.29 1468 (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]. 6.31/6.29 1469 (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]. 6.31/6.29 1470 (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_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)))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1471 (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]. 6.31/6.29 1472 (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)),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))))) = 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))) # label(fact_nat__rec__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1473 (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]. 6.31/6.29 1474 (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]. 6.31/6.29 1475 (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]. 6.31/6.29 1476 (all V_n_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2) = hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))) # label(fact_atLeast__upt) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1477 (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]. 6.31/6.29 1478 (all V_ys_2 all V_Pa_2 all T_d all T_c (c_List_Olist_ONil(T_d) = V_ys_2 <-> 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]. 6.31/6.29 1479 (all V_A_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) & class_Groups_Ocomm__monoid__mult(T_c) -> 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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,T_c),V_f_2),V_A_2))) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1480 (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]. 6.31/6.29 1481 (all V_m_2 all V_n_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)))) # label(fact_less__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1482 (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]. 6.31/6.29 1483 (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]. 6.31/6.29 1484 (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]. 6.31/6.29 1485 (all V_com_H_2 all V_fun_H_2 all V_com_2 all V_fun_2 (V_com_2 = V_com_H_2 & V_fun_H_2 = V_fun_2 <-> c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) = c_Com_Ocom_OWhile(V_fun_2,V_com_2))) # label(fact_com_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1486 (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]. 6.31/6.29 1487 (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_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_Oid(T_c))) # label(fact_inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1488 (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]. 6.31/6.29 1489 (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]. 6.31/6.29 1490 (all V_C_2 all V_J_2 all V_I_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_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__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2)) # label(fact_Sigma__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1491 (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]. 6.31/6.29 1492 (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]. 6.31/6.29 1493 (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_Olex(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_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))) # label(fact_lexord__lex) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1494 (all V_j all V_i all V_m hAPP(hAPP(c_List_Odrop(tc_Nat_Onat),V_m),c_List_Oupt(V_i,V_j)) = c_List_Oupt(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_m),V_j)) # label(fact_drop__upt) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1495 (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]. 6.31/6.29 1496 (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]. 6.31/6.29 1497 (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_Set_Oinsert(T_c),V_a_2),V_A_2) = V_A_2)) # label(fact_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1498 (all V_com_H_2 all V_fun_H_2 all V_loc_H_2 c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1499 (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]. 6.31/6.29 1500 (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]. 6.31/6.29 1501 (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]. 6.31/6.29 1502 (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))) <-> (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))) # label(fact_in__set__conv__decomp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1503 (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]. 6.31/6.29 1504 (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]. 6.31/6.29 1505 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z))) # label(fact_min__max_Osup__assoc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1506 (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]. 6.31/6.29 1507 (all V_I_2 all V_xs_2 all T_c 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))))) = hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2))) # label(fact_set__sublist) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1508 (all V_n all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) = 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))))) # label(fact_power__m1__odd) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1509 (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]. 6.31/6.29 1510 (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,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__append2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1511 (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]. 6.31/6.29 1512 (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]. 6.31/6.29 1513 (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)))))) <-> V_b_2 = hAPP(V_f_2,V_a_2))) # label(fact_vimage__singleton__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1514 (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]. 6.31/6.29 1515 (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]. 6.31/6.29 1516 (all V_x_2 all T_c c_Hilbert__Choice_OEps(T_c,hAPP(c_fequal,V_x_2)) = V_x_2) # label(fact_some__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1517 (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]. 6.31/6.29 1518 (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]. 6.31/6.29 1519 (all V_vs_2 all V_us_2 all V_r_2 all V_ys_2 all V_xs_2 all T_c (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)),V_us_2),V_vs_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_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)),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]. 6.31/6.29 1520 (all V_b_2 all V_a_2 all V_c_2 all T_c all T_e all T_d hAPP(hAPP(V_c_2,V_a_2),V_b_2) = 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))) # label(fact_internal__split__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1521 (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]. 6.31/6.29 1522 (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]. 6.31/6.29 1523 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),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]. 6.31/6.29 1524 (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(hAPP(V_Pa_2,V_Z_2),V_s_2)) & hBOOL(hAPP(V_p_2,V_s_2)))) # label(fact_peek__and__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1525 (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))) -> (c_Orderings_Otop__class_Otop(tc_fun(T_e,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))) -> 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]. 6.31/6.29 1526 (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]. 6.31/6.29 1527 (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_Power_Opower__class_Opower(T_d),V_y_2),hAPP(c_Finite__Set_Ocard(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)))) # label(fact_setprod__constant) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1528 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (V_b_2 = V_a_2 -> c_SetInterval_Oord__class_OatLeastAtMost(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)))))) # label(fact_atLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1529 (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]. 6.31/6.29 1530 (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_Orotate1(T_c),V_xs_2))) # label(fact_rotate1__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1531 (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]. 6.31/6.29 1532 (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)) = 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))))) # label(fact_atLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1533 (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]. 6.31/6.29 1534 (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)))) -> 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,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__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]. 6.31/6.29 1535 (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 -> c_Fun_Ofun__upd(T_d,T_c,V_f_2,V_x_2,V_y_2) = V_f_2)) # label(fact_fun__upd__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1536 (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]. 6.31/6.29 1537 (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]. 6.31/6.29 1538 (all V_A_2 all V_f_2 all T_d all T_c (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_d,T_c,V_f_2),V_A_2))) # label(fact_empty__is__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1539 (all V_x_2 all V_S_2 all V_R_2 all T_c (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)) <-> 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))))) # label(fact_sup__Un__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1540 (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]. 6.31/6.29 1541 (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]. 6.31/6.29 1542 (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]. 6.31/6.29 1543 (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]. 6.31/6.29 1544 (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_Orev(T_c),V_xs_2))))) # label(fact_distinct__rev) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1545 (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)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(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_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1546 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(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)))) # label(fact_divide__self__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1547 (all V_k_2 all V_A_2 all T_c ((exists B_b exists B_B (-hBOOL(hAPP(hAPP(c_member(T_c),B_b),B_B)) & V_k_2 = hAPP(c_Finite__Set_Ocard(T_c),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) & V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_c),B_b),B_B))) <-> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Nat_OSuc,V_k_2))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1548 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_zle__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1549 (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))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B_x)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),V_ns_2))) # label(fact_listsum__eq__0__nat__iff__nat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1550 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_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))))) # label(fact_of__nat__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1551 (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)) -> hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_F_2)))) # label(fact_setsum__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1552 (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),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)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_divide__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1553 (all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> hAPP(c_List_Oset(T_c),V_xs_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_set__empty2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1554 (all V_xs_2 all T_c hAPP(c_Set_OPow(T_c),hAPP(c_List_Oset(T_c),V_xs_2)) = 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)))) # label(fact_sublists__powset) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1555 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1556 (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]. 6.31/6.29 1557 (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]. 6.31/6.29 1558 (all V_g_2 all V_x_2 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_g_2),V_x_2) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),V_x_2) <-> hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_c,T_e),V_x_2)) = hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_c,T_e),V_x_2)))) # label(fact_apsnd__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1559 (all V_r_2 all T_c hAPP(c_Relation_ORange(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2)) = hAPP(c_Relation_ORange(T_c,T_c),V_r_2)) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1560 (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]. 6.31/6.29 1561 (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),c_List_Oremove1(T_c,V_b_2,V_xs_2)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2)))))) # label(fact_in__set__remove1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1562 (all V_code__numeral_H_1 c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_1) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) # label(fact_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1563 (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]. 6.31/6.29 1564 (all T_1 all T_2 (class_Lazy__Sequence_Osmall__lazy(T_2) & class_Lazy__Sequence_Osmall__lazy(T_1) -> class_Lazy__Sequence_Osmall__lazy(tc_prod(T_2,T_1)))) # label(arity_prod__Lazy__Sequence_Osmall__lazy) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1565 (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)))) -> V_x_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_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__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)))))) = 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__remove) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1566 (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]. 6.31/6.29 1567 (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]. 6.31/6.29 1568 (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_exists__code) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1569 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_Opred(V_k_2)),V_l_2)))) # label(fact_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1570 (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)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_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]. 6.31/6.29 1571 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))) <-> 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)))) # label(fact_DomainP__Domain__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1572 (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]. 6.31/6.29 1573 (all V_b all V_a ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_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))))) & (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)) -> (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_b)) -> c_Divides_Odivmod__int(V_a,V_b) = c_Divides_OposDivAlg(V_a,V_b))))) # label(fact_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1574 (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(V_f_2,B_x) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_xa != B_x -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(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(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_setsum__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1575 (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)),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_length__tl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1576 (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]. 6.31/6.29 1577 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__Suc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1578 (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_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_Max_OF__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1579 (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_OatLeastLessThan(T_c,V_l_2),V_u_2)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_l_2)))) # label(fact_ivl__disj__un_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1580 (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)))) -> hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_y_2)) = V_y_2)) # label(fact_of__nat__inverse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1581 (all V_ys all T_a all V_xs ((V_xs = c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_ys) = 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_xs)))) # label(fact_hd__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1582 (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]. 6.31/6.29 1583 (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)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a2_2)) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a1_2)) -> (exists B_xa (hBOOL(hAPP(hAPP(V_R_2,B_x),B_xa)) & 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)))) # label(fact_max__extp_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1584 (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]. 6.31/6.29 1585 (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]. 6.31/6.29 1586 (all V_fa_2 all T_c hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_fa_2),c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_lazy__sequence_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.29 1587 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 <-> 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]. 6.31/6.29 1588 (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]. 6.31/6.29 1589 (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]. 6.31/6.29 1590 (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]. 6.31/6.30 1591 (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]. 6.31/6.30 1592 (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]. 6.31/6.30 1593 (all V_B_2 all V_x_2 all V_A_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)),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_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_subset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1594 (all V_ys_2 all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> V_xs_2 = c_List_Olist_ONil(T_c) & c_List_Olist_ONil(T_c) = V_ys_2)) # label(fact_append__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1595 (all T_a c_Nitpick_Oone__frac(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)))) # label(fact_Nitpick_Oone__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1596 (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]. 6.31/6.30 1597 (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]. 6.31/6.30 1598 (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]. 6.31/6.30 1599 (all V_x all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i) = V_x)) # label(fact_nth__list__update__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1600 (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]. 6.31/6.30 1601 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_l_2))))) # label(fact_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1602 (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]. 6.31/6.30 1603 (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]. 6.31/6.30 1604 (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(V_f_2,V_z_2) = hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_z_2)) & (V_z_2 = V_x_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]. 6.31/6.30 1605 (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 -> V_t_2 != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))) # label(fact_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1606 (all V_a all T_a (class_Rings_Omult__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1607 (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]. 6.31/6.30 1608 (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]. 6.31/6.30 1609 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) # label(fact_min__0L) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1610 (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]. 6.31/6.30 1611 (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]. 6.31/6.30 1612 (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]. 6.31/6.30 1613 (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]. 6.31/6.30 1614 (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]. 6.31/6.30 1615 (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_OBit1(V_l_2))))) # label(fact_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1616 (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]. 6.31/6.30 1617 (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),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)))))))))) # label(fact_setsum_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1618 (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]. 6.31/6.30 1619 (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]. 6.31/6.30 1620 (all V_qr_2 all V_n_2 all V_m_2 (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m_2,V_n_2),V_qr_2)) <-> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)) & (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(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__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),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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))))) & 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)))) # label(fact_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1621 (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)) = 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))) # label(fact_rev__singleton__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1622 (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_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_A_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))))) # label(fact_INT__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1623 (all V_x_2 all T_c V_x_2 = c_Hilbert__Choice_OEps(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2))) # label(fact_some__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1624 (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]. 6.31/6.30 1625 (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]. 6.31/6.30 1626 (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]. 6.31/6.30 1627 (all V_B_2 all V_A_2 all T_d 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__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1628 (all V_r_2 all V_A_2 all T_c (V_A_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)) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) # label(fact_quotient__is__empty2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1629 (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]. 6.31/6.30 1630 (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]. 6.31/6.30 1631 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1632 (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_y_2 = V_x_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]. 6.31/6.30 1633 (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]. 6.31/6.30 1634 (all V_xs_2 all V_x_2 all V_j_2 all V_i_2 (c_List_Oupt(V_i_2,V_j_2) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_x_2),V_xs_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),V_j_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) = V_xs_2 & V_i_2 = V_x_2)) # label(fact_upt__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1635 (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_a_2 = V_b_2 & V_c_2 = V_b_2))) # label(fact_atLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1636 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls)),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)))))) # label(fact_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1637 (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]. 6.31/6.30 1638 (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]. 6.31/6.30 1639 (all V_l_2 all V_k_2 (-(hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2))) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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))) # label(fact_dvd_OgreaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1640 (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]. 6.31/6.30 1641 (all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_q = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1642 (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]. 6.31/6.30 1643 (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_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_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)))))) # label(fact_setsum__diff1__ring) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1644 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a))) # label(fact_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1645 (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_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)) = 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)))) # label(fact_Un__Union__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1646 (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]. 6.31/6.30 1647 (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]. 6.31/6.30 1648 (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]. 6.31/6.30 1649 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,V_n),V_xs) = 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))) # label(fact_rotate__drop__take) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1650 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__self) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1651 (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]. 6.31/6.30 1652 (all V_j_2 all V_i_2 (c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i_2,V_j_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))) # label(fact_upt__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1653 (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]. 6.31/6.30 1654 (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]. 6.31/6.30 1655 (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]. 6.31/6.30 1656 (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]. 6.31/6.30 1657 (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_I27_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1658 (all V_x all V_k all V_i all T_a c_List_Oreplicate(T_a,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_i),V_k),V_x) = hAPP(hAPP(c_List_Otake(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x))) # label(fact_take__replicate) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1659 (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]. 6.31/6.30 1660 (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]. 6.31/6.30 1661 (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]. 6.31/6.30 1662 (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]. 6.31/6.30 1663 (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_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_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))) # label(fact_vimage__Diff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1664 (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]. 6.31/6.30 1665 (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)) & (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(V_f_2,B_x) = 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(T_c,tc_Nat_Onat),V_f_2),V_A_2)))) # label(fact_setsum__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1666 (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 -> 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]. 6.31/6.30 1667 (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]. 6.31/6.30 1668 (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),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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_pos__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1669 (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]. 6.31/6.30 1670 (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]. 6.31/6.30 1671 (all V_xss_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oconcat(T_c,V_xss_2) <-> (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))) -> c_List_Olist_ONil(T_c) = B_x)))) # label(fact_concat__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1672 (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]. 6.31/6.30 1673 (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]. 6.31/6.30 1674 (all V_z all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)))) # label(fact_of__int__minus) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1675 (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(hAPP(c_Set_Oinsert(T_d),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2))) # label(fact_image__constant) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1676 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OatMost(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_OatLeast(T_c),V_x_2)))) # label(fact_image__uminus__atLeast) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1677 (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]. 6.31/6.30 1678 (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(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2))))))) # label(fact_folding__one__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1679 (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]. 6.31/6.30 1680 (all V_Pa_2 all V_a_2 all V_n_2 all T_c (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(V_Pa_2,V_a_2)) <-> (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)))))) # label(fact_Ball__set__replicate) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1681 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2)))))) # label(fact_of__int__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1682 (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]. 6.31/6.30 1683 (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]. 6.31/6.30 1684 (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]. 6.31/6.30 1685 (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(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) = hAPP(V_f_2,V_a_2))))) # label(fact_UN__equiv__class) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1686 (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]. 6.31/6.30 1687 (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]. 6.31/6.30 1688 (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]. 6.31/6.30 1689 (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_drop__butlast) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1690 (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]. 6.31/6.30 1691 (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_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)) = 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)) # label(fact_rel__comp__UNION__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1692 (all V_y_2 all V_x_2 (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)))) <-> 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)))) # label(fact_equiv__intrel__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1693 (all V_I_2 all V_xs_2 all T_c 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)))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Osublist(T_c,V_xs_2,V_I_2))) # label(fact_length__sublist) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1694 (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]. 6.31/6.30 1695 (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) = V_zs_2 <-> 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)) # label(fact_append__eq__conv__conj) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1696 (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]. 6.31/6.30 1697 (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]. 6.31/6.30 1698 (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]. 6.31/6.30 1699 (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]. 6.31/6.30 1700 (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]. 6.31/6.30 1701 (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__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2)) # label(fact_subset__Un__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1702 (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]. 6.31/6.30 1703 (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]. 6.31/6.30 1704 (all V_k all V_v ((V_k = V_v -> c_Random_Oinc__shift(V_v,V_k) = c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) & (V_k != V_v -> 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]. 6.31/6.30 1705 (all T_c all V_B_2 all 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)) & 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)) # label(fact_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1706 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))))) # label(fact_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1707 (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]. 6.31/6.30 1708 (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]. 6.31/6.30 1709 (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),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),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)))))) # label(fact_finite__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1710 (all V_z all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_inf__sup__aci_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1711 (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]. 6.31/6.30 1712 (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]. 6.31/6.30 1713 (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]. 6.31/6.30 1714 (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]. 6.31/6.30 1715 (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]. 6.31/6.30 1716 (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]. 6.31/6.30 1717 (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]. 6.31/6.30 1718 (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]. 6.31/6.30 1719 (all V_xs_2 all V_f_2 all T_c c_List_Opartition(T_c,V_f_2,V_xs_2) = 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))) # label(fact_partition__filter__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1720 (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]. 6.31/6.30 1721 (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_rx)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),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_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1722 (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]. 6.31/6.30 1723 (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]. 6.31/6.30 1724 (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__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1725 (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]. 6.31/6.30 1726 (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]. 6.31/6.30 1727 (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]. 6.31/6.30 1728 (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]. 6.31/6.30 1729 (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]. 6.31/6.30 1730 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)))) # label(fact_diff__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1731 (all V_seed_2 all V_size_2 all V_nrandom_2 all T_c c_New__DSequence_Opos__empty(T_c) = c_New__Random__Sequence_Opos__empty(T_c,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__empty__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1732 (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_Ofinite__psubset(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)))) # label(fact_in__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1733 (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]. 6.31/6.30 1734 (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]. 6.31/6.30 1735 (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]. 6.31/6.30 1736 (all V_x_2 all V_A_2 all V_y_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),V_A_2),V_x_2)) <-> V_x_2 = V_y_2 | hBOOL(hAPP(V_A_2,V_x_2)))) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1737 (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]. 6.31/6.30 1738 (all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1739 (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]. 6.31/6.30 1740 (all V_x_H all V_x all V_xs all T_a all V_i_H all V_i (V_i_H != V_i -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i_H),V_x_H) = hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i_H),V_x_H)),V_i),V_x))) # label(fact_list__update__swap) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1741 (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]. 6.31/6.30 1742 (all V_x all V_xs all T_a all V_k ((hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_k -> c_List_Obutlast(T_a,V_xs) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x))) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) != V_k -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = hAPP(hAPP(c_List_Olist__update(T_a,c_List_Obutlast(T_a,V_xs)),V_k),V_x)))) # label(fact_butlast__list__update) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1743 (all V_v_H_2 all V_v_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_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_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))))) # label(fact_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1744 (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]. 6.31/6.30 1745 (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]. 6.31/6.30 1746 (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]. 6.31/6.30 1747 (all V_Qa_2 all V_Pa_2 all T_d 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_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))) = 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))) # label(fact_SetCompr__Sigma__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1748 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OMin) = c_Int_Opred(V_k)) # label(fact_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1749 (all T_c c_List_Olists(T_c,c_Orderings_Obot__class_Obot(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_lists__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1750 (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]. 6.31/6.30 1751 (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]. 6.31/6.30 1752 (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]. 6.31/6.30 1753 (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))) <-> (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))))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1754 (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]. 6.31/6.30 1755 (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]. 6.31/6.30 1756 (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(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),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_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]. 6.31/6.30 1757 (all V_xs_2 all V_x_2 all T_c c_Enum_Osublists(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = 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)))))) # label(fact_sublists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1758 (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]. 6.31/6.30 1759 (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]. 6.31/6.30 1760 (all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_prod(T_a,T_b))),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))) # label(fact_length__zip) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1761 (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]. 6.31/6.30 1762 (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]. 6.31/6.30 1763 (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]. 6.31/6.30 1764 (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]. 6.31/6.30 1765 (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]. 6.31/6.30 1766 (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]. 6.31/6.30 1767 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1768 (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))) -> V_c_2 != hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_y))))))) # label(fact_SigmaE) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1769 (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]. 6.31/6.30 1770 (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]. 6.31/6.30 1771 (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_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) = 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))) # label(fact_foldl__map) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1772 (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]. 6.31/6.30 1773 (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]. 6.31/6.30 1774 (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_a_2 = V_b_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 6.31/6.30 1775 (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]. 6.31/6.30 1776 (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_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),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),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),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]. 6.31/6.31 1777 (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]. 6.31/6.31 1778 (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,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> c_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))))) # label(fact_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1779 (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]. 6.31/6.31 1780 (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]. 6.31/6.31 1781 (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]. 6.31/6.31 1782 (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]. 6.31/6.31 1783 (all V_a_2 all V_xs_2 all V_x_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_Olist_OCons(T_d),V_x_2),V_xs_2),V_a_2) = hAPP(hAPP(V_f_2,V_x_2),c_List_Ofoldr(T_d,T_c,V_f_2,V_xs_2,V_a_2))) # label(fact_foldr_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1784 (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]. 6.31/6.31 1785 (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]. 6.31/6.31 1786 (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]. 6.31/6.31 1787 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c)) # label(fact_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1788 (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]. 6.31/6.31 1789 (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_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)) = 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]. 6.31/6.31 1790 (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 -> 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)) & (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)))) # label(fact_anamorph_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1791 (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(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)) # label(fact_insert__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1792 (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]. 6.31/6.31 1793 (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]. 6.31/6.31 1794 (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]. 6.31/6.31 1795 (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]. 6.31/6.31 1796 (all V_n_2 hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_lessThan__Suc__eq__insert__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1797 (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]. 6.31/6.31 1798 (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(c_SetInterval_Oord__class_OatLeastLessThan(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_OgreaterThanLessThan(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I18_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1799 (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]. 6.31/6.31 1800 (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]. 6.31/6.31 1801 (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]. 6.31/6.31 1802 (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)),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_length__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1803 (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)) -> 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__absorb1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1804 (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]. 6.31/6.31 1805 (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]. 6.31/6.31 1806 (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]. 6.31/6.31 1807 (all V_n_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_n_2))) <-> 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_x_2)))) # label(fact_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1808 (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]. 6.31/6.31 1809 (all V_g_2 all V_f_2 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,V_f_2,V_g_2) = 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))) # label(fact_map__pair__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1810 (all V_s_2 all V_r_2 all T_c all T_d 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)) = 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))) # label(fact_converse__Int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1811 (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_d != V_c & V_a = V_b -> 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]. 6.31/6.31 1812 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Nitpick_Onat__gcd(hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x)),hAPP(c_Int_Onat,hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_y)))) = hAPP(hAPP(c_Nitpick_Oint__gcd,V_x),V_y)) # label(fact_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1813 (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]. 6.31/6.31 1814 (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),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_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2))))) # label(fact_fun__left__comm_Ofold__rec) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1815 (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]. 6.31/6.31 1816 (all V_r_2 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_c),V_r_2)),hAPP(c_Relation_ORange(T_c,T_c),V_r_2)) = hAPP(c_Relation_OField(T_c),V_r_2)) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1817 (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_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) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2))) # label(fact_bij__vimage__eq__inv__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1818 (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]. 6.31/6.31 1819 (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_m_2 = V_n_2))) # label(fact_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1820 (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]. 6.31/6.31 1821 (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(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]. 6.31/6.31 1822 (all V_xs_2 all V_x_2 all T_c hAPP(c_List_Ofilter(T_c,hAPP(c_fequal,V_x_2)),V_xs_2) = 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)) # label(fact_replicate__length__filter) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1823 (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]. 6.31/6.31 1824 (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]. 6.31/6.31 1825 (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]. 6.31/6.31 1826 (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]. 6.31/6.31 1827 (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),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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2))))) # label(fact_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1828 (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_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Otake(T_b),V_n),V_ys)) = 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))) # label(fact_take__zip) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1829 (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_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)) = 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))) # label(fact_apsnd__apfst__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1830 (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]. 6.31/6.31 1831 (all V_n_2 all V_m_2 ((exists B_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k) = V_n_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1832 (all V_r all V_q all V_b all V_a (V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) -> ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),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]. 6.31/6.31 1833 (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]. 6.31/6.31 1834 (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) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__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]. 6.31/6.31 1835 (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]. 6.31/6.31 1836 (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]. 6.31/6.31 1837 (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))) -> 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_Opsimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1838 (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(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__idem_Osubset__comp__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1839 (all V_A_2 all T_c V_A_2 = 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)))) # label(fact_Diff__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1840 (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),hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_g_2),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)))))) # label(fact_setsum_Oreindex) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1841 (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(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))) # label(fact_inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1842 (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]. 6.31/6.31 1843 (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]. 6.31/6.31 1844 (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]. 6.31/6.31 1845 (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]. 6.31/6.31 1846 (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]. 6.31/6.31 1847 (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]. 6.31/6.31 1848 (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]. 6.31/6.31 1849 (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]. 6.31/6.31 1850 (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]. 6.31/6.31 1851 (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]. 6.31/6.31 1852 (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]. 6.31/6.31 1853 (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]. 6.31/6.31 1854 (all V_i_2 all V_xq_2 hAPP(c_New__DSequence_Opos__not__seq(V_xq_2),V_i_2) = c_Lazy__Sequence_Ohb__not__seq(hAPP(V_xq_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_i_2)))) # label(fact_New__DSequence_Opos__not__seq__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1855 (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]. 6.31/6.31 1856 (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]. 6.31/6.31 1857 (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]. 6.31/6.31 1858 (all V_A_2 all T_d all T_c 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))) = 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)))) # label(fact_Compl__Times__UNIV1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1859 (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]. 6.31/6.31 1860 (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 (hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),B_x),B_y) = V_p_2 -> -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]. 6.31/6.31 1861 (all V_w_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w_2)))))) # label(fact_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1862 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_divide__zero__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1863 (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]. 6.31/6.31 1864 (all V_w all V_z hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1865 (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]. 6.31/6.31 1866 (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]. 6.31/6.31 1867 (all V_v_2 all V_n_2 (hAPP(c_Nat_OSuc,V_n_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_fequal),c_Int_Onat)),V_n_2))))))) # label(fact_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1868 (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]. 6.31/6.31 1869 (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]. 6.31/6.31 1870 (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)))) -> 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__key__triv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1871 (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]. 6.31/6.31 1872 (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]. 6.31/6.31 1873 (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]. 6.31/6.31 1874 (all V_B_2 all V_f_2 all T_d all T_c 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)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)) # label(fact_vimage__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1875 (all V_m V_m = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1876 (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]. 6.31/6.31 1877 (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]. 6.31/6.31 1878 (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]. 6.31/6.31 1879 (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]. 6.31/6.31 1880 (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]. 6.31/6.31 1881 (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_a_2 != V_b_2 -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2))))) # label(fact_insertE) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1882 (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]. 6.31/6.31 1883 (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 (hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_A) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a1_2),V_a2_2) & -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)))) <-> 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]. 6.31/6.31 1884 (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),V_a),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_Otimes__class_Otimes(T_a),V_a),V_b)),V_c))) # label(fact_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1885 (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]. 6.31/6.31 1886 (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_Product__Type_Osnd(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)))) # label(fact_pick_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1887 (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]. 6.31/6.31 1888 (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]. 6.31/6.31 1889 (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]. 6.31/6.31 1890 (all T_c 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))) = hAPP(c_Set_OPow(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Pow__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1891 (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))) & (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)) = V_x))) # label(fact_last_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1892 (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]. 6.31/6.31 1893 (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]. 6.31/6.31 1894 (all V_a_2 all V_b_2 (-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)) -> 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__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1895 (all V_b_2 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(hAPP(c_Set_Oinsert(T_c),V_b_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]. 6.31/6.31 1896 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a))) # label(fact_left__minus) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1897 (all V_b all V_a all T_b all T_a c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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))) # label(fact_pair__box_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1898 (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) = 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))))) # label(fact_image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1899 (all V_n all T_a (class_Groups_Omonoid__mult(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),V_n))) # label(fact_power__one) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1900 (all V_f_2 all V_n_2 all V_m_2 all T_c 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)) = 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)) # label(fact_funpow__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1901 (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]. 6.31/6.31 1902 (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]. 6.31/6.31 1903 (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]. 6.31/6.31 1904 (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]. 6.31/6.31 1905 (all V_xs all V_x all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = 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))) & (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))))) # label(fact_take__Cons_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1906 (all V_f_2 all V_n_2 all T_d all T_c c_Fun_Ocomp(T_d,T_d,T_c,hAPP(hAPP(c_Nat_Ocompow(T_d,T_d),V_n_2),V_f_2)) = 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))) # label(fact_comp__funpow) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1907 (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__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_R_2),V_S_2))) <-> 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)))) # label(fact_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1908 (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]. 6.31/6.31 1909 (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]. 6.31/6.31 1910 (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_X_2 = V_Y_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_X_2),V_Y_2))))) # label(fact_quotient__disj) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1911 (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]. 6.31/6.31 1912 (all V_xs_2 all T_d all T_c (class_Groups_Omonoid__add(T_c) -> c_Groups_Ozero__class_Ozero(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)))) # label(fact_listsum__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1913 (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]. 6.31/6.31 1914 (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_m_2 = V_n_2)) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1915 (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))) -> 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)))))) # label(fact_finite__UNIV__inj__surj) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1916 (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]. 6.31/6.31 1917 (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]. 6.31/6.31 1918 (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_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)) <-> 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))))) # label(fact_in__inv__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1919 (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]. 6.31/6.31 1920 (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_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))) = 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))))) # label(fact_setsum__Un__nat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1921 (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]. 6.31/6.31 1922 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_minus__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1923 (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]. 6.31/6.31 1924 (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]. 6.31/6.31 1925 (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]. 6.31/6.31 1926 (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(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))) <-> 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)))) # label(fact_sup__Un__eq2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1927 (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]. 6.31/6.31 1928 (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]. 6.31/6.31 1929 (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]. 6.31/6.31 1930 (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_Y_2 = V_X_2))))))) # label(fact_quotient__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1931 (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]. 6.31/6.31 1932 (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]. 6.31/6.31 1933 (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)),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)))) # label(fact_ivl__disj__int_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1934 (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),V_A_2)) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1935 (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]. 6.31/6.31 1936 (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_OMax(T_c,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)))))) # label(fact_minus__Max__eq__Min) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1937 (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]. 6.31/6.31 1938 (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]. 6.31/6.31 1939 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_mult__minus1__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1940 (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]. 6.31/6.31 1941 (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]. 6.31/6.31 1942 (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_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = V_A_2) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1943 (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]. 6.31/6.31 1944 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (V_a_H_2 = V_a_2 & V_b_H_2 = V_b_2 <-> 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))) # label(fact_Pair__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1945 (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]. 6.31/6.31 1946 (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_Olist__update(T_a,V_xs),V_i),V_a) = 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))))) # label(fact_upd__conv__take__nth__drop) # label(axiom) # label(non_clause). [assumption]. 6.31/6.31 1947 (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]. 6.31/6.31 1948 (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]. 6.31/6.31 1949 (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]. 6.31/6.31 1950 (all V_Pa_2 all V_n_2 ((exists B_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_m),V_n_2)) & hBOOL(hAPP(V_Pa_2,B_m)))) <-> (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]. 6.31/6.32 1951 (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) & (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_comm__monoid__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1952 (all 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))) = hAPP(c_Nat_Osize__class_Osize(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_nat))) # label(fact_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1953 (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]. 6.31/6.32 1954 (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]. 6.31/6.32 1955 (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]. 6.31/6.32 1956 (all V_R_2 all T_d all T_c hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_c,c_Product__Type_Ofst(T_c,T_d)),V_R_2) = hAPP(c_Relation_ODomain(T_c,T_d),V_R_2)) # label(fact_fst__eq__Domain) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1957 (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]. 6.31/6.32 1958 (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]. 6.31/6.32 1959 (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]. 6.31/6.32 1960 (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_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))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_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)))) # label(fact_UN__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1961 (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]. 6.31/6.32 1962 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Oone__class_Oone(T_a)))) # label(fact_divide__self) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1963 (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]. 6.31/6.32 1964 (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]. 6.31/6.32 1965 (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__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)) = c_New__Random__Sequence_Opos__union(T_c,V_R1_2,V_R2_2,V_nrandom_2,V_size_2,V_seed_2)) # label(fact_pos__union__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1966 (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]. 6.31/6.32 1967 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)) # label(fact_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1968 (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_y_2 & c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> 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]. 6.31/6.32 1969 (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),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]. 6.31/6.32 1970 (all V_n all V_b all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))) # label(fact_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1971 (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]. 6.31/6.32 1972 (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(hAPP(c_member(T_c),B_x),c_Quotient_ORespects(T_c,V_R_2))) & hBOOL(hAPP(V_Pa_2,B_x))))))) # label(fact_bex__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1973 (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]. 6.31/6.32 1974 (all V_a_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,V_xs_2,V_a_2) = 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))) # label(fact_foldr__foldl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1975 (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]. 6.31/6.32 1976 (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]. 6.31/6.32 1977 (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]. 6.31/6.32 1978 (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]. 6.31/6.32 1979 (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]. 6.31/6.32 1980 (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]. 6.31/6.32 1981 (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 (hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_ys) = 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_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]. 6.31/6.32 1982 (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(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_total__on__diff__Id) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1983 (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]. 6.31/6.32 1984 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2) <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2) = V_n_2))) # label(fact_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1985 (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]. 6.31/6.32 1986 (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]. 6.31/6.32 1987 (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]. 6.31/6.32 1988 (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]. 6.31/6.32 1989 (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(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(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)) & (-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))))) # label(fact_setprod__delta) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1990 (all V_zs all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),hAPP(hAPP(c_List_Oappend(T_b),V_ys),V_zs)) = 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))) # label(fact_zip__append2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1991 (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]. 6.31/6.32 1992 (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]. 6.31/6.32 1993 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1994 (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]. 6.31/6.32 1995 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(T_c))) # label(fact_Inf__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1996 (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]. 6.31/6.32 1997 (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_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) = 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))) # label(fact_setsum__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 1998 (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]. 6.31/6.32 1999 (all V_i_2 all V_f_2 all V_x_2 all T_c all T_d 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)) = hAPP(c_New__DSequence_Opos__bind(T_d,T_c,V_x_2,V_f_2),V_i_2)) # label(fact_New__DSequence_Opos__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2000 (all V_v ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v),c_Int_OPls)) -> 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_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2001 (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]. 6.31/6.32 2002 (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]. 6.31/6.32 2003 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2004 (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]. 6.31/6.32 2005 (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_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)),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_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]. 6.31/6.32 2006 (all V_ys all V_y all V_xs all V_x all T_b all T_a 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)) = 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))) # label(fact_zip__Cons__Cons) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2007 (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),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_a_2))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2008 (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_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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_less) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2009 (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_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]. 6.31/6.32 2010 (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]. 6.31/6.32 2011 (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)) -> 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)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Collect__conv__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2012 (all V_A_2 all T_d all T_c 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))))) = 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))))) # label(fact_Compl__Times__UNIV2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2013 (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]. 6.31/6.32 2014 (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]. 6.31/6.32 2015 (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]. 6.31/6.32 2016 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(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)) = hAPP(c_Finite__Set_Ocard(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)))) # label(fact_card__Pow) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2017 (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]. 6.31/6.32 2018 (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]. 6.31/6.32 2019 (all V_xs_2 all V_x_2 all T_c ((exists B_i (V_x_2 = hAPP(hAPP(c_List_Onth(T_c),V_xs_2),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(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]. 6.31/6.32 2020 (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_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(V_B_2,B_x))) <-> 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)))) # label(fact_INTER__UNIV__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2021 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2022 (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]. 6.31/6.32 2023 (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]. 6.31/6.32 2024 (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]. 6.31/6.32 2025 (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]. 6.31/6.32 2026 (all V_x all V_y all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_y))) # label(fact_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2027 (all V_b_2 all V_a_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_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OatLeastLessThan__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2028 (all V_A_2 all T_d all V_I_2 all T_c ((exists B_x (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_A_2,B_x))) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)))) -> 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]. 6.31/6.32 2029 (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]. 6.31/6.32 2030 (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]. 6.31/6.32 2031 (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)))))) <-> V_a_2 != c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2032 (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]. 6.31/6.32 2033 (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(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_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__conv__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2034 (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]. 6.31/6.32 2035 (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]. 6.31/6.32 2036 (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]. 6.31/6.32 2037 (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]. 6.31/6.32 2038 (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_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_Un__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2039 (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)) -> (V_n_2 = V_m_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))))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2040 (all V_m_2 all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> 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]. 6.31/6.32 2041 (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]. 6.31/6.32 2042 (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]. 6.31/6.32 2043 (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]. 6.31/6.32 2044 (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]. 6.31/6.32 2045 (all V_f_2 all V_A_2 all T_c all T_d (class_Rings_Ono__zero__divisors(T_d) & class_Rings_Ocomm__semiring__1(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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)))) <-> 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__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2046 (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_A_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_A_2)),V_g_2))))) # label(fact_folding_Ocommute__left__comp_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2047 (all 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)))) = c_Wellfounded_Ofinite__psubset(T_c)) # label(fact_finite__psubset__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2048 (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]. 6.31/6.32 2049 (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),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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2050 (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]. 6.31/6.32 2051 (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]. 6.31/6.32 2052 (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_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_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__set__diff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2053 (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)) & (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__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2054 (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]. 6.31/6.32 2055 (all V_Pa_2 all T_c c_List_OdropWhile(T_c,V_Pa_2,c_List_Olist_ONil(T_c)) = c_List_Olist_ONil(T_c)) # label(fact_dropWhile_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2056 (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]. 6.31/6.32 2057 (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]. 6.31/6.32 2058 (all V_xs_2 all T_c c_List_Osublist(T_c,V_xs_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = c_List_Olist_ONil(T_c)) # label(fact_sublist__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2059 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) <-> c_List_Olist__all(T_c,V_Pa_2,V_ys_2) & c_List_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_list__all__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2060 (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]. 6.31/6.32 2061 (all V_r_2 all T_c 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))) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_rtrancl__r__diff__Id) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2062 (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]. 6.31/6.32 2063 (all V_B_2 all V_A_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(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)) = 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)) # label(fact_UN__insert) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2064 (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]. 6.31/6.32 2065 (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))) -> hAPP(c_List_Orotate(T_a,V_n),V_xs) = V_xs)) # label(fact_rotate__length01) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2066 (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),V_x))) # label(fact_Sup__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2067 (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)))))) & (-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(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__Diff__singleton__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2068 (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),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (all B_i ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j))) -> hBOOL(hAPP(V_Pa_2,B_i))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (all B_i ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(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]. 6.31/6.32 2069 (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) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2070 (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]. 6.31/6.32 2071 (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]. 6.31/6.32 2072 (all V_ys_2 all V_xs_2 all V_Pa_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_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)))) & 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) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2))) # label(fact_list__all2__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2073 (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]. 6.31/6.32 2074 (all V_A_2 V_A_2 = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) # label(fact_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2075 (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]. 6.31/6.32 2076 (all T_c c_Equiv__Relations_Oequivp(T_c,c_fequal)) # label(fact_identity__equivp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2077 (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]. 6.31/6.32 2078 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> 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]. 6.31/6.32 2079 (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_y_2),V_x_2))))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2080 (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]. 6.31/6.32 2081 (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]. 6.31/6.32 2082 (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]. 6.31/6.32 2083 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)),V_z_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2084 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2085 (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]. 6.31/6.32 2086 (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]. 6.31/6.32 2087 (all V_a_2 all T_c V_a_2 = c_HOL_OThe(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))) # label(fact_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2088 (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]. 6.31/6.32 2089 (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]. 6.31/6.32 2090 (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]. 6.31/6.32 2091 (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]. 6.31/6.32 2092 (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]. 6.31/6.32 2093 (all V_com2_H all V_com1_H c_Com_Ocom_OSKIP != c_Com_Ocom_OSemi(V_com1_H,V_com2_H)) # label(fact_com_Osimps_I12_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2094 (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_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)) = 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))) # label(fact_setsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2095 (all V_x_2 all T_c all V_n_2 (V_n_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2096 (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]. 6.31/6.32 2097 (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))) <-> (all 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(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)))))) # label(fact_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2098 (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]. 6.31/6.32 2099 (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]. 6.31/6.32 2100 (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]. 6.31/6.32 2101 (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]. 6.31/6.32 2102 (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]. 6.31/6.32 2103 (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]. 6.31/6.32 2104 (all V_yq all T_a V_yq = c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a),V_yq)) # label(fact__02) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2105 (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_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_y_2)) = 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)) <-> V_y_2 = V_z_2 | V_x_2 = V_w_2))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2106 (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]. 6.31/6.32 2107 (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]. 6.31/6.32 2108 (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]. 6.31/6.32 2109 (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]. 6.31/6.32 2110 (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]. 6.31/6.32 2111 (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]. 6.31/6.32 2112 (all V_j all V_i hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),c_List_Oupt(V_i,V_j)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i)) # label(fact_length__upt) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2113 (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]. 6.31/6.32 2114 (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]. 6.31/6.32 2115 (all V_n_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) # label(fact_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2116 (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) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2117 (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]. 6.31/6.32 2118 (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]. 6.31/6.32 2119 (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))) & hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2)),V_y_2) = V_x_2)) # label(fact_lexord__take__index__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2120 (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]. 6.31/6.32 2121 (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]. 6.31/6.32 2122 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2123 (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]. 6.31/6.32 2124 (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]. 6.31/6.32 2125 (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]. 6.31/6.32 2126 (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]. 6.31/6.32 2127 (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_y),V_x))) # label(fact_min__max_Osup__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2128 (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),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_less__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2129 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2130 (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]. 6.31/6.32 2131 (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_b_2 = V_a_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]. 6.31/6.32 2132 (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)) -> (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),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_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_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))))))))))))) # label(fact_sup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.32 2133 (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]. 6.31/6.32 2134 (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),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_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)))) # label(fact_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2135 (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]. 6.31/6.33 2136 (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]. 6.31/6.33 2137 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))) # label(fact_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2138 (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]. 6.31/6.33 2139 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_divide__zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2140 (all V_Qa_2 all V_Pa_2 (-hBOOL(V_Pa_2) | -hBOOL(V_Qa_2) | hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)))) # label(help_c__fconj__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2141 (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]. 6.31/6.33 2142 (all V_k c_Int_OPls != c_Int_OBit1(V_k)) # label(fact_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2143 (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]. 6.31/6.33 2144 (all V_R_2 all V_n_2 all V_m_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_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)) = 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)) # label(fact_rel__pow__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2145 (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_x_2 = 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)))) # label(fact_the__inv__f__f) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2146 (all V_Pa_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_OEx(T_d)),hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Pa_2))) = 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)))) # label(fact_Range__Collect__split) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2147 (all V_r_H all V_q_H all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> V_q = V_q_H)))) # label(fact_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2148 (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_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2149 (all V_R_2 all T_c c_Transitive__Closure_Otrancl(T_c,c_Transitive__Closure_Ortrancl(T_c,V_R_2)) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)) # label(fact_rtrancl__trancl__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2150 (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]. 6.31/6.33 2151 (all V_R_2 all 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) = c_Relation_OId(T_c)) # label(fact_relpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2152 (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]. 6.31/6.33 2153 (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]. 6.31/6.33 2154 (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]. 6.31/6.33 2155 (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]. 6.31/6.33 2156 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> V_x = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)))) # label(fact_double__compl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2157 (all V_r_2 all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)) = hAPP(c_Relation_ORange(T_d,T_c),V_r_2)) # label(fact_Domain__converse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2158 (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]. 6.31/6.33 2159 (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]. 6.31/6.33 2160 (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 (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) & hAPP(V_f_2,B_x) = V_z_2)))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2161 (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]. 6.31/6.33 2162 (all T_a (class_Rings_Osemiring__1(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)))) = c_Groups_Oone__class_Oone(T_a))) # label(fact_one__power2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2163 (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]. 6.31/6.33 2164 (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]. 6.31/6.33 2165 (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]. 6.31/6.33 2166 (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]. 6.31/6.33 2167 (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]. 6.31/6.33 2168 (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]. 6.31/6.33 2169 (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_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)) = c_Divides_Odivmod__nat(V_m,V_n)))) # label(fact_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2170 (all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> V_a = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_a))) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2171 (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),c_Orderings_Otop__class_Otop(T_a)),V_x))) # label(fact_sup__top__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2172 (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]. 6.31/6.33 2173 (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 (hAPP(c_Nat_OSuc,B_m) = V_n_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)),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]. 6.31/6.33 2174 (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),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(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_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2175 (all V_ys all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) -> 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]. 6.31/6.33 2176 (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__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))) = 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__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2177 (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 (hAPP(V_f_2,B_x) = c_Groups_Ozero__class_Ozero(T_d) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_d))))) # label(fact_setprod__zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2178 (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]. 6.31/6.33 2179 (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]. 6.31/6.33 2180 (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]. 6.31/6.33 2181 (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]. 6.31/6.33 2182 (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]. 6.31/6.33 2183 (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]. 6.31/6.33 2184 (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]. 6.31/6.33 2185 (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]. 6.31/6.33 2186 (all V_B_2 all T_c 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) = V_B_2) # label(fact_Int__UNIV__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2187 (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]. 6.31/6.33 2188 (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)) -> ((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]. 6.31/6.33 2189 (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]. 6.31/6.33 2190 (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]. 6.31/6.33 2191 (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_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_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))) # label(fact_UN__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2192 (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) -> (hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2) = 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 & hAPP(hAPP(c_List_Omap(tc_prod(T_c,T_d),T_c),c_Product__Type_Ofst(T_c,T_d)),V_zs_2) = V_xs_2))) # label(fact_zip__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2193 (all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,V_R_2) <-> c_Predicate_Otransp(T_c,V_R_2) & c_Predicate_Osymp(T_c,V_R_2) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))))) # label(fact_part__equivp__refl__symp__transp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2194 (all V_B_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_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))))) # label(fact_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2195 (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]. 6.31/6.33 2196 (all V_x_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2) = c_Groups_Oone__class_Oone(T_c) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c)) = V_x_2 | c_Groups_Oone__class_Oone(T_c) = V_x_2))) # label(fact_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2197 (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]. 6.31/6.33 2198 (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]. 6.31/6.33 2199 (all V_x_2 all T_c V_x_2 = c_HOL_OThe(T_c,hAPP(c_fequal,V_x_2))) # label(fact_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2200 (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]. 6.31/6.33 2201 (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]. 6.31/6.33 2202 (all V_B_2 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)),V_A_2),V_B_2) -> V_A_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2203 (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]. 6.31/6.33 2204 (all V_y all V_x hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y))) # label(fact_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2205 (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]. 6.31/6.33 2206 (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]. 6.31/6.33 2207 (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]. 6.31/6.33 2208 (all V_r_2 all T_c 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),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_r_2)) # label(fact_quotient__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2209 (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]. 6.31/6.33 2210 (all V_xs all V_n all T_a 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)) = hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rotate__rev) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2211 (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]. 6.31/6.33 2212 (all V_n_2 hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2) = hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2213 (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_ys_2) = hAPP(c_List_Oset(T_c),V_xs_2) -> V_ys_2 = V_xs_2)))))))) # label(fact_map__sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2214 (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]. 6.31/6.33 2215 (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),V_B_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)),V_B_2))) # label(fact_Diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2216 (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]. 6.31/6.33 2217 (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]. 6.31/6.33 2218 (all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oorder(T_d) & class_Enum_Oenum(T_c) -> (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]. 6.31/6.33 2219 (all V_B_2 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)),V_A_2),V_B_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_B_2)) -> B_xa != B_x)))))) # label(fact_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2220 (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_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)) = 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))) # label(fact_setsum__product) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2221 (all V_k all V_a all V_b 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),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)) # label(fact_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2222 (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)) & (-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_OF__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2223 (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) -> 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]. 6.31/6.33 2224 (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]. 6.31/6.33 2225 (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]. 6.31/6.33 2226 (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]. 6.31/6.33 2227 (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]. 6.31/6.33 2228 (all V_j_2 all V_i_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_i_2,V_j_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_i_2,V_j_2))))) # label(fact_setsum__set__upto__conv__listsum__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2229 (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]. 6.31/6.33 2230 (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]. 6.31/6.33 2231 (all V_R_2 all T_c 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) = V_R_2) # label(fact_rel__pow__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2232 (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))) -> ((all B_i all B_j (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)) -> hBOOL(hAPP(hAPP(V_Pa_2,B_i),B_j)))) <-> 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)))))) # label(fact_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2233 (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)) = 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)))) & (-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_Olist_ONil(tc_Nat_Onat) = 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]. 6.31/6.33 2234 (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))) & (all B_x all B_y (hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)) <-> hBOOL(hAPP(hAPP(V_R_2,B_y),B_y)) & hAPP(V_R_2,B_x) = hAPP(V_R_2,B_y) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_x)))))) # label(fact_part__equivp__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2235 (all T_c all V_Pa_2 (hBOOL(hAPP(V_Pa_2,c_Hilbert__Choice_OEps(T_c,V_Pa_2))) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))) # label(fact_some__eq__ex) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2236 (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]. 6.31/6.33 2237 (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_OINFI(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_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))) # label(fact_converse__INTER) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2238 (all V_f_2 all T_c all T_d c_List_Olist_ONil(T_c) = c_List_Omaps(T_d,T_c,V_f_2,c_List_Olist_ONil(T_d))) # label(fact_maps__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2239 (all V_y_2 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(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]. 6.31/6.33 2240 (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]. 6.31/6.33 2241 (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)) -> (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__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2242 (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__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)) = 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)) # label(fact_neg__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2243 (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),V_a_2),V_b_2)) <-> 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)))))) # label(fact_neg__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2244 (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]. 6.31/6.33 2245 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (V_a_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) -> (V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_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]. 6.31/6.33 2246 (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]. 6.31/6.33 2247 (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]. 6.31/6.33 2248 (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]. 6.31/6.33 2249 (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]. 6.31/6.33 2250 (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_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y) = 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)))) # label(fact_divide__diff__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2251 (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]. 6.31/6.33 2252 (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]. 6.31/6.33 2253 (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]. 6.31/6.33 2254 (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) -> c_Groups_Ozero__class_Ozero(T_a) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))) # label(fact_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2255 (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]. 6.31/6.33 2256 (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]. 6.31/6.33 2257 (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]. 6.31/6.33 2258 (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]. 6.31/6.33 2259 (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))) -> 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__betw__imp__surj) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2260 (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]. 6.31/6.33 2261 (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(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) = 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))))) # label(fact_the__inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2262 (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]. 6.31/6.33 2263 (all V_m_2 c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2264 (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]. 6.31/6.33 2265 (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]. 6.31/6.33 2266 (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(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)) = 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)))) # label(fact_card__SigmaI) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2267 (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) -> 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))) & (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),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(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]. 6.31/6.33 2268 (all V_d_2 all V_m_2 ((exists B_q V_m_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_d_2),B_q)) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,V_d_2))) # label(fact_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2269 (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]. 6.31/6.33 2270 (all T_c all V_as_2 (c_List_Olist_ONil(T_c) != V_as_2 -> 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]. 6.31/6.33 2271 (all V_x_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)) <-> hBOOL(hAPP(c_Quotient_ORespects(T_c,V_R_2),V_x_2)))) # label(fact_Respects__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2272 (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]. 6.31/6.33 2273 (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]. 6.31/6.33 2274 (all V_lazy__sequence_2 all V_a_2 all V_fa_2 all T_c 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)) = 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)))) # label(fact_lazy__sequence_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2275 (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]. 6.31/6.33 2276 (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]. 6.31/6.33 2277 (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]. 6.31/6.33 2278 (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]. 6.31/6.33 2279 (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]. 6.31/6.33 2280 (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]. 6.31/6.33 2281 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2282 (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]. 6.31/6.33 2283 (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]. 6.31/6.33 2284 (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__inf__class_Oinf(T_c),V_B_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)),V_B_2),V_A_2)))) # label(fact_inf__Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2285 (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]. 6.31/6.33 2286 (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]. 6.31/6.33 2287 (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]. 6.31/6.33 2288 (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),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_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2)))))) # label(fact_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2289 (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]. 6.31/6.33 2290 (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]. 6.31/6.33 2291 (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]. 6.31/6.33 2292 (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]. 6.31/6.33 2293 (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]. 6.31/6.33 2294 (all V_Pa_2 all T_c all T_d (class_Enum_Oenum(T_c) & class_Enum_Oenum(T_d) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(tc_prod(T_c,T_d)),V_Pa_2)) <-> 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)))))))) # label(fact_enum__all__prod__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2295 (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]. 6.31/6.33 2296 (all T_c all V_a_2 all V_Pa_2 ((-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(c_fequal,V_a_2))),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)))))) # label(fact_Collect__conv__if2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2297 (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]. 6.31/6.33 2298 (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]. 6.31/6.33 2299 (all V_z all V_w 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))) = 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)))) # label(fact_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2300 (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(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),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2)))) # label(fact_fun__left__comm_Ofold__fun__comm) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2301 (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)) -> hAPP(V_B_2,B_x) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_INTER__UNIV__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2302 (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_A_2 = V_B_2 -> (c_Nat__Transfer_Onat__set(V_B_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hAPP(V_g_2,B_x) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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]. 6.31/6.33 2303 (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]. 6.31/6.33 2304 (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_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]. 6.31/6.33 2305 (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]. 6.31/6.33 2306 (all V_xs_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (V_xs_2 != c_List_Olist_ONil(T_c) -> 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]. 6.31/6.33 2307 (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]. 6.31/6.33 2308 (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__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_Set_Oinsert(T_c),V_a_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2))) # label(fact_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2309 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2310 (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]. 6.31/6.33 2311 (all V_ys all V_xs all T_a all V_n ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n) = hAPP(hAPP(c_List_Onth(T_a),V_ys),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n)))) # label(fact_nth__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2312 (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]. 6.31/6.33 2313 (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]. 6.31/6.33 2314 (all V_S_2 all V_R_2 all T_c (V_S_2 = 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) = hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2))) # label(fact_pred__equals__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2315 (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]. 6.31/6.33 2316 (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]. 6.31/6.33 2317 (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]. 6.31/6.33 2318 (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]. 6.31/6.33 2319 (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]. 6.31/6.33 2320 (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(c_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2)))) # label(fact_ivl__disj__un_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2321 (all V_x all V_z all V_y all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (V_y != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),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)))) # label(fact_add__num__frac) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2322 (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]. 6.31/6.33 2323 (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]. 6.31/6.33 2324 (all V_nat_H_2 all V_nat_2 (V_nat_H_2 = V_nat_2 <-> hAPP(c_Nat_OSuc,V_nat_H_2) = hAPP(c_Nat_OSuc,V_nat_2))) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2325 (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]. 6.31/6.33 2326 (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]. 6.31/6.33 2327 (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]. 6.31/6.33 2328 (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__sup__class_Osup(T_c)),V_A_2) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))) # label(fact_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2329 (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]. 6.31/6.33 2330 (all V_z all V_y all V_x 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)) = 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))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2331 (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)) <-> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2) = V_y_2))) # label(fact_min__max_Ole__iff__sup) # label(axiom) # label(non_clause). [assumption]. 6.31/6.33 2332 (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]. 6.31/6.33 2333 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c 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)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_filter__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2334 (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_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]. 6.31/6.34 2335 (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]. 6.31/6.34 2336 (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]. 6.31/6.34 2337 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2338 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_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]. 6.31/6.34 2339 (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)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_c_2)) = 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))) # label(fact_Int__atLeastAtMostL2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2340 (all V_A_2 all V_f_2 all T_c c_HOL_OThe(T_c,c_Finite__Set_Ofold1Set(T_c,V_f_2,V_A_2)) = hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2)) # label(fact_fold1__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2341 (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]. 6.31/6.34 2342 (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]. 6.31/6.34 2343 (all V_k c_Code__Numeral_OSuc__code__numeral(V_k) = hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Nat_OSuc,hAPP(c_Code__Numeral_Onat__of,V_k)))) # label(fact_Suc__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2344 (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]. 6.31/6.34 2345 (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]. 6.31/6.34 2346 (all V_R_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),c_Quotient_ORespects(T_c,V_R_2))))) # label(fact_in__respects) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2347 (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]. 6.31/6.34 2348 (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]. 6.31/6.34 2349 (all V_q all V_r all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_r),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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]. 6.31/6.34 2350 (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]. 6.31/6.34 2351 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))))) # label(fact_transfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2352 (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]. 6.31/6.34 2353 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) # label(fact_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2354 (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]. 6.31/6.34 2355 (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 -> V_M_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))))) # label(fact_INF__const) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2356 (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]. 6.31/6.34 2357 (all V_y all V_ys all V_x all V_xs all T_a hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_y) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys))) # label(fact_list__update__length) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2358 (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]. 6.31/6.34 2359 (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]. 6.31/6.34 2360 (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]. 6.31/6.34 2361 (all V_g_2 all T_c all T_d V_g_2 = hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),V_g_2)) # label(fact_id__o) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2362 (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]. 6.31/6.34 2363 (all V_f_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_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_setsum__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2364 (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,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_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))))) # label(fact_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2365 (all V_n_2 all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(c_Nat_OSuc,V_n_2) <-> hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_fequal),c_Int_Onat)),V_n_2))))))) # label(fact_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2366 (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)) & (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(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]. 6.31/6.34 2367 (all V_r_2 all V_q_2 all T_c (c_Nitpick_Oless__eq__frac(T_c,V_q_2,V_r_2) <-> 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))))) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2368 (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]. 6.31/6.34 2369 (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]. 6.31/6.34 2370 (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]. 6.31/6.34 2371 (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]. 6.31/6.34 2372 (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]. 6.31/6.34 2373 (all V_k all V_l ((V_l != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_OposDivAlg(hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l)) = c_Divides_Opdivmod(V_k,V_l)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = 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)) = c_Divides_Opdivmod(V_k,V_l)))) # label(fact_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2374 (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]. 6.31/6.34 2375 (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))) <-> (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)))))) # label(fact_Image__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2376 (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_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_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]. 6.31/6.34 2377 (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]. 6.31/6.34 2378 (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]. 6.31/6.34 2379 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_y_2 & c_Groups_Ozero__class_Ozero(T_c) = V_x_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]. 6.31/6.34 2380 (all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2381 (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_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__separator__converseE) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2382 (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))) -> 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) = V_g_2)) # label(fact_o__inv__o__cancel) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2383 (all V_y_2 all V_x_2 all T_c (class_Int_Oring__char__0(T_c) & class_Int_Onumber__ring(T_c) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2) <-> V_y_2 = V_x_2))) # label(fact_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2384 (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),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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2385 (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(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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_atMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2386 (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]. 6.31/6.34 2387 (all V_n_2 all V_m_2 all V_u_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)),V_u_2)),V_n_2)))))) # label(fact_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2388 (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]. 6.31/6.34 2389 (all V_Pa_2 all V_B_2 all V_A_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(V_Pa_2,B_x)))) | (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]. 6.31/6.34 2390 (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_x),V_z)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_z),V_x))))) # label(fact_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2391 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)),V_c)))) # label(fact_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2392 (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 <-> V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))) # label(fact_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2393 (all V_xs_2 all V_f_2 all T_d all V_n_2 all T_c hAPP(c_List_Orotate(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(c_List_Orotate(T_d,V_n_2),V_xs_2))) # label(fact_rotate__map) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2394 (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_c_2 = V_a_2))))) # label(fact_atLeastLessThan__inj_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2395 (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]. 6.31/6.34 2396 (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]. 6.31/6.34 2397 (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)) -> (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_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2398 (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__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)) = 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)) # label(fact_pos__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2399 (all V_Pa_2 all T_c c_HOL_OThe(T_c,V_Pa_2) = c_Enum_Oenum__the(T_c,V_Pa_2)) # label(fact_enum__the__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2400 (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]. 6.31/6.34 2401 (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 <-> 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))))) # label(fact_card__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2402 (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]. 6.31/6.34 2403 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),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]. 6.31/6.34 2404 (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]. 6.31/6.34 2405 (all V_v_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)),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 <-> 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))) # label(fact_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2406 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)))) # label(fact_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2407 (all V_b_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),V_b_2) = 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))))))) # label(fact_Sup__binary) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2408 (all V_z2 all V_z1 all V_w 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)) = 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))) # label(fact_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2409 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)))))) # label(fact_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2410 (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]. 6.31/6.34 2411 (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]. 6.31/6.34 2412 (all V_a all V_b (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) & (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))))) # label(fact_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2413 (all V_n_2 all V_xs_2 all V_x_2 all T_c hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_n_2) = c_Nat_Onat_Onat__case(T_c,V_x_2,hAPP(c_List_Onth(T_c),V_xs_2),V_n_2)) # label(fact_nth_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2414 (all V_y_2 all V_x_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_y_2 & c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> 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__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2415 (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_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))))) # label(fact_min__max_OSup__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2416 (all V_va all V_v all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_v),V_va) = c_List_Osplice(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_v),V_va),c_List_Olist_ONil(T_a))) # label(fact_splice_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2417 (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]. 6.31/6.34 2418 (all V_j all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> c_List_Olist_ONil(tc_Int_Oint) = c_List_Oupto(V_i,V_j)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),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)) = c_List_Oupto(V_i,V_j)))) # label(fact_upto_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2419 (all V_xss_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oconcat(T_c,V_xss_2)) = 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))) # label(fact_length__concat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2420 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2421 (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]. 6.31/6.34 2422 (all V_xs_2 all V_f_2 all T_d all T_c 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)) = 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))) # label(fact_transpose__map__map) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2423 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2424 (all V_l_2 all V_k_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_k_2,V_l_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_k_2,V_l_2))))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2425 (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]. 6.31/6.34 2426 (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]. 6.31/6.34 2427 (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]. 6.31/6.34 2428 (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]. 6.31/6.34 2429 (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) -> (V_xs = hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_zs) -> hAPP(hAPP(c_List_Oappend(T_a),V_ys),V_zs) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_Cons__eq__appendI) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2430 (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]. 6.31/6.34 2431 (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]. 6.31/6.34 2432 (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]. 6.31/6.34 2433 (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]. 6.31/6.34 2434 (all V_n all V_a all T_a (class_Rings_Oring__1__no__zero__divisors(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2435 (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_OBit0(V_l_2))))) # label(fact_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2436 (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]. 6.31/6.34 2437 (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]. 6.31/6.34 2438 (all V_B_2 all V_A_2 all V_I_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_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_B_2)) = 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))) # label(fact_Sigma__Diff__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2439 (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]. 6.31/6.34 2440 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (V_b_2 = V_b_H_2 & V_a_2 = V_a_H_2 <-> c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2) = c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_H_2,V_b_H_2))) # label(fact_pair__box_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2441 (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_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))) <-> 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)))) # label(fact_pred__comp__rel__comp__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2442 (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]. 6.31/6.34 2443 (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))) -> V_xs_2 = c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2)))) # label(fact_insort__insert__triv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2444 (all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Nat_OSuc,hAPP(c_Int_Onat,V_z)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)))) # label(fact_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2445 (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]. 6.31/6.34 2446 (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_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != hAPP(c_Finite__Set_Ocard(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__fun__UNIVD1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2447 (all V_m_2 hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) # label(fact_int__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2448 (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))) -> (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__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2449 (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]. 6.31/6.34 2450 (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]. 6.31/6.34 2451 (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]. 6.31/6.34 2452 (all V_g_2 all V_f_2 all T_d all T_c all T_e all T_f 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)) = 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))) # label(fact_snd__comp__map__pair) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2453 (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),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2)))))) # label(fact_dvd__minus__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2454 (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(hAPP(c_member(T_c),B_x),V_A_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),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))))))))))) # label(fact_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2455 (all V_R_2 all T_c all T_d c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_OImage(T_d,T_c,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_Image__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2456 (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_c = V_b -> 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]. 6.31/6.34 2457 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_x) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y))) # label(fact_min__max_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2458 (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]. 6.31/6.34 2459 (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]. 6.31/6.34 2460 (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]. 6.31/6.34 2461 (all V_m all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = 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)))) # label(fact_times__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2462 (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_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) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2)))) # label(fact_INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2463 (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]. 6.31/6.34 2464 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)))) # label(fact_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2465 (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]. 6.31/6.34 2466 (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]. 6.31/6.34 2467 (all V_c all V_b all V_a all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> V_b = V_c))) # label(fact_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2468 (all V_nat_H hAPP(c_Nat_OSuc,V_nat_H) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2469 (all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)) <-> hAPP(V_R_2,B_y) = hAPP(V_R_2,B_x))))) # label(fact_equivp__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2470 (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(V_Pa_2,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(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))))))) # label(fact_list__ex1__iff__raw) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2471 (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) -> hAPP(V_Abs_2,hAPP(V_Rep_2,V_x_2)) = V_x_2)) # label(fact_type__definition_ORep__inverse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2472 (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]. 6.31/6.34 2473 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_rx)),V_ry))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2474 (all V_c all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)))))))) # label(fact_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2475 (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]. 6.31/6.34 2476 (all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_of__int__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2477 (all V_x all V_y ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Nat__Transfer_Otsub(V_x,V_y)))) # label(fact_tsub__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2478 (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]. 6.31/6.34 2479 (all V_r_2 all V_q_2 all T_c (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))) <-> c_Nitpick_Oless__frac(T_c,V_q_2,V_r_2))) # label(fact_Nitpick_Oless__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2480 (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]. 6.31/6.34 2481 (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_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_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))))))) # label(fact_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2482 (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_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2483 (all V_xs all V_x all T_a 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)) = V_xs) # label(fact_drop__1__Cons) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2484 (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]. 6.31/6.34 2485 (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]. 6.31/6.34 2486 (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]. 6.31/6.34 2487 (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]. 6.31/6.34 2488 (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(hAPP(c_Groups_Ominus__class_Ominus(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),V_a_2),V_b_2))))) # label(fact_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2489 (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]. 6.31/6.34 2490 (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]. 6.31/6.34 2491 (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]. 6.31/6.34 2492 (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]. 6.31/6.34 2493 (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]. 6.31/6.34 2494 (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]. 6.31/6.34 2495 (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]. 6.31/6.34 2496 (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)) -> (hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2) = 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)) & 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))))))) # label(fact_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2497 (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]. 6.31/6.34 2498 (all V_n all V_m (V_n = V_m -> 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]. 6.31/6.34 2499 (all V_i_2 all V_xq_2 all V_f_2 all T_c all T_d c_New__DSequence_Opos__map(T_d,T_c,V_f_2,V_xq_2,V_i_2) = c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,hAPP(V_xq_2,V_i_2))) # label(fact_New__DSequence_Opos__map__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2500 (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),V_b_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_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__neg) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2501 (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]. 6.31/6.34 2502 (all T_d all T_c all V_t_2 all V_s_2 (V_t_2 = 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) & hAPP(c_Product__Type_Ofst(T_c,T_d),V_t_2) = hAPP(c_Product__Type_Ofst(T_c,T_d),V_s_2))) # label(fact_Pair__fst__snd__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2503 (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]. 6.31/6.34 2504 (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]. 6.31/6.34 2505 (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]. 6.31/6.34 2506 (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]. 6.31/6.34 2507 (all V_B_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_B_2 = 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)) -> 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)))))) # label(fact_snd__image__times) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2508 (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]. 6.31/6.34 2509 (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]. 6.31/6.34 2510 (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_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__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2511 (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(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_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2512 (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]. 6.31/6.34 2513 (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]. 6.31/6.34 2514 (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]. 6.31/6.34 2515 (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]. 6.31/6.34 2516 (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]. 6.31/6.34 2517 (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_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2518 (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]. 6.31/6.34 2519 (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_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) = 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))) # label(fact_Image__singleton) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2520 (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]. 6.31/6.34 2521 (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]. 6.31/6.34 2522 (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]. 6.31/6.34 2523 (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]. 6.31/6.34 2524 (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]. 6.31/6.34 2525 (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]. 6.31/6.34 2526 (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]. 6.31/6.34 2527 (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]. 6.31/6.34 2528 (all V_a all V_b all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_nonzero__minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2529 (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(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(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.34 2530 (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))),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,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2531 (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]. 6.31/6.35 2532 (all V_n all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2533 (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_OtakeWhile(T_c,V_Pa_2,V_xs_2) = V_xs_2)) # label(fact_takeWhile__eq__all__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2534 (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]. 6.31/6.35 2535 (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]. 6.31/6.35 2536 (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]. 6.31/6.35 2537 (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]. 6.31/6.35 2538 (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]. 6.31/6.35 2539 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_compl__sup) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2540 (all V_r_2 all V_A_2 all T_c (c_Relation_Orefl__on(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_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)))))) # label(fact_refl__on__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2541 (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]. 6.31/6.35 2542 (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]. 6.31/6.35 2543 (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]. 6.31/6.35 2544 (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]. 6.31/6.35 2545 (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]. 6.31/6.35 2546 (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]. 6.31/6.35 2547 (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]. 6.31/6.35 2548 (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]. 6.31/6.35 2549 (all T_c hAPP(c_List_Omap(T_c,T_c),c_COMBI(T_c)) = c_Fun_Oid(tc_List_Olist(T_c))) # label(fact_List_Omap_Oidentity) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2550 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Oall__interval__nat(V_Pa_2,V_i_2,V_j_2) <-> c_List_Olist__all(tc_Nat_Onat,V_Pa_2,c_List_Oupt(V_i_2,V_j_2)))) # label(fact_list__all__iff__all__interval__nat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2551 (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]. 6.31/6.35 2552 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))) # label(fact_tl__take) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2553 (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]. 6.31/6.35 2554 (all V_w hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w)) # label(fact_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2555 (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(T_c)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),V_i_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)))))) # label(fact_transpose__column__length) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2556 (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]. 6.31/6.35 2557 (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]. 6.31/6.35 2558 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(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))) -> (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_i != B_j -> hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_j) != hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i)))))))) # label(fact_distinct__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2559 (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]. 6.31/6.35 2560 (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]. 6.31/6.35 2561 (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]. 6.31/6.35 2562 (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_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_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2563 (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]. 6.31/6.35 2564 (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]. 6.31/6.35 2565 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)))) # label(fact_min__max_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2566 (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]. 6.31/6.35 2567 (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]. 6.31/6.35 2568 (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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (V_k != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2569 (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)),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__if0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2570 (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),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),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_y_2),V_z_2))))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2571 (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]. 6.31/6.35 2572 (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]. 6.31/6.35 2573 (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),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> 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(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))))) # label(fact_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2574 (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),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a_H),V_b_H),V_c))))) # label(fact_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2575 (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_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(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_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2576 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k))) # label(fact_number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2577 (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]. 6.31/6.35 2578 (all V_xss_2 all 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))) = 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)))) # label(fact_length__remdups__concat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2579 (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]. 6.31/6.35 2580 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))))) # label(fact_transfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2581 (all V_w all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) -> V_z = V_w))) # label(fact_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2582 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Obutlast(T_a,V_xs)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_length__butlast) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2583 (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]. 6.31/6.35 2584 (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]. 6.31/6.35 2585 (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]. 6.31/6.35 2586 (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 != V_A_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__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2587 (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_2,V_l_2),V_i_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_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]. 6.31/6.35 2588 (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]. 6.31/6.35 2589 (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]. 6.31/6.35 2590 (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]. 6.31/6.35 2591 (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),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),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__right__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2592 (all V_x all V_i all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)) -> V_xs = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x))) # label(fact_list__update__beyond) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2593 (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)),c_SetInterval_Oord__class_OatLeastAtMost(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_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2594 (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]. 6.31/6.35 2595 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_b))) # label(fact_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2596 (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_B_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_B_2)),V_g_2)))))) # label(fact_folding_Ocommute__left__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2597 (all V_s_2 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_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]. 6.31/6.35 2598 (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]. 6.31/6.35 2599 (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]. 6.31/6.35 2600 (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]. 6.31/6.35 2601 (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]. 6.31/6.35 2602 (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(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)))))) # label(fact_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2603 (all V_A_2 all V_y_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> 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) = 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))) # label(fact_setsum__constant) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2604 (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]. 6.31/6.35 2605 (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]. 6.31/6.35 2606 (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]. 6.31/6.35 2607 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),c_List_Oconcat(T_c,V_xs_2)) = 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))) # label(fact_set__concat) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2608 (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]. 6.31/6.35 2609 (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_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(c_Code__Numeral_Oint__of,V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Code__Numeral_Oint__of,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_k),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))),c_Groups_Oone__class_Oone(tc_Int_Oint)))) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k -> 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]. 6.31/6.35 2610 (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]. 6.31/6.35 2611 (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]. 6.31/6.35 2612 (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]. 6.31/6.35 2613 (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]. 6.31/6.35 2614 (all V_a1_2 all V_a2_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_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,V_a1_2) <-> 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]. 6.31/6.35 2615 (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)),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_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]. 6.31/6.35 2616 (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),B_a),B_A) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_X_2) -> (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]. 6.31/6.35 2617 (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]. 6.31/6.35 2618 (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]. 6.31/6.35 2619 (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]. 6.31/6.35 2620 (all V_y_2 all V_x_2 (hAPP(c_Int_ORep__Integ,V_y_2) = hAPP(c_Int_ORep__Integ,V_x_2) <-> V_x_2 = V_y_2)) # label(fact_Rep__Integ__inject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2621 (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]. 6.31/6.35 2622 (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]. 6.31/6.35 2623 (all V_n all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)))) # label(fact_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2624 (all V_n_2 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_Nat_OSuc,V_n_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]. 6.31/6.35 2625 (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]. 6.31/6.35 2626 (all V_Qa_2 all V_Pa_2 all T_d 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_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))) = 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)))) # label(fact_Collect__split) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2627 (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]. 6.31/6.35 2628 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_b != V_a -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2629 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Nitpick_Onorm__frac__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = c_Nitpick_Onorm__frac(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (-(c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2) -> hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Nitpick_Oint__gcd,V_a_2),V_b_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2))),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b_2))) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2 | V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)))))) # label(fact_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2630 (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_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)) = 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))))))) # label(fact_setprod__Un__one) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2631 (all V_m_2 all V_n_2 (V_n_2 = V_m_2 <-> hAPP(c_Code__Numeral_Oof__nat,V_n_2) = hAPP(c_Code__Numeral_Oof__nat,V_m_2))) # label(fact_Code__Numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2632 (all V_A_2 all V_I_2 all T_d all T_c exists B_f (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))) & 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)))) # label(fact_Ex__inj__on__UNION__Sigma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2633 (all V_A_2 all T_c c_Relation_OId__on(T_c,V_A_2) = 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))))) # label(fact_Id__on__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2634 (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]. 6.31/6.35 2635 (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]. 6.31/6.35 2636 (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]. 6.31/6.35 2637 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (V_y != c_Groups_Ozero__class_Ozero(T_a) -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_w),V_z)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_w),V_y))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)))))) # label(fact_add__frac__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2638 (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]. 6.31/6.35 2639 (all T_c c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) = hAPP(c_Set_OPow(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Pow__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2640 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = V_x)) # label(fact_sup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2641 (all V_C_2 all V_B_2 all V_A_2 all T_c (hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),V_B_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_A_2),V_C_2) -> 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]. 6.31/6.35 2642 (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]. 6.31/6.35 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)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)))) # label(fact_div__geq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2644 (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]. 6.31/6.35 2645 (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]. 6.31/6.35 2646 (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]. 6.31/6.35 2647 (all V_Pa_2 all T_d all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> c_Quotient_OBabs(T_c,T_d,c_Quotient_ORespects(T_c,V_R_2),V_Pa_2) = V_Pa_2)) # label(fact_babs__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2648 (all V_S_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) & class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_S_2))) <-> (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))))))) # label(fact_less__Sup__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2649 (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_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)) = hAPP(c_Random_Oiterate(T_d,T_c,V_k_2,V_f_2),V_x_2)) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k_2 -> 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]. 6.31/6.35 2650 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_tl__drop) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2651 (all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)) # label(fact_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2652 (all V_zs_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (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))) <-> 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))))) # label(fact_sorted__many__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2653 (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]. 6.31/6.35 2654 (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]. 6.31/6.35 2655 (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]. 6.31/6.35 2656 (all V_u_2 all V_l_2 hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),V_l_2)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_l_2,V_u_2))) # label(fact_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2657 (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_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)) = 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))) # label(fact_setprod__timesf) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2658 (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]. 6.31/6.35 2659 (all V_xs all V_x all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs)))) # label(fact_listsum__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2660 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2661 (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]. 6.31/6.35 2662 (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]. 6.31/6.35 2663 (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]. 6.31/6.35 2664 (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]. 6.31/6.35 2665 (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)) -> (V_xs_2 != c_List_Olist_ONil(T_c) -> 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)) & (c_List_Olist_ONil(T_c) = V_xs_2 -> 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]. 6.31/6.35 2666 (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]. 6.31/6.35 2667 (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(hAPP(c_Finite__Set_Ofold(T_c,T_c,V_f_2),V_a_2),V_A_2) = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_folding__one__idem_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2668 (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]. 6.31/6.35 2669 (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_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2670 (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]. 6.31/6.35 2671 (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]. 6.31/6.35 2672 (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]. 6.31/6.35 2673 (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),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))) # label(fact_card__greaterThanAtMost__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2674 (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]. 6.31/6.35 2675 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Opred(V_w)))) # label(fact_number__of__pred) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2676 (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]. 6.31/6.35 2677 (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_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]. 6.31/6.35 2678 (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]. 6.31/6.35 2679 (all V_v all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = 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))) # label(fact_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2680 (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_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Diff__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2681 (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]. 6.31/6.35 2682 (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]. 6.31/6.35 2683 (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]. 6.31/6.35 2684 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Otop__class_Otop(T_a)))) # label(fact_inf__top__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2685 (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]. 6.31/6.35 2686 (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__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__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_order__fun_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2687 (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]. 6.31/6.35 2688 (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]. 6.31/6.35 2689 (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,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))) = 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__inter) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2690 (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]. 6.31/6.35 2691 (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]. 6.31/6.35 2692 (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]. 6.31/6.35 2693 (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]. 6.31/6.35 2694 (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_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) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_f_2)))) # label(fact_SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2695 (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]. 6.31/6.35 2696 (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]. 6.31/6.35 2697 (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]. 6.31/6.35 2698 (all V_n_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),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)))) # label(fact_atMost__upto) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2699 (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]. 6.31/6.35 2700 (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]. 6.31/6.35 2701 (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]. 6.31/6.35 2702 (all V_xs all V_m all V_n all T_a 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)) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Odrop(T_a),V_m),V_xs))) # label(fact_take__drop) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2703 (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)),V_n) = 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)))) # label(fact_power__minus) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2704 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> 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)) = 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]. 6.31/6.35 2705 (all T_d all T_c c_FunDef_Orp__inv__image(T_c,T_d) = 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)))) # label(fact_rp__inv__image__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2706 (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(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))) = 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)))) # label(fact_listsum__addf) # label(axiom) # label(non_clause). [assumption]. 6.31/6.35 2707 (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]. 6.31/6.35 2708 (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]. 6.31/6.35 2709 (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]. 6.31/6.35 2710 (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]. 6.31/6.35 2711 (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]. 6.31/6.35 2712 (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]. 6.31/6.35 2713 (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]. 6.31/6.35 2714 (all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2715 (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]. 6.31/6.36 2716 (all V_k_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))))) -> V_A_2 = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))))) # label(fact_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2717 (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]. 6.31/6.36 2718 (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]. 6.31/6.36 2719 (all V_c_2 all V_b_2 all V_a_2 (V_b_2 = V_c_2 & V_b_2 = 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))) # label(fact_dvd_OatLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2720 (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(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)) = 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))) # label(fact_vimage__Compl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2721 (all V_A_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_Set_Oinsert(T_c),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_a_2),V_A_2)) # label(fact_insert__is__Un) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2722 (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]. 6.31/6.36 2723 (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]. 6.31/6.36 2724 (all V_b hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2725 (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)))) <-> 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,V_fs_2))) & 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]. 6.31/6.36 2726 (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]. 6.31/6.36 2727 (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]. 6.31/6.36 2728 (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]. 6.31/6.36 2729 (all V_A_2 all T_c 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)) = V_A_2) # label(fact_Collect__mem__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2730 (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]. 6.31/6.36 2731 (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]. 6.31/6.36 2732 (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) -> V_x_2 = hAPP(hAPP(V_f_2,V_x_2),V_x_2))) # label(fact_folding__image__simple__idem_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2733 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> (exists B_n exists B_f V_A_2 = 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)))))) # label(fact_finite__conv__nat__seg__image) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2734 (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]. 6.31/6.36 2735 (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(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]. 6.31/6.36 2736 (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_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)))))) # label(fact_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2737 (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_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2738 (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]. 6.31/6.36 2739 (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]. 6.31/6.36 2740 (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_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)))) # label(fact_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2741 (all V_ns_2 all V_m_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_List_Oset(tc_Nat_Onat),V_ns_2))) -> B_x = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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))) # label(fact_sum__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2742 (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]. 6.31/6.36 2743 (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]. 6.31/6.36 2744 (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(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)))) = 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)))) -> 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))))) # label(fact_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2745 (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]. 6.31/6.36 2746 (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]. 6.31/6.36 2747 (all V_y all V_n all V_x (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_y,V_n) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y),V_x)) -> (exists B_q hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),B_q)) = V_x)))) # label(fact_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2748 (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]. 6.31/6.36 2749 (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]. 6.31/6.36 2750 (all V_x_2 all V_n_2 all T_c all V_f_2 hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),hAPP(V_f_2,V_x_2)) = hAPP(V_f_2,hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),V_x_2))) # label(fact_funpow__swap1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2751 (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]. 6.31/6.36 2752 (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(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n)) & (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]. 6.31/6.36 2753 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) <-> 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)))) # label(fact_wfP__wf__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2754 (all V_v 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),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v))) # label(fact_of__int__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2755 (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]. 6.31/6.36 2756 (all V_B_2 all V_a_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_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__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Un__insert__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2757 (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]. 6.31/6.36 2758 (all V_f_2 all V_x_2 all T_c all T_d all V_i_2 ((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))) & (V_i_2 = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> c_Lazy__Sequence_Ohit__bound(T_c) = hAPP(c_New__DSequence_Oneg__decr__bind(T_d,T_c,V_x_2,V_f_2),V_i_2)))) # label(fact_New__DSequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2759 (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]. 6.31/6.36 2760 (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) -> (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(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)))) & 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))))) # label(fact_equiv__class__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2761 (all V_xs_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),V_xs_2) = 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))))) # label(fact_listsum__setsum__nth) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2762 (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]. 6.31/6.36 2763 (all V_u_2 hAPP(c_Int_Onat,V_u_2) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))) # label(fact_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2764 (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]. 6.31/6.36 2765 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Otake(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))) # label(fact_take__1__Cons) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2766 (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]. 6.31/6.36 2767 (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]. 6.31/6.36 2768 (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]. 6.31/6.36 2769 (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]. 6.31/6.36 2770 (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(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_x_2)))))) # label(fact_folding__idem_Oin__comp__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2771 (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]. 6.31/6.36 2772 (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(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)))) # label(fact_scomp__unfold) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2773 (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]. 6.31/6.36 2774 (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]. 6.31/6.36 2775 (all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> 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(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)))) # label(fact_ivl__disj__un_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2776 (all V_q all V_p c_Divides_Odivmod__int(V_p,V_q) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_p),V_q)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_p,V_q))) # label(fact_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2777 (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]. 6.31/6.36 2778 (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]. 6.31/6.36 2779 (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]. 6.31/6.36 2780 (all V_a_2 all V_b_2 all T_c (class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(T_c) -> (V_a_2 = c_Groups_Ozero__class_Ozero(T_c) <-> V_b_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_a_2)))) # label(fact_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2781 (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(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) = 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)))) # label(fact_of__nat__setprod) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2782 (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),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(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_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__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_le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2783 (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]. 6.31/6.36 2784 (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]. 6.31/6.36 2785 (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)))) -> 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)))))) = c_Big__Operators_Olinorder__class_OMax(T_c,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)))) -> c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_2)))) # label(fact_Max_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2786 (all T_c all V_c_2 all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) -> (c_Hilbert__Choice_OEps(T_c,V_Pa_2) = V_c_2 -> hBOOL(hAPP(V_Pa_2,V_c_2))))) # label(fact_exE__some) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2787 (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]. 6.31/6.36 2788 (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]. 6.31/6.36 2789 (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_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_I53_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2790 (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]. 6.31/6.36 2791 (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]. 6.31/6.36 2792 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = c_Groups_Oone__class_Oone(tc_Nat_Onat) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2793 (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_a_2 = V_b_2 | hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2794 (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]. 6.31/6.36 2795 (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]. 6.31/6.36 2796 (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]. 6.31/6.36 2797 (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]. 6.31/6.36 2798 (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]. 6.31/6.36 2799 (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]. 6.31/6.36 2800 (all V_n_2 all V_m_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2801 (all V_v_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_2)))) # label(fact_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2802 (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]. 6.31/6.36 2803 (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]. 6.31/6.36 2804 (all V_w_2 all T_c (class_Int_Oring__char__0(T_c) & class_Int_Onumber__ring(T_c) -> (c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),c_Int_OBit0(V_w_2))) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) # label(fact_iszero__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2805 (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]. 6.31/6.36 2806 (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(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_image__INT) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2807 (all V_f_2 all V_A_2 all 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_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]. 6.31/6.36 2808 (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]. 6.31/6.36 2809 (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(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)))))) = 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)) & (-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]. 6.31/6.36 2810 (all V_z_2 all T_c (class_Rings_Osemiring__1(T_c) -> (c_Int_Oiszero(T_c,V_z_2) <-> V_z_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_iszero__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2811 (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 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)) = 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))) # label(fact_fun__upd__comp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2812 (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]. 6.31/6.36 2813 (all V_l all V_k 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))) = c_Divides_Opdivmod(V_k,V_l)) # label(fact_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2814 (all V_xs_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> 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)) = 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))))) # label(fact_uminus__listsum__map) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2815 (all V_b all V_a c_Divides_Odiv__class_Omod(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,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b))) # label(fact_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2816 (all V_l c_Int_OPls != c_Int_OBit1(V_l)) # label(fact_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2817 (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(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)) = 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))) # label(fact_listsum__mult__const) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2818 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)))) # label(fact_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2819 (all V_Pa_2 all T_c -(exists 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_bex__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2820 (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_a_2 = V_b_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_a_2 != V_b_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2821 (all V_a all T_a (class_Groups_Omonoid__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__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2822 (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]. 6.31/6.36 2823 (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]. 6.31/6.36 2824 (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]. 6.31/6.36 2825 (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]. 6.31/6.36 2826 (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]. 6.31/6.36 2827 (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]. 6.31/6.36 2828 (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]. 6.31/6.36 2829 (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]. 6.31/6.36 2830 (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]. 6.31/6.36 2831 (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)))) -> 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_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(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_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2) = V_x_2))))) # label(fact_min__max_OInf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2832 (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]. 6.31/6.36 2833 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> hAPP(c_List_Ohd(tc_Nat_Onat),c_List_Oupt(V_i,V_j)) = V_i)) # label(fact_hd__upt) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2834 (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_B_2 = V_A_2)))) # label(fact_card__seteq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2835 (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]. 6.31/6.36 2836 (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]. 6.31/6.36 2837 (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)) -> (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_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_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)) = 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))))))))) # label(fact_fold1__Un) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2838 (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),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_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))))))) # label(fact_setprod__Un__Int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2839 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> 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)) = 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))))) # label(fact_diff__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2840 (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)),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)))) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__un_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2841 (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]. 6.31/6.36 2842 (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]. 6.31/6.36 2843 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)) = hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_j),c_List_Olist_ONil(tc_Nat_Onat))))) # label(fact_upt__Suc__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2844 (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]. 6.31/6.36 2845 (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_Ozip(T_c,T_d),hAPP(hAPP(c_List_Omap(T_e,T_c),V_f_2),V_xs_2)),V_ys_2) = 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))) # label(fact_zip__map1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2846 (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]. 6.31/6.36 2847 (all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_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),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]. 6.31/6.36 2848 (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]. 6.31/6.36 2849 (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_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]. 6.31/6.36 2850 (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) -> 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) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))))) # label(fact_union__comp__emptyL) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2851 (all V_k c_Int_OBit1(c_Int_Opred(V_k)) = c_Int_Opred(c_Int_OBit0(V_k))) # label(fact_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2852 (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]. 6.31/6.36 2853 (all V_r_2 all V_q_2 all T_c c_Nitpick_Oplus__frac(T_c,V_q_2,V_r_2) = 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)))) # label(fact_Nitpick_Oplus__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2854 (all V_A_2 all T_c 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)) = V_A_2) # label(fact_double__complement) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2855 (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)) -> 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_m_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I19_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2856 (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]. 6.31/6.36 2857 (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]. 6.31/6.36 2858 (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),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(T_c),V_a_2),V_b_2)))))) # label(fact_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2859 (all V_n V_n != hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2860 (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]. 6.31/6.36 2861 (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]. 6.31/6.36 2862 (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]. 6.31/6.36 2863 (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]. 6.31/6.36 2864 (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]. 6.31/6.36 2865 (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]. 6.31/6.36 2866 (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]. 6.31/6.36 2867 (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]. 6.31/6.36 2868 (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_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)) = 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]. 6.31/6.36 2869 (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__decomp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2870 (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]. 6.31/6.36 2871 (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]. 6.31/6.36 2872 (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_x_2 = V_z_2)) # label(fact_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2873 (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]. 6.31/6.36 2874 (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]. 6.31/6.36 2875 (all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),c_List_Olist_ONil(T_b)) = c_List_Olist_ONil(tc_prod(T_a,T_b))) # label(fact_zip_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2876 (all V_g_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),V_x_2) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_g_2),V_x_2) <-> hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)) = hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)))) # label(fact_apfst__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2877 (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]. 6.31/6.36 2878 (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]. 6.31/6.36 2879 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2))))) # label(fact_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2880 (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]. 6.31/6.36 2881 (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]. 6.31/6.36 2882 (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]. 6.31/6.36 2883 (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) -> (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))) <-> (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)))))))) # label(fact_le__INF__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2884 (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_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Min_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2885 (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]. 6.31/6.36 2886 (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_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]. 6.31/6.36 2887 (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_w_2 != V_z_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]. 6.31/6.36 2888 (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]. 6.31/6.36 2889 (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]. 6.31/6.36 2890 (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_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))))))))) # label(fact_card__Suc__Diff1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2891 (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]. 6.31/6.36 2892 (all V_ys_2 all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_append__self__conv2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2893 (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_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),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_setsum_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2894 (all V_xy all V_i all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),hAPP(c_Product__Type_Ofst(T_a,T_b),V_xy))),hAPP(hAPP(c_List_Olist__update(T_b,V_ys),V_i),hAPP(c_Product__Type_Osnd(T_a,T_b),V_xy))) = hAPP(hAPP(c_List_Olist__update(tc_prod(T_a,T_b),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i),V_xy)) # label(fact_update__zip) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2895 (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]. 6.31/6.36 2896 (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_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)) = 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)))))))) # label(fact_min__max_OSup__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2897 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n) = V_m)) # label(fact_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2898 (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]. 6.31/6.36 2899 (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]. 6.31/6.36 2900 (all V_ts_2 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_zs_2),V_ts_2) <-> (exists B_us (hAPP(hAPP(c_List_Oappend(T_c),V_zs_2),B_us) = V_xs_2 & hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ys_2) = V_ts_2 | V_zs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),B_us) & V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ts_2))))) # label(fact_append__eq__append__conv2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2901 (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]. 6.31/6.36 2902 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2903 (all V_y_2 all V_xs_2 all V_x_2 all T_c (V_x_2 = V_y_2 | hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_y_2)) <-> 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)))) # label(fact_member__rec_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2904 (all V_a_2 all V_xs_2 all T_c (class_Groups_Oab__semigroup__add(T_c) -> c_List_Ofoldr(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_xs_2,V_a_2) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_xs_2))) # label(fact_foldr__conv__foldl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.36 2905 (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]. 6.31/6.37 2906 (all V_lazy__sequence_H_2 all V_a_H_2 all V_lazy__sequence_2 all V_a_2 all T_c (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) <-> V_lazy__sequence_H_2 = V_lazy__sequence_2 & V_a_2 = V_a_H_2)) # label(fact_lazy__sequence_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2907 (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]. 6.31/6.37 2908 (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]. 6.31/6.37 2909 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_a_2))))) # label(fact_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2910 (all V_xs_2 all V_n_2 all 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_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)))) = hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_On__lists(T_c,V_n_2,V_xs_2))) # label(fact_set__n__lists) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2911 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2912 (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]. 6.31/6.37 2913 (all V_z_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),V_z_2)))) # label(fact_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2914 (all V_a all V_y all V_x all T_b all T_a (V_a = hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y)) -> V_x = V_a)) # label(fact_fst__eqD) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2915 (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]. 6.31/6.37 2916 (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(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2917 (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]. 6.31/6.37 2918 (all V_n_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)),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),V_m_2),V_n_2)))) # label(fact_zless__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2919 (all V_a_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (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)))) <-> V_a_2 = c_Groups_Oone__class_Oone(T_c) | V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c))))) # label(fact_power2__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2920 (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]. 6.31/6.37 2921 (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),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool))),V_R_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_rel__comp__empty1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2922 (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]. 6.31/6.37 2923 (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]. 6.31/6.37 2924 (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]. 6.31/6.37 2925 (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_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]. 6.31/6.37 2926 (all V_A_2 all T_c c_List_Olists(T_c,V_A_2) = 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))) # label(fact_lists__eq__set) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2927 (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_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_I34_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2928 (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_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))))) # label(fact_setsum__reindex__id) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2929 (all V_m_2 (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)))) <-> 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)))))))) # label(fact_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2930 (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_Otranspose(T_c,V_xs_2) = c_List_Olist_ONil(tc_List_Olist(T_c)))) # label(fact_transpose__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2931 (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]. 6.31/6.37 2932 (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]. 6.31/6.37 2933 (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]. 6.31/6.37 2934 (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]. 6.31/6.37 2935 (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]. 6.31/6.37 2936 (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]. 6.31/6.37 2937 (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]. 6.31/6.37 2938 (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]. 6.31/6.37 2939 (all V_w all T_a (class_Int_Oring__char__0(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(V_w))))) # label(fact_iszero__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2940 (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__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)) = 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_Int__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2941 (all V_B_2 all V_A_2 all T_d all T_c (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 <-> 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),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2)))) # label(fact_Times__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2942 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_m)) # label(fact_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2943 (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_ys_2 = V_xs_2)) # label(fact_list__eq__iff__nth__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2944 (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]. 6.31/6.37 2945 (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]. 6.31/6.37 2946 (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]. 6.31/6.37 2947 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),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)))))) # label(fact_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2948 (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_g_2),V_A_2) = hAPP(hAPP(V_F_2,V_h_2),V_A_2))))) # label(fact_folding__image_Ocong) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2949 (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]. 6.31/6.37 2950 (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),V_a_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)))))) # label(fact_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2951 (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]. 6.31/6.37 2952 (all V_A_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_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]. 6.31/6.37 2953 (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))) <-> V_y_2 = V_x_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_Otrancl(T_c,V_R_2))))) # label(fact_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2954 (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),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)))) <-> 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))))) # label(fact_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2955 (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]. 6.31/6.37 2956 (all V_i_2 all V_f_2 all V_x_2 all T_c all T_d hAPP(c_New__DSequence_Oneg__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,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))) # label(fact_New__DSequence_Oneg__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2957 (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(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2) = hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2958 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Nat_OSuc,V_n)))) # label(fact_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2959 (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)))) <-> V_xs_2 = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_filter__id__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2960 (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]. 6.31/6.37 2961 (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]. 6.31/6.37 2962 (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]. 6.31/6.37 2963 (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)),V_x) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_compl__sup__top) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2964 (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]. 6.31/6.37 2965 (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]. 6.31/6.37 2966 (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(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)) = 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)))) # label(fact_Image__Int__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2967 (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_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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)))))) # label(fact_neg__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2968 (all V_k_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2969 (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]. 6.31/6.37 2970 (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]. 6.31/6.37 2971 (all V_xs_2 all T_c 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))) = c_Random_Oselect(T_c,V_xs_2)) # label(fact_select__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2972 (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]. 6.31/6.37 2973 (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]. 6.31/6.37 2974 (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_OgreaterThanLessThan(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__int_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2975 (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]. 6.31/6.37 2976 (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),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_a_2)) # label(fact_Inf__singleton) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2977 (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]. 6.31/6.37 2978 (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]. 6.31/6.37 2979 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_c_2 | 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_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2)))))) # label(fact_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2980 (all V_ys_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (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,V_xs_2) <-> 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]. 6.31/6.37 2981 (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_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)) = hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_length__insort) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2982 (all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2983 (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)),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)) = c_Transitive__Closure_Otrancl(T_c,V_r_2)) # label(fact_trancl__unfold) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2984 (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)) -> ((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]. 6.31/6.37 2985 (all V_xs_2 all T_c all V_ys_2 (V_xs_2 = c_List_Olist_ONil(T_c) <-> V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_self__append__conv2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2986 (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_enum__ex) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2987 (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))) -> hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2) = c_List_Olinorder__class_Oinsort__insert__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]. 6.31/6.37 2988 (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]. 6.31/6.37 2989 (all V_Ga_2 all V_F_2 all T_c (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))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)))) # label(fact_finite__Un) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2990 (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_Oone__class_Oone(tc_Nat_Onat) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> 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_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2991 (all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = c_Transitive__Closure_Otrancl(T_c,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]. 6.31/6.37 2992 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_drop__tl) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2993 (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]. 6.31/6.37 2994 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) <-> c_Relation_Ototal__on(T_c,V_A_2,V_r_2))) # label(fact_total__on__converse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2995 (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]. 6.31/6.37 2996 (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_a2_2)) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_a2_2 & (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_a1_2)))) # label(fact_max__extp_Oequations) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 2997 (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]. 6.31/6.37 2998 (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]. 6.31/6.37 2999 (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]. 6.31/6.37 3000 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3001 (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),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_c))) # label(fact_setprod__empty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3002 (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(hAPP(c_List_Onth(T_c),V_xs_2),V_l_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)))) # label(fact_pick__same) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3003 (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]. 6.31/6.37 3004 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (c_Groups_Osgn__class_Osgn(T_c,V_a_2) = c_Groups_Oone__class_Oone(T_c) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_sgn__1__pos) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3005 (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]. 6.31/6.37 3006 (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)),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)) = 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))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3007 (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) & V_B_2 = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_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]. 6.31/6.37 3008 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__mult__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3009 (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]. 6.31/6.37 3010 (all V_w_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3011 (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]. 6.31/6.37 3012 (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]. 6.31/6.37 3013 (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)) -> (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_B_2 = V_A_2))) # label(fact_inj__on__Un__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3014 (all V_b all V_a ((-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))) = c_Divides_OposDivAlg(V_a,V_b)) & (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)))) # label(fact_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3015 (all V_f_2 all V_A_2 all V_B_2 all T_c all T_d (class_Groups_Oordered__ab__semigroup__add(T_d) & class_Groups_Ocomm__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_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]. 6.31/6.37 3016 (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]. 6.31/6.37 3017 (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]. 6.31/6.37 3018 (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]. 6.31/6.37 3019 (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]. 6.31/6.37 3020 (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]. 6.31/6.37 3021 (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]. 6.31/6.37 3022 (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_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))) = 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)))) # label(fact_SUP__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3023 (all V_ys_2 all V_xs_2 all T_c 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)) = hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_set__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3024 (all V_n all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_n))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3025 (all V_N_2 ((exists B_m all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_x),B_m)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2)))) # label(fact_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3026 (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(V_F_2,hAPP(hAPP(c_Set_Oinsert(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))))) # label(fact_folding__one_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3027 (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_y_2)) | 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),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]. 6.31/6.37 3028 (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]. 6.31/6.37 3029 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i))) # label(fact_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3030 (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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,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__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)))) # label(fact_transfer__int__nat__sum__prod2_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3031 (all V_xs_2 all V_x_2 all V_n_2 all T_c hAPP(hAPP(c_List_Otake(T_c),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = 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)) # label(fact_take__Cons) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3032 (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_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) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3033 (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]. 6.31/6.37 3034 (all V_A_2 all 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)) <-> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_card__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3035 (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]. 6.31/6.37 3036 (all V_c all V_a all V_b all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_c)))) # label(fact_min__max_Oinf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3037 (all V_lazy__sequence_H all V_a_H all T_a c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a) != c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a_H,V_lazy__sequence_H)) # label(fact_lazy__sequence_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3038 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_x_2),hAPP(c_Set_OPow(T_c),V_A_2))) <-> 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)))) # label(fact_Powp__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3039 (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),V_xs_2) = c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))) # label(fact_remove1__filter__not) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3040 (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]. 6.31/6.37 3041 (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]. 6.31/6.37 3042 (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]. 6.31/6.37 3043 (all V_n_2 all V_m_2 (c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n_2 & c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) # label(fact_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3044 (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]. 6.31/6.37 3045 (all V_a_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(c_fequal,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_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3046 (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]. 6.31/6.37 3047 (all V_A_2 all T_c (-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__infinite) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3048 (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_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) = 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))))) # label(fact_setprod__dividef) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3049 (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]. 6.31/6.37 3050 (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]. 6.31/6.37 3051 (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]. 6.31/6.37 3052 (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_OatMost(T_c),V_k_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2))) # label(fact_Compl__atMost) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3053 (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]. 6.31/6.37 3054 (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]. 6.31/6.37 3055 (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]. 6.31/6.37 3056 (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 -> V_b_2 = hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_a_2,V_b_2),V_x_2)) & (V_x_2 != V_a_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)))) # label(fact_fun__upd__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3057 (all V_code__numeral_H_2 all V_code__numeral_2 (c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2) = c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_2) <-> V_code__numeral_H_2 = V_code__numeral_2)) # label(fact_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3058 (all V_xs all T_a all V_ys (V_ys != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = c_List_Olast(T_a,V_ys))) # label(fact_last__appendR) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3059 (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]. 6.31/6.37 3060 (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]. 6.31/6.37 3061 (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]. 6.31/6.37 3062 (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]. 6.31/6.37 3063 (all V_xs all T_a 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) = c_List_Obutlast(T_a,V_xs)) # label(fact_butlast__conv__take) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3064 (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_OgreaterThanLessThan(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_c_2,V_d_2)) = 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)))) # label(fact_Int__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3065 (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)),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_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)),V_xs_2),c_List_Olists(T_c,V_A_2))))) # label(fact_append__in__lists__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3066 (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]. 6.31/6.37 3067 (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]. 6.31/6.37 3068 (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]. 6.31/6.37 3069 (all V_g_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),V_g_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Groups_Oone__class_Oone(T_c))) # label(fact_setprod_Oempty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3070 (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]. 6.31/6.37 3071 (all V_l (c_List_Onat__list(V_l) -> c_List_Oembed__list(c_List_Oreturn__list(V_l)) = V_l)) # label(fact_transfer__nat__int__list__return__embed) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3072 (all V_x_2 all V_A_2 all T_c (class_Groups_Oab__semigroup__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)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_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_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) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3073 (all V_Qa_2 all V_Pa_2 (-hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)) | hBOOL(V_Qa_2) | hBOOL(V_Pa_2))) # label(help_c__fdisj__3) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3074 (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]. 6.31/6.37 3075 (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]. 6.31/6.37 3076 (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]. 6.31/6.37 3077 (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]. 6.31/6.37 3078 (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]. 6.31/6.37 3079 (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_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)) & (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))))) -> V_b_2 = c_Partial__Function_Oflat__lub(T_c,V_b_2,V_A_2)))) # label(fact_flat__lub__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3080 (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_d,T_c,T_e,V_f_2),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))) # label(fact_comp__cong) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3081 (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_fun_2 = V_fun_H_2 & V_com2_2 = V_com2_H_2 & V_com1_2 = V_com1_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]. 6.31/6.37 3082 (all V_c all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)),V_c)))) # label(fact_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3083 (all V_r_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_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)) = 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))) # label(fact_Field__insert) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3084 (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]. 6.31/6.37 3085 (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]. 6.31/6.37 3086 (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]. 6.31/6.37 3087 (all V_c all V_b all V_v all T_b (class_Rings_Oring(T_b) & class_Int_Onumber(T_b) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),V_b),V_c)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_b),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v)),V_c)))) # label(fact_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3088 (all V_n_2 all 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) <-> V_m_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) & V_n_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = V_m_2 & hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = V_n_2)) # label(fact_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3089 (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(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_Fun_Oid(T_d))) # label(fact_folding_Oempty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3090 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3091 (all V_l hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l))) # label(fact_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3092 (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]. 6.31/6.37 3093 (all V_z_2 all V_z_H_2 ((hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z_H_2)) -> hAPP(c_Int_Onat,V_z_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z_2)),hAPP(c_Int_Onat,V_z_H_2))) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z_H_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z_2)),hAPP(c_Int_Onat,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))))) # label(fact_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3094 (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]. 6.31/6.37 3095 (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]. 6.31/6.37 3096 (all V_x_2 all V_k_2 all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_k_2),V_x_2) <-> V_xs_2 = c_List_Olist_ONil(T_c))) # label(fact_list__update__nonempty) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3097 (all V_size_2 all V_nrandom_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)),c_DSequence_Oempty(T_c)) = c_Random__Sequence_Oempty(T_c,V_nrandom_2,V_size_2)) # label(fact_Random__Sequence_Oempty__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3098 (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(V_f_2,V_a1_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)))))))) # label(fact_UN__equiv__class2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3099 (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)),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) = 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))) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.37 3100 (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]. 6.31/6.37 3101 (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]. 6.31/6.37 3102 (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),V_a_2),V_b_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)) <-> 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]. 6.31/6.37 3103 (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(hAPP(c_Groups_Ominus__class_Ominus(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),V_a_2),V_b_2))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3104 (all V_a_2 all V_c_2 all V_b_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 -> c_Groups_Ozero__class_Ozero(T_c) = V_a_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]. 6.31/6.38 3105 (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]. 6.31/6.38 3106 (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]. 6.31/6.38 3107 (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_e,T_e,T_d,hAPP(V_f_2,B_x)),V_h_2) = hAPP(c_Fun_Ocomp(T_d,T_e,T_d,V_h_2),hAPP(V_g_2,B_x)))) -> 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]. 6.31/6.38 3108 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3109 (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]. 6.31/6.38 3110 (all V_l_2 (c_List_Onat__list(V_l_2) <-> c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),V_l_2)))) # label(fact_nat__list__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3111 (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]. 6.31/6.38 3112 (all T_1 all T_2 (class_Enum_Oenum(T_2) & class_Enum_Oenum(T_1) -> class_Enum_Oenum(tc_sum(T_2,T_1)))) # label(arity_sum__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3113 (all V_k V_k = hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_k))) # label(fact_of__nat__nat__of) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3114 (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]. 6.31/6.38 3115 (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]. 6.31/6.38 3116 (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]. 6.31/6.38 3117 (all T_d all T_c all V_Pa_2 ((all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (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)))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3118 (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_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__commute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3119 (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]. 6.31/6.38 3120 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w)))) # label(fact_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3121 (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_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]. 6.31/6.38 3122 (all V_xs_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> 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) = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_List_Oset(T_c),V_xs_2)))) # label(fact_Inf__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3123 (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)) <-> 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)))) # label(fact_inj__on__Un) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3124 (all V_l_2 all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_k_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)) & (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 -> 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__code) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3125 (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]. 6.31/6.38 3126 (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),V_A_2)) # label(fact_Int__absorb) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3127 (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_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]. 6.31/6.38 3128 (all V_n_2 all V_m_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_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)))) # label(fact_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3129 (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]. 6.31/6.38 3130 (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 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)) = 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))) # label(fact_map__pair_Ocomp) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3131 (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]. 6.31/6.38 3132 (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]. 6.31/6.38 3133 (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) -> (V_c_2 != V_d_2 & V_a_2 != V_b_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)) != 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))))) # label(fact_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3134 (all V_a all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield(T_a) -> V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_divide__numeral__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3135 (all V_XS_2 all V_A_2 all T_c c_List_Oset__Cons(T_c,V_A_2,V_XS_2) = 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))))))) # label(fact_set__Cons__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3136 (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)) -> (hAPP(V_f_2,V_a_2) != c_Groups_Ozero__class_Ozero(T_d) -> (-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)))))) & (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)))))))))) # label(fact_setprod__diff1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3137 (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]. 6.31/6.38 3138 (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]. 6.31/6.38 3139 (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]. 6.31/6.38 3140 (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]. 6.31/6.38 3141 (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))) -> 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_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)))) # label(fact_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3142 (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]. 6.31/6.38 3143 (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]. 6.31/6.38 3144 (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_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_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]. 6.31/6.38 3145 (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]. 6.31/6.38 3146 (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]. 6.31/6.38 3147 (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]. 6.31/6.38 3148 (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]. 6.31/6.38 3149 (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(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)) = 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)))))) # label(fact_filter__insort) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3150 (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_OlessThan(T_c),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_i_2),V_k_2))))) # label(fact_lessThan__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3151 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_minus__divide__left) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3152 (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]. 6.31/6.38 3153 (all V_xs all V_n all T_a c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs))) # label(fact_butlast__drop) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3154 (all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_of__int__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3155 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3156 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a)),V_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3157 (all V_zs all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_zs) = 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)))) # label(fact_zip__append1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3158 (all V_l_2 (c_Int_OBit1(V_l_2) = c_Int_OMin <-> V_l_2 = c_Int_OMin)) # label(fact_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3159 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_y)))) # label(fact_dvd__neg__div) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3160 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k_2),V_n_2)) -> (V_n_2 = V_m_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3161 (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)) <-> V_xs_2 = c_List_Olist_ONil(T_c))) # label(fact_list__all2__Nil2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3162 (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,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))) # label(fact_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3163 (all V_n_2 all V_k_2 all 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) <-> V_m_2 = V_n_2 | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2)) # label(fact_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3164 (all V_n_2 all V_k_2 all V_m_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3165 (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(T_c),V_y_2),V_x_2)) <-> V_y_2 = V_x_2))) # label(fact_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3166 (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]. 6.31/6.38 3167 (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]. 6.31/6.38 3168 (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]. 6.31/6.38 3169 (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)))) -> hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),hAPP(c_List_Oset(T_c),V_xs_2))) | V_y_2 = V_x_2)) # label(fact_set__ConsD) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3170 (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]. 6.31/6.38 3171 (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)) -> ((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)) & 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(V_Pa_2,B_i),B_j)))) <-> 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)))))) # label(fact_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3172 (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]. 6.31/6.38 3173 (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]. 6.31/6.38 3174 (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]. 6.31/6.38 3175 (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]. 6.31/6.38 3176 (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]. 6.31/6.38 3177 (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]. 6.31/6.38 3178 (all V_y all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs) = 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)) # label(fact_list__update__code_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3179 (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]. 6.31/6.38 3180 (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]. 6.31/6.38 3181 (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]. 6.31/6.38 3182 (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)) -> hAPP(V_Rep_2,hAPP(V_Abs_2,V_y_2)) = V_y_2))) # label(fact_type__definition_OAbs__inverse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3183 (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]. 6.31/6.38 3184 (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]. 6.31/6.38 3185 (all V_A_2 all V_I_2 all T_d all V_B_2 all T_c 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)) = 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))) # label(fact_Un__INT__distrib) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3186 (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]. 6.31/6.38 3187 (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]. 6.31/6.38 3188 (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(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) = hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_i_2)))) # label(fact_nth__map__upt) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3189 (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]. 6.31/6.38 3190 (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)),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)) -> 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) = 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))) # label(fact_UN__finite2__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3191 (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]. 6.31/6.38 3192 (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]. 6.31/6.38 3193 (all V_q all T_a all T_b (class_Rings_Oinverse(T_b) & class_Rings_Oring__1(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]. 6.31/6.38 3194 (all T_c c_List_Omember(T_c) = c_List_Oset(T_c)) # label(fact_member__set) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3195 (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]. 6.31/6.38 3196 (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]. 6.31/6.38 3197 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),V_y_2)))))) # label(fact_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3198 (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]. 6.31/6.38 3199 (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]. 6.31/6.38 3200 (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]. 6.31/6.38 3201 (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]. 6.31/6.38 3202 (all V_zs_2 all V_x_2 all T_c (class_Groups_Omonoid__add(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_x_2),V_zs_2) = 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)))) # label(fact_foldl__absorb0) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3203 (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]. 6.31/6.38 3204 (all V_k_2 hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_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))))) # label(fact_greaterThan__Suc) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3205 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> V_a = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_power__one__right) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3206 (all V_b all V_a all T_b all T_a V_a = hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b))) # label(fact_fst__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3207 (all V_Y_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_Y_2)) -> (V_Y_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_Y_2)),c_FunDef_Omax__strict))))) # label(fact_smax__emptyI) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3208 (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]. 6.31/6.38 3209 (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]. 6.31/6.38 3210 (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_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_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)) & (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__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3211 (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_List_Odistinct(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)) & V_A_2 = hAPP(c_List_Oset(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]. 6.31/6.38 3212 (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]. 6.31/6.38 3213 (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_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_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3214 (all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a) = V_a)) # label(fact_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3215 (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_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))))) # label(fact_The__split__eq) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3216 (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]. 6.31/6.38 3217 (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 (V_a4_2 = hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) & V_a2_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) | (exists B_x exists B_y (hAPP(hAPP(V_a3_2,V_a1_2),V_a5_2) = hAPP(hAPP(V_a3_2,B_x),B_y) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a2_2)) & 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))) <-> 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]. 6.31/6.38 3218 (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]. 6.31/6.38 3219 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c)),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_I25_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3220 (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]. 6.31/6.38 3221 (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_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_a_2),V_b_2))))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3222 (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),hAPP(c_fequal,V_y_2))))) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3223 (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),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),c_Predicate_Oconversep(T_c,T_c,V_r_2)))))) # label(fact_acyclicP__converse) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3224 (all V_m c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mod__1) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3225 (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]. 6.31/6.38 3226 (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]. 6.31/6.38 3227 (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]. 6.31/6.38 3228 (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]. 6.31/6.38 3229 (all V_Pa_2 all V_f_2 all T_c all T_d 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)) = c_Random__Sequence_Omap(T_d,T_c,V_f_2,V_Pa_2)) # label(fact_Random__Sequence_Omap__def) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3230 (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]. 6.31/6.38 3231 (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]. 6.31/6.38 3232 (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]. 6.31/6.38 3233 (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]. 6.31/6.38 3234 (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]. 6.31/6.38 3235 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) -> c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(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)))) & (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(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]. 6.31/6.38 3236 (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]. 6.31/6.38 3237 (all V_m all 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)) = 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_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3238 (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_Oapfst(T_f,T_c,T_e,V_g_2),V_x_2)) = 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)))) # label(fact_apsnd__apfst) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3239 (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]. 6.31/6.38 3240 (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]. 6.31/6.38 3241 (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]. 6.31/6.38 3242 (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]. 6.31/6.38 3243 (all V_j_2 all V_i_2 all T_c (class_Rings_Oring__1(T_c) -> 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)) = 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)))))))) # label(fact_of__int) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3244 (all V_A_2 all T_c V_A_2 = hAPP(c_Relation_ORange(T_c,T_c),c_Relation_OId__on(T_c,V_A_2))) # label(fact_Range__Id__on) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3245 (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]. 6.31/6.38 3246 (all V_xss_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oconcat(T_c,V_xss_2) <-> (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))) -> c_List_Olist_ONil(T_c) = B_x)))) # label(fact_Nil__eq__concat__conv) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3247 (all V_A_2 all T_c c_List_Osublist(T_c,c_List_Olist_ONil(T_c),V_A_2) = c_List_Olist_ONil(T_c)) # label(fact_sublist__nil) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3248 (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]. 6.31/6.38 3249 (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]. 6.31/6.38 3250 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> (hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2) <-> V_B_2 = V_A_2)))) # label(fact_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3251 (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_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__idem) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3252 (all V_x_2 all V_n_2 all T_c (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> c_List_Oreplicate(T_c,V_n_2,V_x_2) = c_List_Olist_ONil(T_c))) # label(fact_empty__replicate) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3253 (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_Nat_Onat),V_n_2),V_k_2))) <-> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (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)))))))) # label(fact_split__div) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3254 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_ys)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) # label(fact_drop__append) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3255 (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_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__mult__class_Osetprod(T_d,T_e),V_h_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_setprod__reindex) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3256 (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_b),V_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_b))) # label(fact_sup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3257 (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]. 6.31/6.38 3258 (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) -> (hAPP(V_Rep_2,V_y_2) = hAPP(V_Rep_2,V_x_2) <-> V_x_2 = V_y_2))) # label(fact_type__definition_ORep__inject) # label(axiom) # label(non_clause). [assumption]. 6.31/6.38 3259 (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]. 6.31/6.38 3260 (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]. 6.31/6.38 3261 (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]. 6.31/6.38 3262 (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]. 6.42/6.38 3263 (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__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)) & (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2)))) # label(fact_list__ex1__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.38 3264 (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]. 6.42/6.38 3265 (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),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_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 6.42/6.38 3266 (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]. 6.42/6.38 3267 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.42/6.38 3268 (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__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))) # label(fact_pair__box_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.38 3269 (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]. 6.42/6.38 3270 (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]. 6.42/6.38 3271 (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) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m) -> V_m = V_n))) # label(fact_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3272 (all T_c (class_Orderings_Olinorder(T_c) & class_Lattices_Osemilattice__sup(T_c) -> c_Orderings_Oord__class_Omax(T_c) = c_Lattices_Osemilattice__sup__class_Osup(T_c))) # label(fact_sup__max) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3273 (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]. 6.42/6.39 3274 (all V_xs_2 all T_c (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 = c_List_Olist_ONil(T_c))) # label(fact_length__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3275 (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_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_z)))) # label(fact_inf__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3276 (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]. 6.42/6.39 3277 (all V_r_2 all T_c all T_d (hAPP(c_Relation_ORange(T_d,T_c),V_r_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)) = V_r_2)) # label(fact_Range__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3278 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_x)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3279 (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]. 6.42/6.39 3280 (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 -> (V_ys_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)) -> 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]. 6.42/6.39 3281 (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))) -> hAPP(V_g_2,B_x) = c_Groups_Ozero__class_Ozero(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_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)) = 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))))))) # label(fact_setsum_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3282 (all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_a_2),c_List_Olists(T_c,V_A_2))) <-> (exists B_a exists B_l (V_a_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_a),B_l) & 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))))) | c_List_Olist_ONil(T_c) = V_a_2)) # label(fact_lists_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3283 (all V_y all 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_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))))) # label(fact_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3284 (all V_ys_2 all V_xs_2 all T_c (c_List_Olist_ONil(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))) # label(fact_Nil__is__append__conv) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3285 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(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_Oabs__class_Oabs(T_c),V_a_2)))))) # label(fact_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3286 (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]. 6.42/6.39 3287 (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)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3288 (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]. 6.42/6.39 3289 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ORange(T_c,T_c),c_Relation_OId(T_c))) # label(fact_Range__Id) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3290 (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]. 6.42/6.39 3291 (all V_ys_2 all V_Pa_2 all V_xs_2 all V_x_2 all T_c ((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 & V_xs_2 = hAPP(c_List_Ofilter(T_c,V_Pa_2),B_vs) & hBOOL(hAPP(V_Pa_2,V_x_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(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))) # label(fact_Cons__eq__filter__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3292 (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)) -> 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,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))))) # label(fact_Min_Oinsert) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3293 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_less__minus__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3294 (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)) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(tc_Nat_Onat))) <-> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_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]. 6.42/6.39 3295 (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(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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2))))) # label(fact_greaterThan__subset__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3296 (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 (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_Nat_Onat),B_i),V_n_2)) & V_k_2 = hAPP(V_f_2,B_i)))))))) # label(fact_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3297 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3298 (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]. 6.42/6.39 3299 (all V_Pa_2 ((all B_x hBOOL(hAPP(V_Pa_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))) <-> (all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3300 (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_m_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) <-> V_m_2 = V_n_2))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3301 (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)),c_List_Olistsp(T_c,V_A_2)),c_List_Olistsp(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)),V_A_2),V_B_2))) # label(fact_listsp__inf__eq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3302 (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]. 6.42/6.39 3303 (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_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))) & (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))))) # label(fact_INT__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3304 (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]. 6.42/6.39 3305 (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_b = V_a)))))) # label(fact_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3306 (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)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) -> 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)))) # label(fact_map__snd__zip) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3307 (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]. 6.42/6.39 3308 (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]. 6.42/6.39 3309 (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)) -> 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]. 6.42/6.39 3310 (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]. 6.42/6.39 3311 (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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_b_2),V_a_2)),V_e_2)),V_d_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),V_a_2),V_e_2)),V_c_2)))) # label(fact_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3312 (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]. 6.42/6.39 3313 (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]. 6.42/6.39 3314 (all V_S_2 (c_Nat__Transfer_Onat__set(V_S_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)))))) # label(fact_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3315 (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]. 6.42/6.39 3316 (all V_ys all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys) != hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) -> V_xs != V_ys)) # label(fact_neq__if__length__neq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3317 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2))))) # label(fact_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3318 (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]. 6.42/6.39 3319 (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]. 6.42/6.39 3320 (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]. 6.42/6.39 3321 (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_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))) <-> 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)))) # label(fact_SUP__UN__eq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3322 (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]. 6.42/6.39 3323 (all V_n_2 all V_m_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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3324 (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]. 6.42/6.39 3325 (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]. 6.42/6.39 3326 (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]. 6.42/6.39 3327 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y))))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3328 (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_d)),V_ys_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) & c_List_Olist__all2(T_c,T_d,V_Pa_2,V_us_2,V_vs_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))))) # label(fact_list__all2__append) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3329 (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]. 6.42/6.39 3330 (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]. 6.42/6.39 3331 (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]. 6.42/6.39 3332 (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]. 6.42/6.39 3333 (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]. 6.42/6.39 3334 (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]. 6.42/6.39 3335 (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]. 6.42/6.39 3336 (all V_b 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)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_m)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3337 (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]. 6.42/6.39 3338 (all V_r_H all V_q_H all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> V_r = V_r_H)))) # label(fact_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3339 (all V_v all V_b all V_a all T_b (class_Rings_Oring(T_b) & class_Int_Onumber(T_b) -> hAPP(hAPP(c_Groups_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))) = 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)))) # label(fact_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3340 (all V_p_2 all T_c all T_d 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) = hAPP(c_Product__Type_Osnd(T_d,T_c),V_p_2)) # label(fact_snd__def) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3341 (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_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)) = 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))) # label(fact_map__update) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3342 (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]. 6.42/6.39 3343 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_d (V_B_2 = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_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)) = hAPP(c_Set_OPow(T_c),V_B_2))) # label(fact_image__Pow__surj) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3344 (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]. 6.42/6.39 3345 (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]. 6.42/6.39 3346 (all V_A_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),hAPP(c_COMBK(T_c,T_d),c_Groups_Ozero__class_Ozero(T_c))),V_A_2))) # label(fact_setsum__0) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3347 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_z)))) # label(fact_sup__assoc) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3348 (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]. 6.42/6.39 3349 (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]. 6.42/6.39 3350 (all 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)) <-> 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))))) # label(fact_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3351 (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]. 6.42/6.39 3352 (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]. 6.42/6.39 3353 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (hAPP(V_f_2,V_x_2) = V_y_2 <-> c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2) = V_f_2)) # label(fact_fun__upd__idem__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3354 (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_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)),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)))) # label(fact_Int__Inter__image) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3355 (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_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]. 6.42/6.39 3356 (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_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))))) <-> 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)))) # label(fact_Image__subset__eq) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3357 (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]. 6.42/6.39 3358 (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_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)) = 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)) & (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]. 6.42/6.39 3359 (all V_xs all V_x all T_a all V_v ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = V_x) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_nth__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3360 (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]. 6.42/6.39 3361 (all V_b_2 all V_a_2 all 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)),c_Relation_OId(T_c))))) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3362 (all T_c c_Wellfounded_Omeasure(T_c) = hAPP(c_Relation_Oinv__image(tc_Nat_Onat,T_c),c_Wellfounded_Oless__than)) # label(fact_measure__def) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3363 (all V_n all V_m ((V_n != hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,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)) & (hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = V_n -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)))) # label(fact_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3364 (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]. 6.42/6.39 3365 (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]. 6.42/6.39 3366 (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 ((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)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))))))) # label(fact_list__ex1__iff) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3367 (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]. 6.42/6.39 3368 (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(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)) = 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))))) # label(fact_listsum__const__mult) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3369 (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]. 6.42/6.39 3370 (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]. 6.42/6.39 3371 (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]. 6.42/6.39 3372 (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]. 6.42/6.39 3373 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (V_y_2 = V_x_2 <-> hAPP(c_Int_Onat,V_y_2) = hAPP(c_Int_Onat,V_x_2))))) # label(fact_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3374 (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]. 6.42/6.39 3375 (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_y),V_x))) # label(fact_inf__commute) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3376 (all V_m all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v))))) # label(fact_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3377 (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]. 6.42/6.39 3378 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> V_xs_2 = 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)))) # label(fact_remove1__insort) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3379 (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]. 6.42/6.39 3380 (all V_j_2 all V_i_2 all V_k_2 hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))) # label(fact_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3381 (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]. 6.42/6.39 3382 (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]. 6.42/6.39 3383 (all V_Q all V_P all T_b all T_a V_P = hAPP(hAPP(c_COMBK(T_a,T_b),V_P),V_Q)) # label(help_c__COMBK__1) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3384 (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) -> V_x_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)))))) # label(fact_folding__one_Osingleton) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3385 (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]. 6.42/6.39 3386 (all V_B_2 all V_A_2 all V_C_2 all T_d all T_c 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__extend__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3387 (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_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_ab__diff__minus) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3388 (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]. 6.42/6.39 3389 (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]. 6.42/6.39 3390 (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(T_c),B_x),V_x_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]. 6.42/6.39 3391 (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]. 6.42/6.39 3392 (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]. 6.42/6.39 3393 (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]. 6.42/6.39 3394 (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]. 6.42/6.39 3395 (all V_ys_2 all T_c all V_xs_2 (c_List_Olist_ONil(T_c) = V_ys_2 <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = V_xs_2)) # label(fact_self__append__conv) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3396 (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]. 6.42/6.39 3397 (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)) -> ((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]. 6.42/6.39 3398 (all T_c c_Relation_OId(T_c) = hAPP(c_Relation_Oconverse(T_c,T_c),c_Relation_OId(T_c))) # label(fact_converse__Id) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3399 (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]. 6.42/6.39 3400 (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]. 6.42/6.39 3401 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_sup__left__idem) # label(axiom) # label(non_clause). [assumption]. 6.42/6.39 3402 (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]. 6.42/6.39 3403 (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]. 6.43/6.39 3404 (all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = V_a) # label(fact_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3405 (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_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__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Int__Diff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3406 (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_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),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),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]. 6.43/6.39 3407 (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]. 6.43/6.39 3408 (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(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)) = 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))))) # label(fact_listsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3409 (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(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(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),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]. 6.43/6.39 3410 (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]. 6.43/6.39 3411 (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]. 6.43/6.39 3412 (all V_A_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_x_2),V_A_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))))) # label(fact_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3413 (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]. 6.43/6.39 3414 (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]. 6.43/6.39 3415 (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]. 6.43/6.39 3416 (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]. 6.43/6.39 3417 (all V_k all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_k)),V_j) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3418 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3419 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3420 (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]. 6.43/6.39 3421 (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]. 6.43/6.39 3422 (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]. 6.43/6.39 3423 (all T_c c_Relation_Otrans(T_c,c_Relation_OId(T_c))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3424 (all V_z all V_z_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_z_H)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H))))) # label(fact_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3425 (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]. 6.43/6.39 3426 (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]. 6.43/6.39 3427 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3428 (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_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) <-> c_Fun_Obij__betw(T_d,T_e,V_f_H_2,V_A_H_2,V_A_H_H_2)))) # label(fact_bij__betw__comp__iff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3429 (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,hAPP(c_List_Orev(T_c),V_xs_2),hAPP(c_List_Orev(T_d),V_ys_2)) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2))) # label(fact_list__all2__rev) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3430 (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]. 6.43/6.39 3431 (all V_B_2 all V_A_2 all T_c 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)) = 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)) # label(fact_Compl__Diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3432 (all V_y_2 all V_x_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2))) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_y_2 -> V_x_2 = c_Nitpick_Onat__gcd(V_x_2,V_y_2)) & (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]. 6.43/6.39 3433 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_a))) # label(fact_min__max_Osup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3434 (all T_a (class_Rings_Olinordered__idom(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3435 (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]. 6.43/6.39 3436 (all V_xq_2 all V_s_2 all T_c hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_s_2),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(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))) # label(fact_lazy__sequence__size__code) # label(axiom) # label(non_clause). [assumption]. 6.43/6.39 3437 (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]. 6.43/6.40 3438 (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_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2)))) # label(fact_ivl__disj__un_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3439 (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]. 6.43/6.40 3440 (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]. 6.43/6.40 3441 (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)) -> c_List_Oupt(V_i,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_k)) = 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))))) # label(fact_upt__add__eq__append) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3442 (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]. 6.43/6.40 3443 (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),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(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_z_2))))) # label(fact_max__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3444 (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]. 6.43/6.40 3445 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3446 (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]. 6.43/6.40 3447 (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_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2)))) <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2)))) # label(fact_eq__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3448 (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]. 6.43/6.40 3449 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(c_Product__Type_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))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) # label(fact_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3450 (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]. 6.43/6.40 3451 (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]. 6.43/6.40 3452 (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]. 6.43/6.40 3453 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Groups_Oone__class_Oone(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_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2))))))) # label(fact_eq__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3454 (all V_Pa_2 all T_c (c_FunDef_Oreduction__pair(T_c,V_Pa_2) <-> 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))))) # label(fact_reduction__pair__def) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3455 (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]. 6.43/6.40 3456 (all V_com2_H_2 all V_com1_H_2 all V_com2_2 all V_com1_2 (c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) = c_Com_Ocom_OSemi(V_com1_2,V_com2_2) <-> V_com1_2 = V_com1_H_2 & V_com2_2 = V_com2_H_2)) # label(fact_com_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3457 (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]. 6.43/6.40 3458 (all V_f_2 all T_d all T_c all V_xs_2 (c_List_Olist_ONil(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)) = hAPP(V_f_2,c_List_Olast(T_c,V_xs_2)))) # label(fact_last__map) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3459 (all T_c all V_ys_2 all V_xs_2 (c_List_Olist__all2(T_c,T_c,c_fequal,V_xs_2,V_ys_2) <-> V_ys_2 = V_xs_2)) # label(fact_list__all2__eq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3460 (all V_xs all V_i all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__take) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3461 (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]. 6.43/6.40 3462 (all T_c (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)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_finite__option__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3463 (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]. 6.43/6.40 3464 (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]. 6.43/6.40 3465 (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_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) # label(fact_listrel__eq__len) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3466 (all V_Pa_2 all V_n_2 ((exists B_x (hBOOL(hAPP(V_Pa_2,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))))) <-> (exists B_m (hBOOL(hAPP(V_Pa_2,B_m)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2)))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3467 (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]. 6.43/6.40 3468 (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)) -> 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)) = V_A_2))) # label(fact_double__diff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3469 (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]. 6.43/6.40 3470 (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]. 6.43/6.40 3471 (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]. 6.43/6.40 3472 (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 -> 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)) = 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))))))))))))) # label(fact_inf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3473 (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]. 6.43/6.40 3474 (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),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)),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_divide__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3475 (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]. 6.43/6.40 3476 (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),V_B_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)))) # label(fact_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3477 (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]. 6.43/6.40 3478 (all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> V_A_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)))) # label(fact_Union__quotient) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3479 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d c_List_Omaps(T_d,T_c,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2)) = 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))) # label(fact_maps__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3480 (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]. 6.43/6.40 3481 (all V_k c_Int_OBit0(V_k) != c_Int_OMin) # label(fact_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3482 (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]. 6.43/6.40 3483 (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) = hAPP(V_b_2,V_a_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))))) # label(fact_setsum__delta_H) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3484 (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]. 6.43/6.40 3485 (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]. 6.43/6.40 3486 (all V_w all V_z hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3487 (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__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)))) <-> (-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_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__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),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)))))) # label(fact_le__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3488 (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]. 6.43/6.40 3489 (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]. 6.43/6.40 3490 (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]. 6.43/6.40 3491 (all V_n_2 all V_m_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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2)))) # label(fact_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3492 (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]. 6.43/6.40 3493 (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]. 6.43/6.40 3494 (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]. 6.43/6.40 3495 (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]. 6.43/6.40 3496 (all V_b all V_a all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> (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_eq__divide__imp) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3497 (all V_a_2 all T_c (class_Rings_Olinordered__idom(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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_sgn__1__neg) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3498 (all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> 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) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3499 (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]. 6.43/6.40 3500 (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]. 6.43/6.40 3501 (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]. 6.43/6.40 3502 (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]. 6.43/6.40 3503 (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]. 6.43/6.40 3504 (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]. 6.43/6.40 3505 (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]. 6.43/6.40 3506 (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]. 6.43/6.40 3507 (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) <-> (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))))))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3508 (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(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(c_Nat_OSuc,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))))) # label(fact_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3509 (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_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) <-> 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)))))))) # label(fact_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3510 (all V_f_2 all T_e all T_d all T_c V_f_2 = 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))) # label(fact_curry__split) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3511 (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]. 6.43/6.40 3512 (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]. 6.43/6.40 3513 (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]. 6.43/6.40 3514 (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]. 6.43/6.40 3515 (all V_z all V_x all V_y all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (V_y != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y)))) # label(fact_add__frac__num) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3516 (all V_xs all V_n all T_a hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_length__drop) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3517 (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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3518 (all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OMin),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)) # label(fact_mult__Min) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3519 (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]. 6.43/6.40 3520 (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]. 6.43/6.40 3521 (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]. 6.43/6.40 3522 (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]. 6.43/6.40 3523 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d 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)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_prod_Orecs) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3524 (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]. 6.43/6.40 3525 (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) -> (hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2) = V_g_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]. 6.43/6.40 3526 (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]. 6.43/6.40 3527 (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]. 6.43/6.40 3528 (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_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)) = 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)))) # label(fact_times__divide__times__eq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3529 (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),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b))) # label(fact_mod__minus__eq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3530 (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_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)) = 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)) # label(fact_UN__o) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3531 (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]. 6.43/6.40 3532 (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 -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2)) & (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Set_Oinsert(T_d),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2)))) # label(fact_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3533 (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]. 6.43/6.40 3534 (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(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)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)),V_a_2),V_A_2)))) # label(fact_fold1__eq__fold__idem) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3535 (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]. 6.43/6.40 3536 (all T_c 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))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Union__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3537 (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]. 6.43/6.40 3538 (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]. 6.43/6.40 3539 (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]. 6.43/6.40 3540 (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]. 6.43/6.40 3541 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (hBOOL(hAPP(V_Pa_2,V_x_2)) & c_List_Olist__all(T_c,V_Pa_2,V_xs_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]. 6.43/6.40 3542 (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_x_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_x_2))))) # label(fact_folding_Ocommute__comp_H) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3543 (all V_R_2 all T_d all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_d),V_R_2),c_Relation_OId(T_d)) = V_R_2) # label(fact_R__O__Id) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3544 (all V_y all V_i all T_a hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_y) = c_List_Olist_ONil(T_a)) # label(fact_list__update__code_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3545 (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_S_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2)))))) # label(fact_setprod__mono__one__right) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3546 (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]. 6.43/6.40 3547 (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]. 6.43/6.40 3548 (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]. 6.43/6.40 3549 (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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),V_e_2)),V_c_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__iff1) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3550 (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]. 6.43/6.40 3551 (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]. 6.43/6.40 3552 (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]. 6.43/6.40 3553 (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]. 6.43/6.40 3554 (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_Groups_Ozero__class_Ozero(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)))) # label(fact_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3555 (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]. 6.43/6.40 3556 (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,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))) # label(fact_mod__geq) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3557 (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_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))) = 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))) # label(fact_Int__UN__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3558 (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]. 6.43/6.40 3559 (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 (V_v_2 = hAPP(hAPP(V_f_2,V_x_2),B_y) -> -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]. 6.43/6.40 3560 (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]. 6.43/6.40 3561 (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]. 6.43/6.40 3562 (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]. 6.43/6.40 3563 (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]. 6.43/6.40 3564 (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)) -> 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]. 6.43/6.40 3565 (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]. 6.43/6.40 3566 (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)))) -> 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))) = c_Divides_OposDivAlg(V_a_2,V_b_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2) = c_Divides_OposDivAlg(V_a_2,V_b_2)))) # label(fact_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3567 (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]. 6.43/6.40 3568 (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]. 6.43/6.40 3569 (all T_c c_Fun_Oid(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))) # label(fact_inv__id) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3570 (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]. 6.43/6.40 3571 (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]. 6.43/6.40 3572 (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]. 6.43/6.40 3573 (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]. 6.43/6.40 3574 (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]. 6.43/6.40 3575 (all V_code__numeral_2 all V_f2_2 all V_f1_2 all T_c 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)) = 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))) # label(fact_code__numeral_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3576 (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]. 6.43/6.40 3577 (all V_f_2 all T_c all T_e all T_d c_Product__Type_Omap__pair(T_c,T_c,T_d,T_e,c_Fun_Oid(T_c),V_f_2) = c_Product__Type_Oapsnd(T_d,T_e,T_c,V_f_2)) # label(fact_apsnd__def) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3578 (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]. 6.43/6.40 3579 (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]. 6.43/6.40 3580 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(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_Oabs__class_Oabs(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3581 (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),c_Set_Oinsert(T_c)),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_union__fold__insert) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3582 (all V_A_2 all T_d all T_c 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),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_Sigma__empty2) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3583 (all V_n all V_z all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)),V_n))) # label(fact_of__int__power) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3584 (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]. 6.43/6.40 3585 (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]. 6.43/6.40 3586 (all V_z_2 all V_Pa_2 all T_c all T_e all T_d 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)) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_Pa_2),V_z_2)) # label(fact_split__beta) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3587 (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]. 6.43/6.40 3588 (all V_g_2 all V_f_2 ((all B_x hAPP(V_f_2,B_x) = hAPP(V_g_2,B_x)) -> V_g_2 = V_f_2)) # label(fact_ext) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3589 (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(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]. 6.43/6.40 3590 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__top(T_c) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_y_2) = c_Orderings_Otop__class_Otop(T_c) <-> c_Orderings_Otop__class_Otop(T_c) = V_y_2 & c_Orderings_Otop__class_Otop(T_c) = V_x_2))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3591 (all V_m all V_x 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) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_x),V_m)) # label(fact_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3592 (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))) -> 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__nonzero) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3593 (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),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),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__div__neg) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3594 (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]. 6.43/6.40 3595 (all V_a all V_b (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))))) # label(fact_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 6.43/6.40 3596 (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]. 6.43/6.40 3597 (all V_A_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))) <-> (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)))))) # label(fact_in__lists__conv__set) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3598 (all V_v_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2)))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3599 (all V_y_2 all V_x_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)),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)))) <-> V_x_2 != c_Groups_Ozero__class_Ozero(T_c) | V_y_2 != c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3600 (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) -> (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]. 6.44/6.40 3601 (all V_n all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)))),V_n) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m)),V_n)) # label(fact_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3602 (all V_B_2 all V_A_2 all T_c hAPP(c_Relation_OImage(T_c,T_c,c_Relation_OId__on(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_Image__Id__on) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3603 (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]. 6.44/6.40 3604 (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]. 6.44/6.40 3605 (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,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)))) # label(fact_Min_OF__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3606 (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]. 6.44/6.40 3607 (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]. 6.44/6.40 3608 (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]. 6.44/6.40 3609 (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]. 6.44/6.40 3610 (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]. 6.44/6.40 3611 (all V_a_2 all V_times_2 all V_one_2 all T_c 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)) = V_one_2) # label(fact_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3612 (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]. 6.44/6.40 3613 (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]. 6.44/6.40 3614 (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]. 6.44/6.40 3615 (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]. 6.44/6.40 3616 (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]. 6.44/6.40 3617 (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_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_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)) # label(fact_INT__extend__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3618 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n))) # label(fact_power__abs) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3619 (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_b_2)),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)),V_b_2))))) # label(fact_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3620 (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]. 6.44/6.40 3621 (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]. 6.44/6.40 3622 (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]. 6.44/6.40 3623 (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]. 6.44/6.40 3624 (all V_y_2 all V_x_2 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,V_x_2)),V_y_2) = 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))) # label(fact_apfst__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.40 3625 (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_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)) -> 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]. 6.44/6.40 3626 (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]. 6.44/6.40 3627 (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]. 6.44/6.41 3628 (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]. 6.44/6.41 3629 (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]. 6.44/6.41 3630 (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_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))) & (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_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]. 6.44/6.41 3631 (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]. 6.44/6.41 3632 (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_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) = 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))) # label(fact_UN__extend__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3633 (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]. 6.44/6.41 3634 (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]. 6.44/6.41 3635 (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]. 6.44/6.41 3636 (all V_f_2 all V_r_2 all T_c all T_d hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),V_r_2),V_f_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(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)))) # label(fact_inv__image__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3637 (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]. 6.44/6.41 3638 (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_Olist_OCons(T_c),V_a_2),V_xs_2) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_a_2),V_xs_2)))) # label(fact_insort__is__Cons) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3639 (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]. 6.44/6.41 3640 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_a_2 = V_b_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))) # label(fact_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3641 (all V_A_2 all T_c (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))))) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3642 (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]. 6.44/6.41 3643 (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]. 6.44/6.41 3644 (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]. 6.44/6.41 3645 (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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Compl__partition) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3646 (all V_r_2 all T_c hAPP(c_List_Olexn(T_c,V_r_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool))) # label(fact_lexn_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3647 (all V_xs_2 all V_n_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct__rotate) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3648 (all V_B_2 all V_A_2 (V_B_2 = V_A_2 <-> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2))) # label(fact_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3649 (all V_ys all V_xs all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oconcat(T_a,V_xs)),c_List_Oconcat(T_a,V_ys)) = c_List_Oconcat(T_a,hAPP(hAPP(c_List_Oappend(tc_List_Olist(T_a)),V_xs),V_ys))) # label(fact_concat__append) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3650 (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]. 6.44/6.41 3651 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Orev(T_c),V_xs_2)) = hAPP(c_List_Orev(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_rev__filter) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3652 (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]. 6.44/6.41 3653 (all V_n_2 all V_x_2 all T_c (class_Fields_Ofield(T_c) -> (V_x_2 != c_Groups_Oone__class_Oone(T_c) -> 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]. 6.44/6.41 3654 (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]. 6.44/6.41 3655 (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]. 6.44/6.41 3656 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs))) # label(fact_take__tl) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3657 (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]. 6.44/6.41 3658 (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]. 6.44/6.41 3659 (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]. 6.44/6.41 3660 (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]. 6.44/6.41 3661 (all V_xs_2 all V_f_2 all V_Pa_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,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = 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)))) # label(fact_filter__sort) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3662 (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]. 6.44/6.41 3663 (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_j != V_i -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j)) & (V_j = V_i -> V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j)))) # label(fact_nth__list__update) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3664 (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]. 6.44/6.41 3665 (all V_b_2 all V_a_2 all V_fb_2 all V_fa_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_fa_2,V_a_2)),hAPP(V_fb_2,V_b_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = 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))) # label(fact_pair__box_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3666 (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]. 6.44/6.41 3667 (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]. 6.44/6.41 3668 (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)),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)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I20_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3669 (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]. 6.44/6.41 3670 (all T_c all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_equalityD2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3671 (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]. 6.44/6.41 3672 (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_e,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_list_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_list_2))) # label(fact_List_Omap_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3673 (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_k,V_x)),c_List_Oreplicate(T_a,V_n,V_x)) = 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))) # label(fact_append__replicate__commute) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3674 (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]. 6.44/6.41 3675 (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)) -> 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]. 6.44/6.41 3676 (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]. 6.44/6.41 3677 (all V_l_2 (c_Int_OPls = c_Int_OBit0(V_l_2) <-> c_Int_OPls = V_l_2)) # label(fact_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3678 (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]. 6.44/6.41 3679 (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]. 6.44/6.41 3680 (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]. 6.44/6.41 3681 (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 -> 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__idem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3682 (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]. 6.44/6.41 3683 (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),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2)))) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))))) # label(fact_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3684 (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_c_2),V_a_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_c_2),V_a_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_d_2))))))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3685 (all V_n_2 all V_m_2 all 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))) <-> 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)))) # label(fact_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3686 (all V_k_2 (c_Int_OBit1(V_k_2) = c_Int_OMin <-> V_k_2 = c_Int_OMin)) # label(fact_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3687 (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__fun__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3688 (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]. 6.44/6.41 3689 (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(c_Groups_Ouminus__class_Ouminus(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))) # label(fact_disjoint__eq__subset__Compl) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3690 (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]. 6.44/6.41 3691 (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]. 6.44/6.41 3692 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))) # label(fact_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3693 (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]. 6.44/6.41 3694 (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]. 6.44/6.41 3695 (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]. 6.44/6.41 3696 (all V_n_2 all V_m_2 hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),c_Nat_OSuc),c_List_Oupt(V_m_2,V_n_2)) = c_List_Oupt(hAPP(c_Nat_OSuc,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_map__Suc__upt) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3697 (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]. 6.44/6.41 3698 (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]. 6.44/6.41 3699 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3700 (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]. 6.44/6.41 3701 (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]. 6.44/6.41 3702 (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]. 6.44/6.41 3703 (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__eq(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)))))) # label(fact_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3704 (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_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_power__diff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3705 (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)))) <-> (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))))))) # label(fact_all__set__conv__all__nth) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3706 (all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Otop__class_Otop(T_a))) # label(fact_compl__bot__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3707 (all V_n all V_a all T_a (class_Power_Opower(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),hAPP(c_Nat_OSuc,V_n)))) # label(fact_power__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3708 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a) -> V_c = V_b))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3709 (all 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)) = c_Divides_OposDivAlg(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b)) # label(fact_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3710 (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]. 6.44/6.41 3711 (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]. 6.44/6.41 3712 (all V_y_2 all V_x_2 (hAPP(c_Code__Numeral_Onat__of,V_x_2) = hAPP(c_Code__Numeral_Onat__of,V_y_2) <-> V_x_2 = V_y_2)) # label(fact_code__numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3713 (all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) # label(fact_length__rev) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3714 (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]. 6.44/6.41 3715 (all V_f_2 all V_x_2 all T_c all T_d all V_i_2 ((V_i_2 != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> 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)) = hAPP(c_New__DSequence_Opos__decr__bind(T_d,T_c,V_x_2,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]. 6.44/6.41 3716 (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]. 6.44/6.41 3717 (all V_n all V_i all V_m hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_i)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n)),V_i)) # label(fact_min__diff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3718 (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]. 6.44/6.41 3719 (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]. 6.44/6.41 3720 (all V_r_2 all T_c (c_Relation_Oantisym(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)) -> B_y = B_x))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3721 (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]. 6.44/6.41 3722 (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]. 6.44/6.41 3723 (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]. 6.44/6.41 3724 (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]. 6.44/6.41 3725 (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) -> (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_B_2),V_D_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_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]. 6.44/6.41 3726 (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]. 6.44/6.41 3727 (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)) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,B_x)))) -> 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_f_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))))) # label(fact_listsum__map__filter) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3728 (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]. 6.44/6.41 3729 (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_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_g_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_g_2,B_x) = hAPP(V_f_2,B_x))))) # label(fact_map__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3730 (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 (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)))) & (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(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]. 6.44/6.41 3731 (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]. 6.44/6.41 3732 (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__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)) = 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)) # label(fact_New__Random__Sequence_Opos__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3733 (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]. 6.44/6.41 3734 (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__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3735 (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]. 6.44/6.41 3736 (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__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_b_1))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> V_y = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a_1,V_b_1))))) # label(fact_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3737 (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_fun2_2 = V_fun2_H_2 & V_com_2 = V_com_H_2 & V_fun1_2 = V_fun1_H_2)) # label(fact_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3738 (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__iff2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3739 (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]. 6.44/6.41 3740 (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]. 6.44/6.41 3741 (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]. 6.44/6.41 3742 (all T_d all T_c c_Orderings_Otop__class_Otop(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),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_UNIV__Times__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3743 (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]. 6.44/6.41 3744 (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]. 6.44/6.41 3745 (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]. 6.44/6.41 3746 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ODomain(T_c,T_c),c_Relation_OId(T_c))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3747 (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]. 6.44/6.41 3748 (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]. 6.44/6.41 3749 (all V_l c_Int_OMin != c_Int_OBit0(V_l)) # label(fact_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3750 (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]. 6.44/6.41 3751 (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]. 6.44/6.41 3752 (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) -> 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]. 6.44/6.41 3753 (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]. 6.44/6.41 3754 (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]. 6.44/6.41 3755 (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]. 6.44/6.41 3756 (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]. 6.44/6.41 3757 (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_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)) = 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)))) # label(fact_fold__image__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3758 (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]. 6.44/6.41 3759 (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]. 6.44/6.41 3760 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Osgn__class_Osgn(T_a,c_Groups_Osgn__class_Osgn(T_a,V_a)) = c_Groups_Osgn__class_Osgn(T_a,V_a))) # label(fact_sgn__sgn) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3761 (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]. 6.44/6.41 3762 (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),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),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))))) # label(fact_pos__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3763 (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]. 6.44/6.41 3764 (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)))) -> V_n_2 = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) # label(fact_length__n__lists__elem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3765 (all V_vs all V_ys all V_us all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_us) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_vs) -> 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]. 6.44/6.41 3766 (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]. 6.44/6.41 3767 (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]. 6.44/6.41 3768 (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]. 6.44/6.41 3769 (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_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]. 6.44/6.41 3770 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_f1_2 all T_c all T_d hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2) = 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))) # label(fact_triple_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3771 (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]. 6.44/6.41 3772 (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_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_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]. 6.44/6.41 3773 (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]. 6.44/6.41 3774 (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]. 6.44/6.41 3775 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> 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)))) = V_x_2)) # label(fact_min__max_OInf__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3776 (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]. 6.44/6.41 3777 (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]. 6.44/6.41 3778 (all V_m all V_y all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_y,V_m)),V_m)) # label(fact_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3779 (all V_n all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) # label(fact_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3780 (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]. 6.44/6.41 3781 (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]. 6.44/6.41 3782 (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]. 6.44/6.41 3783 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)))))) # label(fact_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3784 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3785 (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)) -> 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)) = V_xs_2))))) # label(fact_insort__key__remove1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3786 (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]. 6.44/6.41 3787 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_a)))) # label(fact_div__self) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3788 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_k_2)))) # label(fact_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3789 (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]. 6.44/6.41 3790 (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,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_Oremdups(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_Oremdups(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = c_List_Oremdups(T_c,V_xs_2)))) # label(fact_remdups_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3791 (all V_xs_2 all V_p_2 all T_c hAPP(c_List_Ofilter(T_c,V_p_2),c_List_Oconcat(T_c,V_xs_2)) = 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))) # label(fact_filter__concat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3792 (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]. 6.44/6.41 3793 (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]. 6.44/6.41 3794 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord_Omin(T_c,c_Orderings_Oord__class_Oless__eq(T_c)) = c_Orderings_Oord__class_Omin(T_c))) # label(fact_min__ord__min) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3795 (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]. 6.44/6.41 3796 (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]. 6.44/6.41 3797 (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]. 6.44/6.41 3798 (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_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_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))))))) # label(fact_setsum__diff__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3799 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),B_x),B_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(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)) -> c_Big__Operators_Olattice__class_OInf__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_OInf__fin(T_c,V_N_2))))))) # label(fact_Inf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3800 (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))) -> c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))) & (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]. 6.44/6.41 3801 (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_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) = 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))) # label(fact_split__twice) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3802 (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]. 6.44/6.41 3803 (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]. 6.44/6.41 3804 (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 & c_List_Olinorder__class_Osorted(T_c,B_x) & hBOOL(hAPP(c_List_Odistinct(T_c),B_x)) & (all B_y (c_List_Olinorder__class_Osorted(T_c,B_y) & hBOOL(hAPP(c_List_Odistinct(T_c),B_y)) & hAPP(c_List_Oset(T_c),B_y) = V_A_2 -> B_y = B_x))))))) # label(fact_finite__sorted__distinct__unique) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3805 (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]. 6.44/6.41 3806 (all V_i_1 all V_x all T_a hAPP(c_New__DSequence_Oneg__single(T_a,V_x),V_i_1) = c_Lazy__Sequence_Ohb__single(T_a,V_x)) # label(fact_New__DSequence_Oneg__single__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.41 3807 (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]. 6.44/6.42 3808 (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]. 6.44/6.42 3809 (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]. 6.44/6.42 3810 (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]. 6.44/6.42 3811 (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]. 6.44/6.42 3812 (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)) & hAPP(V_f_2,B_i) = V_k_2)))))) # label(fact_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3813 (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]. 6.44/6.42 3814 (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]. 6.44/6.42 3815 (all V_w_2 all V_m_2 (hAPP(c_Int_Onat,V_w_2) = V_m_2 <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = 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)))) # label(fact_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3816 (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_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(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_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3817 (all T_c c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_card_Oempty) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3818 (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]. 6.44/6.42 3819 (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]. 6.44/6.42 3820 (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_H_2 = V_com_2 & V_fun_2 = V_fun_H_2 & V_loc_H_2 = V_loc_2 <-> c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) = c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2))) # label(fact_com_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3821 (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]. 6.44/6.42 3822 (all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_n)) -> V_i = 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)))) # label(fact_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3823 (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]. 6.44/6.42 3824 (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]. 6.44/6.42 3825 (all V_x all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(c_List_Ohd(T_a),c_List_Oreplicate(T_a,V_n,V_x)) = V_x)) # label(fact_hd__replicate) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3826 (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))) -> ((B_y != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 6.44/6.42 3827 (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_y = B_z)))))) # label(fact_single__valued__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3828 (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]. 6.44/6.42 3829 (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]. 6.44/6.42 3830 (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]. 6.44/6.42 3831 (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]. 6.44/6.42 3832 (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]. 6.44/6.42 3833 (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)) -> 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)) = V_S_2))) # label(fact_inv__into__image__cancel) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3834 (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]. 6.44/6.42 3835 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (c_List_Olist__ex(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__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3836 (all V_v_2 all V_i_2 all V_xs_2 all V_x_2 all T_c 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) = 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)) # label(fact_list__update_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3837 (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)),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__eq__Union__image) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3838 (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]. 6.44/6.42 3839 (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 -> 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)) = 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))))))))) # label(fact_inf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3840 (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]. 6.44/6.42 3841 (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)) -> c_List_Oupto(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))))) # label(fact_upto_Opsimps) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3842 (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_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_a_2))))) # label(fact_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3843 (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]. 6.44/6.42 3844 (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]. 6.44/6.42 3845 (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]. 6.44/6.42 3846 (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]. 6.44/6.42 3847 (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]. 6.44/6.42 3848 (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]. 6.44/6.42 3849 (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]. 6.44/6.42 3850 (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),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__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),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]. 6.44/6.42 3851 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)))) # label(fact_min__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3852 (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]. 6.44/6.42 3853 (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)) -> c_Orderings_Obot__class_Obot(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)) & (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_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_vimage__const) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3854 (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]. 6.44/6.42 3855 (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]. 6.44/6.42 3856 (all T_d all T_c c_Fun_Oid(tc_prod(T_c,T_d)) = 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))) # label(fact_split__Pair) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3857 (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(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)) & (-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)))) # label(fact_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3858 (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]. 6.44/6.42 3859 (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]. 6.44/6.42 3860 (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]. 6.44/6.42 3861 (all V_S_2 all V_R_2 all T_d all T_c (V_S_2 = 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_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))) # label(fact_pred__equals__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3862 (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]. 6.44/6.42 3863 (all V_xs_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2)) = c_List_Ofoldr(tc_List_Olist(T_c),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_c)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xs_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_length__transpose) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3864 (all V_m all V_n hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,V_n),V_m) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_minus__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3865 (all V_xs_2 all T_c V_xs_2 = hAPP(hAPP(c_List_Omap(T_c,T_c),c_COMBI(T_c)),V_xs_2)) # label(fact_map__ident) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3866 (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]. 6.44/6.42 3867 (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]. 6.44/6.42 3868 (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]. 6.44/6.42 3869 (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]. 6.44/6.42 3870 (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]. 6.44/6.42 3871 (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]. 6.44/6.42 3872 (all V_n V_n = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_max__0R) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3873 (all T_1 class_Nat_Osize(tc_List_Olist(T_1))) # label(arity_List__Olist__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3874 (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]. 6.44/6.42 3875 (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]. 6.44/6.42 3876 (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]. 6.44/6.42 3877 (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]. 6.44/6.42 3878 (all V_xs_2 all V_Pa_2 all T_c ((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))))) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex__length) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3879 (all V_x all V_y ((V_y != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 6.44/6.42 3880 (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]. 6.44/6.42 3881 (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]. 6.44/6.42 3882 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2))) <-> 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)))))) # label(fact_le__minus__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3883 (all V_n_2 all 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)) <-> V_m_2 != V_n_2)) # label(fact_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3884 (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]. 6.44/6.42 3885 (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]. 6.44/6.42 3886 (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]. 6.44/6.42 3887 (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]. 6.44/6.42 3888 (all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__minus) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3889 (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_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))))) # label(fact_pred__subset__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3890 (all V_w_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_w_2)))))) # label(fact_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3891 (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),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(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)) = c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3892 (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]. 6.44/6.42 3893 (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]. 6.44/6.42 3894 (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_x_2 = V_y_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]. 6.44/6.42 3895 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_d 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)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),hAPP(c_Relation_ORange(T_d,T_c),V_r_2))) # label(fact_Range__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3896 (all V_A_2 all V_b_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (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))) <-> (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))))))) # label(fact_le__Inf__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3897 (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]. 6.44/6.42 3898 (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)) -> 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))) & (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)) = V_c_2))) # label(fact_UN__constant) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3899 (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]. 6.44/6.42 3900 (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,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)) & (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)) = c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))) # label(fact_dropWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3901 (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]. 6.44/6.42 3902 (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_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)) = 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)) # label(fact_apfst__compose) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3903 (all V_n V_n != hAPP(c_Nat_OSuc,V_n)) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3904 (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]. 6.44/6.42 3905 (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_Power_Opower__class_Opower(T_d),V_c_2),hAPP(c_Finite__Set_Ocard(T_c),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))))) # label(fact_setprod__gen__delta) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3906 (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]. 6.44/6.42 3907 (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]. 6.44/6.42 3908 (all T_a c_List_Olist_ONil(T_a) = hAPP(c_List_Otl(T_a),c_List_Olist_ONil(T_a))) # label(fact_tl_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3909 (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]. 6.44/6.42 3910 (all V_k all V_b all V_a all T_a (class_Rings_Odvd(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_k) = V_a -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_b),V_a))))) # label(fact_dvdI) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3911 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3912 (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]. 6.44/6.42 3913 (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(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__max_OSup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3914 (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_T2 != B_T1 -> (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]. 6.44/6.42 3915 (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)))) -> 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))) & (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,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_2)))) # label(fact_Min_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3916 (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(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))) <-> 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)))) # label(fact_inf__Int__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3917 (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]. 6.44/6.42 3918 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_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)) = V_a)) # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3919 (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) -> (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))) <-> (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)))))))) # label(fact_less__SUP__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3920 (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_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_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]. 6.44/6.42 3921 (all V_r_2 all T_c all T_d hAPP(c_Relation_ODomain(T_c,T_d),V_r_2) = hAPP(c_Relation_ORange(T_d,T_c),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2))) # label(fact_Range__converse) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3922 (all V_f2_2 all V_f1_2 all T_c c_Nat_Onat_Onat__case(T_c,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_f1_2) # label(fact_nat__case__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3923 (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]. 6.44/6.42 3924 (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]. 6.44/6.42 3925 (all V_j_2 all V_i_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)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2)),hAPP(c_Nat_OSuc,V_j_2))) # label(fact_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3926 (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]. 6.44/6.42 3927 (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]. 6.44/6.42 3928 (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]. 6.44/6.42 3929 (all V_br all V_bq all V_b all V_ar all V_aq all V_c all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_aq),V_ar))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_bq),V_br))) -> (V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br),V_c)))))))) # label(fact_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3930 (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(T_c),V_y_2),V_x_2)) | V_y_2 = V_x_2))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3931 (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,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_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3932 (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_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3933 (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]. 6.44/6.42 3934 (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))) -> c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_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))) & (-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]. 6.44/6.42 3935 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_finite__Pow__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3936 (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]. 6.44/6.42 3937 (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_d,T_c),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_e,T_c),V_g_2),V_ys_2) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_e)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2))) # label(fact_map__eq__imp__length__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3938 (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]. 6.44/6.42 3939 (all V_ys_2 all V_xs_2 all T_c (hAPP(c_List_Orev(T_c),V_ys_2) = hAPP(c_List_Orev(T_c),V_xs_2) <-> V_xs_2 = V_ys_2)) # label(fact_rev__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3940 (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]. 6.44/6.42 3941 (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)) -> c_Groups_Ozero__class_Ozero(T_d) = 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))))) # label(fact_setsum__delta) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3942 (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))),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)) = 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)) # label(fact_UN__UN__finite__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3943 (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_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),hAPP(V_f_2,V_x_2)))) # label(fact_inv__f__f) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3944 (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]. 6.44/6.42 3945 (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]. 6.44/6.42 3946 (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]. 6.44/6.42 3947 (all V_xs all V_x all T_a all V_n ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_List_Odrop(T_a),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_Odrop(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),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]. 6.44/6.42 3948 (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]. 6.44/6.42 3949 (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]. 6.44/6.42 3950 (all V_x all T_a all V_xs ((V_xs != c_List_Olist_ONil(T_a) -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Obutlast(T_a,V_xs)) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) & (c_List_Olist_ONil(T_a) = V_xs -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olist_ONil(T_a)))) # label(fact_butlast_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3951 (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]. 6.44/6.42 3952 (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]. 6.44/6.42 3953 (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),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)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3954 (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]. 6.44/6.42 3955 (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_OatMost(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__atMost) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3956 (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]. 6.44/6.42 3957 (all V_n all V_m hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) # label(fact_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3958 (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]. 6.44/6.42 3959 (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]. 6.44/6.42 3960 (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]. 6.44/6.42 3961 (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)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2) = 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)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2) <-> V_m_2 = hAPP(c_Int_Onat,V_w_2))) # label(fact_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3962 (all V_l_2 all V_Pa_2 all T_c (class_Rings_Odvd(T_c) & class_Rings_Osemiring__0(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(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)))) & hBOOL(hAPP(V_Pa_2,B_x))))))) # label(fact_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3963 (all V_f_2 all T_d all T_c all V_S_2 (V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) -> ((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]. 6.44/6.42 3964 (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]. 6.44/6.42 3965 (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]. 6.44/6.42 3966 (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),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)))) -> 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)))) # label(fact_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3967 (all V_x all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_x)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_x)) = V_x)) # label(fact_mult__sgn__abs) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3968 (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]. 6.44/6.42 3969 (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_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)) = hAPP(hAPP(V_f_2,V_a_2),V_b_2)) # label(fact_split__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3970 (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]. 6.44/6.42 3971 (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))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)) = c_List_Olist_ONil(tc_Nat_Onat)))) # label(fact_upt__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3972 (all V_k_2 hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),V_k_2) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_atLeast__Suc__greaterThan) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3973 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b))) # label(fact_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3974 (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]. 6.44/6.42 3975 (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]. 6.44/6.42 3976 (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]. 6.44/6.42 3977 (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]. 6.44/6.42 3978 (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]. 6.44/6.42 3979 (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]. 6.44/6.42 3980 (all V_k all V_n all V_m 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) = 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))) # label(fact_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3981 (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]. 6.44/6.42 3982 (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]. 6.44/6.42 3983 (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]. 6.44/6.42 3984 (all V_v_2 all V_f_2 all V_a_2 all T_c 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)) = 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)))) # label(fact_nat__case__number__of) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3985 (all V_w all V_z all V_x all V_y all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_z),V_w)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_w)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_x),V_y)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_w),V_z)))))) # label(fact_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3986 (all 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))) = V_m) # label(fact_div__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3987 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_xs) = hAPP(hAPP(c_List_Onth(T_a),V_xs),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_hd__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.44/6.42 3988 (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]. 6.44/6.43 3989 (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]. 6.44/6.43 3990 (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]. 6.44/6.43 3991 (all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> 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)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_abs__divide) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 3992 (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]. 6.44/6.43 3993 (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]. 6.44/6.43 3994 (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)),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(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)) -> 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]. 6.44/6.43 3995 (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)) & (-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_List_Oinsert__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 3996 (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]. 6.44/6.43 3997 (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),V_z_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)))) # label(fact_fun__left__comm__idem_Ofun__left__idem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 3998 (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]. 6.44/6.43 3999 (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]. 6.44/6.43 4000 (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_d,T_c,V_g_2,hAPP(hAPP(c_List_Omap(T_e,T_d),V_f_2),V_xs_2),V_a_2) = 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)) # label(fact_foldr__map) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4001 (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]. 6.44/6.43 4002 (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]. 6.44/6.43 4003 (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))) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))))) # label(fact_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4004 (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]. 6.44/6.43 4005 (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),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),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__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4006 (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]. 6.44/6.43 4007 (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_OSup__fin(T_c,c_Orderings_Oord__class_Omax(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)))) -> 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),V_A_2)))))) # label(fact_min__max_OSup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4008 (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_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4009 (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]. 6.44/6.43 4010 (all V_k_2 all V_f_2 (hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(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]. 6.44/6.43 4011 (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]. 6.44/6.43 4012 (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]. 6.44/6.43 4013 (all V_y all V_x (hAPP(c_Nat_OSuc,V_x) = hAPP(c_Nat_OSuc,V_y) -> V_x = V_y)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4014 (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_arith__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4015 (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]. 6.44/6.43 4016 (all T_c c_Relation_Oantisym(T_c,c_Relation_OId(T_c))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4017 (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)) -> 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) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4018 (all V_B_2 all V_A_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),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)) = 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))) # label(fact_Compl__INT) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4019 (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)) -> (V_A_2 = V_B_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)) = 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))))) # label(fact_Times__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4020 (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(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))))) # label(fact_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4021 (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]. 6.44/6.43 4022 (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]. 6.44/6.43 4023 (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]. 6.44/6.43 4024 (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]. 6.44/6.43 4025 (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_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2) = c_Divides_Odivmod__nat(V_m_2,V_n_2)) & (-(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))) -> c_Divides_Odivmod__nat(V_m_2,V_n_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat)),c_Nat_OSuc)),c_Divides_Odivmod__nat(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2),V_n_2))))) # label(fact_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4026 (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]. 6.44/6.43 4027 (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_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))))) # label(fact_INT__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4028 (all V_u_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Int_Onat,V_u_2))) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2))) # label(fact_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4029 (all V_j_2 all V_i_2 all V_Pa_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) <-> c_List_Olist__ex(tc_Int_Oint,V_Pa_2,c_List_Oupto(V_i_2,V_j_2)))) # label(fact_list__ex__iff__not__all__inverval__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4030 (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__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4031 (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]. 6.44/6.43 4032 (all V_b_2 c_Divides_Oadjust(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)))))))) # label(fact_adjust__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4033 (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]. 6.44/6.43 4034 (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]. 6.44/6.43 4035 (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)),V_A_2),V_B_2) = 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)))))) # label(fact_UN__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4036 (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]. 6.44/6.43 4037 (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]. 6.44/6.43 4038 (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]. 6.44/6.43 4039 (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]. 6.44/6.43 4040 (all V_r_2 all V_A_2 all T_c (c_Relation_Otrans(T_c,V_r_2) & c_Relation_Osym(T_c,V_r_2) & c_Relation_Orefl__on(T_c,V_A_2,V_r_2) <-> c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2))) # label(fact_equiv__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4041 (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]. 6.44/6.43 4042 (all V_x all V_z all V_y all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)),V_x) = 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)))) # label(fact_min__max_Osup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4043 (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]. 6.44/6.43 4044 (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]. 6.44/6.43 4045 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_drop__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4046 (all V_ys_2 all V_y_2 all V_xs_2 all V_Pa_2 all T_c (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) <-> -hBOOL(hAPP(V_Pa_2,V_y_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)) = V_xs_2)) # label(fact_dropWhile__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4047 (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]. 6.44/6.43 4048 (all V_y_2 all V_n_2 all V_x_2 all V_m_2 all T_c (V_m_2 = V_n_2 & (V_m_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_y_2 = V_x_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]. 6.44/6.43 4049 (all T_a c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a))) # label(fact_list_Osize_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4050 (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]. 6.44/6.43 4051 (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]. 6.44/6.43 4052 (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]. 6.44/6.43 4053 (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_y_2) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2) <-> V_y_2 = V_x_2))) # label(fact_atLeast__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4054 (all V_A_2 all T_c c_Relation_OId__on(T_c,V_A_2) = hAPP(c_Relation_Oconverse(T_c,T_c),c_Relation_OId__on(T_c,V_A_2))) # label(fact_converse__Id__on) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4055 (all V_l_2 all V_k_2 (-(hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2))) <-> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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))) # label(fact_dvd_OgreaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4056 (all V_zs_2 all V_ys_2 all V_xs_2 all V_x_2 all T_c (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) <-> V_ys_2 = c_List_Olist_ONil(T_c) & 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))))) # label(fact_Cons__eq__append__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4057 (all V_b_2 all V_a_2 all V_Pa_2 (-((exists 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(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)))) <-> 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__asm) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4058 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = V_a))) # label(fact_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4059 (all V_A_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),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)) = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)))) # label(fact_Inf__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4060 (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)),c_Orderings_Otop__class_Otop(T_c)),V_A_2) = c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)))) # label(fact_Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4061 (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]. 6.44/6.43 4062 (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]. 6.44/6.43 4063 (all V_n_2 all V_m_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) <-> V_m_2 = V_n_2)) # label(fact_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4064 (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_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_subset__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4065 (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]. 6.44/6.43 4066 (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]. 6.44/6.43 4067 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OPls) = V_k) # label(fact_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4068 (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]. 6.44/6.43 4069 (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))) -> 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))) = c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4070 (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]. 6.44/6.43 4071 (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 -> ((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]. 6.44/6.43 4072 (all V_n_2 all V_R_2 all T_c 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)) = 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_rel__pow__commute) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4073 (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]. 6.44/6.43 4074 (all V_u_2 all V_l_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_l_2)),V_u_2) = c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4075 (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]. 6.44/6.43 4076 (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]. 6.44/6.43 4077 (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),c_Int_OPls)))) <-> c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)))) # label(fact_eq__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4078 (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]. 6.44/6.43 4079 (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]. 6.44/6.43 4080 (all V_x_2 all T_c (class_Groups_Oone(T_c) -> (c_Groups_Oone__class_Oone(T_c) = V_x_2 <-> c_Groups_Oone__class_Oone(T_c) = V_x_2))) # label(fact_one__reorient) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4081 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4082 (all V_xs all T_a all V_ys ((c_List_Olist_ONil(T_a) != V_ys -> hAPP(hAPP(c_List_Oappend(T_a),V_xs),c_List_Obutlast(T_a,V_ys)) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))) & (V_ys = c_List_Olist_ONil(T_a) -> c_List_Obutlast(T_a,V_xs) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))))) # label(fact_butlast__append) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4083 (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) -> hAPP(c_List_Orotate(T_a,V_n),V_xs) = V_xs)) # label(fact_rotate__id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4084 (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]. 6.44/6.43 4085 (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]. 6.44/6.43 4086 (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]. 6.44/6.43 4087 (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]. 6.44/6.43 4088 (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]. 6.44/6.43 4089 (all V_v_2 all V_n_2 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)) = 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))))) # label(fact_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4090 (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]. 6.44/6.43 4091 (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)) -> V_X_2 = hAPP(c_Relation_OImage(T_c,T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_X_2))) # label(fact_Image__closed__trancl) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4092 (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]. 6.44/6.43 4093 (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]. 6.44/6.43 4094 (all V_w_2 (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))) <-> 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_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4095 (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))))) -> 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)) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4096 (all V_w all V_v hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))) # label(fact_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4097 (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]. 6.44/6.43 4098 (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]. 6.44/6.43 4099 (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_Omin(T_a),V_x),V_y)) = 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)))) # label(fact_minus__min__eq__max) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4100 (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_x_2) = V_y_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]. 6.44/6.43 4101 (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]. 6.44/6.43 4102 (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 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4103 (all V_B_2 all V_A_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_B_2)) -> hBOOL(hAPP(V_A_2,B_x)))) <-> hBOOL(hAPP(c_Predicate_OPowp(T_c,V_A_2),V_B_2)))) # label(fact_Powp__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4104 (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]. 6.44/6.43 4105 (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(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(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))))) # label(fact_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4106 (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_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))) = 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))))) # label(fact_UN__Times__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4107 (all V_B_2 all V_A_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(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),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2))) # label(fact_Int__Union) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4108 (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]. 6.44/6.43 4109 (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]. 6.44/6.43 4110 (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]. 6.44/6.43 4111 (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(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),V_b_2),V_b_H_2)),V_s_2)) & V_a_H_2 = V_a_2 | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_a_H_2)),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),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))))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4112 (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(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]. 6.44/6.43 4113 (all V_r_2 all T_c (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))) <-> c_Predicate_Otransp(T_c,V_r_2))) # label(fact_transp__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4114 (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]. 6.44/6.43 4115 (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]. 6.44/6.43 4116 (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]. 6.44/6.43 4117 (all V_v all V_u all T_b (class_Int_Onumber(T_b) & class_Orderings_Oord(T_b) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_u)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_v))) -> hAPP(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]. 6.44/6.43 4118 (all V_M_2 all T_c ((exists 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),B_x)))) -> 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]. 6.44/6.43 4119 (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]. 6.44/6.43 4120 (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]. 6.44/6.43 4121 (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]. 6.44/6.43 4122 (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]. 6.44/6.43 4123 (all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremdups(T_a,V_xs) = V_xs)) # label(fact_distinct__remdups__id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4124 (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_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))))))) & (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_OMax(T_c,V_A_2)))))) # label(fact_Max_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4125 (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(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)))))))) # label(fact_card__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4126 (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]. 6.44/6.43 4127 (all V_x all V_z all V_y 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_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_z),V_x)) = 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))) # label(fact_inf__sup__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4128 (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]. 6.44/6.43 4129 (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]. 6.44/6.43 4130 (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]. 6.44/6.43 4131 (all V_y all V_x all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x)),V_y)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_y)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_power2__diff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4132 (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(tc_fun(T_c,T_d)),V_g_2),V_f_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]. 6.44/6.43 4133 (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]. 6.44/6.43 4134 (all V_i all V_d ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_Oempty(tc_Int_Oint) = c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_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)))) # label(fact_small__lazy_H_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4135 (all V_c all V_b all V_a 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) = 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))) # label(fact_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4136 (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]. 6.44/6.43 4137 (all V_c all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)) -> (V_c = V_b -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4138 (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]. 6.44/6.43 4139 (all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_take__Nil) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4140 (all V_xs_2 all V_x_2 all V_n_2 all T_c hAPP(hAPP(c_List_Odrop(T_c),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = 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)) # label(fact_drop__Cons) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4141 (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]. 6.44/6.43 4142 (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),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))) # label(fact_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4143 (all T_c all V_ys_2 all V_xs_2 (V_xs_2 = V_ys_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_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_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_list__eq__iff__zip__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4144 (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]. 6.44/6.43 4145 (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]. 6.44/6.43 4146 (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]. 6.44/6.43 4147 (all V_b all V_a hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_Odivmod__int(V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) # label(fact_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4148 (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_x_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_y_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_y_2 = V_x_2)))) # label(fact_inv__into__injective) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4149 (all V_ys all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Otl(T_a),V_xs)),V_ys))) # label(fact_tl__append2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4150 (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]. 6.44/6.43 4151 (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(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)))) # label(fact_Un__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4152 (all V_S_2 all V_g_2 all V_f_2 all V_Pa_2 all T_c all T_d 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))))) = 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)) # label(fact_if__image__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4153 (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]. 6.44/6.43 4154 (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]. 6.44/6.43 4155 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit1(V_l_2))))) # label(fact_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4156 (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]. 6.44/6.43 4157 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> V_m = V_n))) # label(fact_le__antisym) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4158 (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]. 6.44/6.43 4159 (all V_b_2 all V_a_2 all V_f1_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_f1_2),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_a_2),V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_prod_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.43 4160 (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]. 6.44/6.43 4161 (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 V_z_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n)))))) # label(fact_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4162 (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]. 6.44/6.44 4163 (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]. 6.44/6.44 4164 (all V_k_2 all T_c (class_Rings_Oring__1(T_c) -> (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(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)) & (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))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_k_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_of__int__num) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4165 (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]. 6.44/6.44 4166 (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]. 6.44/6.44 4167 (all V_b all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))) # label(fact_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4168 (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)) -> (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_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]. 6.44/6.44 4169 (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]. 6.44/6.44 4170 (all V_B_2 all V_A_2 all V_f_2 all T_e all T_d all T_c 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))) = 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))) # label(fact_vimage__Times) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4171 (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_Osgn__class_Osgn(T_c,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_sgn__less) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4172 (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),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__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)))))) # label(fact_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4173 (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]. 6.44/6.44 4174 (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(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,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_d),V_B_2))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4175 (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]. 6.44/6.44 4176 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4177 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_plus__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4178 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n_2)),hAPP(c_Code__Numeral_Onat__of,V_m_2))))) # label(fact_less__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4179 (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]. 6.44/6.44 4180 (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)) -> (hAPP(c_Int_Onat,V_z_H_2) = hAPP(c_Int_Onat,V_z_2) <-> V_z_2 = V_z_H_2)))) # label(fact_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4181 (all V_n all V_m all V_u all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i)),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)))) # label(fact_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4182 (all V_y_2 all V_x_2 all V_r_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))),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)) <-> 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))))) # label(fact_INF__INT__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4183 (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]. 6.44/6.44 4184 (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]. 6.44/6.44 4185 (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_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4186 (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]. 6.44/6.44 4187 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))))) # label(fact_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4188 (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]. 6.44/6.44 4189 (all V_ys_2 all V_xs_2 all T_c (c_List_Oconcat(T_c,V_xs_2) = c_List_Oconcat(T_c,V_ys_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_ys_2 = V_xs_2)))) # label(fact_concat__injective) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4190 (all V_x all V_z all V_y 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_y),V_z)),V_x) = 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)))) # label(fact_sup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4191 (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]. 6.44/6.44 4192 (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]. 6.44/6.44 4193 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)))) # label(fact_inf__compl__bot) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4194 (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) -> 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]. 6.44/6.44 4195 (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_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__eq__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4196 (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]. 6.44/6.44 4197 (all V_xs all V_x all T_b all T_a (hAPP(c_List_Orotate1(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),V_xs)) = hAPP(hAPP(c_List_Oappend(T_b),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),c_List_Olist_ONil(T_b))) & c_List_Olist_ONil(T_a) = hAPP(c_List_Orotate1(T_a),c_List_Olist_ONil(T_a)))) # label(fact_rotate__simps) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4198 (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,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)))) # label(fact_last__drop) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4199 (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]. 6.44/6.44 4200 (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]. 6.44/6.44 4201 (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]. 6.44/6.44 4202 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_List_Olast(T_a,c_List_Oreplicate(T_a,V_n,V_x)) = V_x)) # label(fact_last__replicate) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4203 (all V_B_2 all T_c 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) = V_B_2) # label(fact_Un__empty__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4204 (all V_nat_2 all V_f2_2 all V_f1_2 all T_c 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)) = hAPP(c_Nat_Onat_Onat__rec(T_c,V_f1_2,V_f2_2),hAPP(c_Nat_OSuc,V_nat_2))) # label(fact_nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4205 (all V_r_2 all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_c (V_x_2 = 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_xs_2),V_ys_2)),c_List_Olistrel1(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)),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),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))))) # label(fact_snoc__listrel1__snoc__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4206 (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)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xa),B_x) != V_t_2)))) -> V_t_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),V_D_2))))) # label(fact_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4207 (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]. 6.44/6.44 4208 (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]. 6.44/6.44 4209 (all V_r_2 all T_c c_List_Olex(T_c,V_r_2) = 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))) # label(fact_lex__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4210 (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_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)) = 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))))) # label(fact_setsum__diff__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4211 (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]. 6.44/6.44 4212 (all V_ys all T_a V_ys = hAPP(hAPP(c_List_Oappend(T_a),c_List_Olist_ONil(T_a)),V_ys)) # label(fact_append__Nil) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4213 (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]. 6.44/6.44 4214 (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_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_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]. 6.44/6.44 4215 (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_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),hAPP(V_f_2,V_x_2))))) # label(fact_inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4216 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) # label(fact_length__rotate) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4217 (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]. 6.44/6.44 4218 (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]. 6.44/6.44 4219 (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))))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n | 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]. 6.44/6.44 4220 (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]. 6.44/6.44 4221 (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]. 6.44/6.44 4222 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> 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)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z)))) # label(fact_of__int__mult) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4223 (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]. 6.44/6.44 4224 (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]. 6.44/6.44 4225 (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_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)))) & (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) = V_x))) # label(fact_nth__Cons_H) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4226 (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]. 6.44/6.44 4227 (all V_p_2 all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),V_p_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_c),c_COMBK(T_c,T_d)),V_p_2)) # label(fact_fst__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4228 (all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_List_Oset(T_c),c_List_Olist_ONil(T_c))) # label(fact_List_Oset_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4229 (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]. 6.44/6.44 4230 (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]. 6.44/6.44 4231 (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]. 6.44/6.44 4232 (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]. 6.44/6.44 4233 (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]. 6.44/6.44 4234 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4235 (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_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__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4236 (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]. 6.44/6.44 4237 (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]. 6.44/6.44 4238 (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]. 6.44/6.44 4239 (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]. 6.44/6.44 4240 (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_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),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__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4241 (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]. 6.44/6.44 4242 (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]. 6.44/6.44 4243 (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_xa != B_x -> 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),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))))) # label(fact_insert__partition) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4244 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = V_n) # label(fact_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4245 (all V_y_2 all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> 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)) = 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)))) # label(fact_image__uminus__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4246 (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]. 6.44/6.44 4247 (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(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)) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2))) # label(fact_map__pair__simp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4248 (all V_z_H all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_z_H))))) # label(fact_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4249 (all V_ys all V_y all V_xs all V_x all T_a c_List_Osplice(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_ys)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Osplice(T_a,V_xs,V_ys)))) # label(fact_splice_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4250 (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) = 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))) # label(fact_card_Oeq__fold__g) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4251 (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]. 6.44/6.44 4252 (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]. 6.44/6.44 4253 (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_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_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_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))))) # label(fact_insort__key_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4254 (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]. 6.44/6.44 4255 (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_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_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_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4256 (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]. 6.44/6.44 4257 (all V_x_2 all V_A_2 all V_L_2 all T_e all T_c all T_d 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)))))) = c_Partial__Function_Ofun__lub(T_d,T_c,T_e,V_L_2,V_A_2,V_x_2)) # label(fact_fun__lub__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4258 (all V_n hAPP(c_Nat_OSuc,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4259 (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]. 6.44/6.44 4260 (all T_c all V_B_2 all V_A_2 (V_A_2 = V_B_2 -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_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]. 6.44/6.44 4261 (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) -> (V_b_2 = V_a_2 <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2) = c_Groups_Oone__class_Oone(T_c))))) # label(fact_right__inverse__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4262 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4263 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))))) # label(fact_diff__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4264 (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(T_d,tc_HOL_Obool)),V_A_2),V_B_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))))))) # label(fact_Times__subset__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4265 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> V_a = V_b))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4266 (all V_B_2 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)),V_C_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_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))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4267 (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]. 6.44/6.44 4268 (all V_y_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_y_2 = 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)))) # label(fact_surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4269 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_d all T_c 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)) = hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),c_List_OtakeWhile(T_d,V_Pa_2,V_ys_2))) # label(fact_zip__takeWhile__snd) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4270 (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_OBit1(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_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4271 (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]. 6.44/6.44 4272 (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]. 6.44/6.44 4273 (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),V_xs_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))))) # label(fact_filter__insort__triv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4274 (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]. 6.44/6.44 4275 (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]. 6.44/6.44 4276 (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]. 6.44/6.44 4277 (all V_A_2 all V_Pa_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_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)))) = 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)) # label(fact_Image__Collect__split) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4278 (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]. 6.44/6.44 4279 (all V_C_2 all V_B_2 all V_A_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_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),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_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)))) # label(fact_Times__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4280 (all V_b_2 all 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__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4281 (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]. 6.44/6.44 4282 (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]. 6.44/6.44 4283 (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]. 6.44/6.44 4284 (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]. 6.44/6.44 4285 (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_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)) = 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_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) = 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))))) # label(fact_INT__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4286 (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]. 6.44/6.44 4287 (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]. 6.44/6.44 4288 (all V_R_2 all V_T_2 all V_S_2 all T_d all T_e 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_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)) = 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)) # label(fact_rel__comp__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4289 (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_OatLeastLessThan(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))))) # label(fact_ivl__disj__un_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4290 (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]. 6.44/6.44 4291 (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_Y_2 = V_X_2))))))) # label(fact_quotient__eqI) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4292 (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]. 6.44/6.44 4293 (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]. 6.44/6.44 4294 (all T_c c_Nitpick_Onum(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))) # label(fact_Nitpick_Onum__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4295 (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]. 6.44/6.44 4296 (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]. 6.44/6.44 4297 (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_List_Oset(tc_Nat_Onat),c_List_Oupt(V_m_2,V_n_2))) = 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))))) # label(fact_setsum__set__upt__conv__listsum__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4298 (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]. 6.44/6.44 4299 (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_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)) = 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)) # label(fact_UN__UN__flatten) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4300 (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_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))) = 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)))) # label(fact_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4301 (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]. 6.44/6.44 4302 (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]. 6.44/6.44 4303 (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_y_2) = hAPP(V_f_2,V_z_2)))) # label(fact_congruentD) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4304 (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]. 6.44/6.44 4305 (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]. 6.44/6.44 4306 (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]. 6.44/6.44 4307 (all V_xs_2 all V_x_2 all V_Qa_2 all T_c hAPP(c_List_Ofilter(T_c,V_Qa_2),c_List_Oremove1(T_c,V_x_2,V_xs_2)) = c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2))) # label(fact_filter__remove1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4308 (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]. 6.44/6.44 4309 (all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_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))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))))) # label(fact_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4310 (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 -> 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)) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))))) -> 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) = 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))))))) # label(fact_fold__image__UN__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4311 (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]. 6.44/6.44 4312 (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_H != V_b))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4313 (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]. 6.44/6.44 4314 (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_a),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]. 6.44/6.44 4315 (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]. 6.44/6.44 4316 (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_Int_Oring__1__class_Oof__int(T_a),V_k) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))))))) # label(fact_of__int__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4317 (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]. 6.44/6.44 4318 (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]. 6.44/6.44 4319 (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]. 6.44/6.44 4320 (all V_n_2 all V_m_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)) = c_List_Oupt(V_m_2,V_n_2)) # label(fact_sorted__list__of__set__range) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4321 (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]. 6.44/6.44 4322 (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]. 6.44/6.44 4323 (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]. 6.44/6.44 4324 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4325 (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_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))) & (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))))) # label(fact_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4326 (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]. 6.44/6.44 4327 (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]. 6.44/6.44 4328 (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]. 6.44/6.44 4329 (all V_A_2 all T_c (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> (exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))))) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4330 (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 = V_x_2 <-> hAPP(V_f_2,V_x_2) = hAPP(V_f_2,V_y_2)))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.44 4331 (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]. 6.44/6.44 4332 (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]. 6.44/6.44 4333 (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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_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)),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))))) # label(fact_nat__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4334 (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]. 6.44/6.45 4335 (all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 <-> 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]. 6.44/6.45 4336 (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),V_x),V_y),V_m) = 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)) # label(fact_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4337 (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_COMBC(T_d,T_e,T_c),V_Pa_2),V_Qa_2),V_R_2) = hAPP(hAPP(V_Pa_2,V_R_2),V_Qa_2)) # label(help_c__COMBC__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4338 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> 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__eqn) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4339 (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_e,V_f_2),hAPP(c_Set_Ovimage(T_e,T_d,V_g_2),V_x_2)) = 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)) # label(fact_vimage__compose) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4340 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_divide__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4341 (all V_k_2 all V_n_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,c_Divides_Odiv__class_Omod(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 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(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 (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_j))))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hBOOL(hAPP(V_Pa_2,V_n_2))))) # label(fact_split__zmod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4342 (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]. 6.44/6.45 4343 (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]. 6.44/6.45 4344 (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]. 6.44/6.45 4345 (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]. 6.44/6.45 4346 (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]. 6.44/6.45 4347 (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]. 6.44/6.45 4348 (all V_r_2 all T_c hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = c_Transitive__Closure_Ortrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2))) # label(fact_rtrancl__converse) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4349 (all V_y_2 all V_f_2 all T_d all T_c all V_x_2 all V_z_2 (V_x_2 != V_z_2 -> hAPP(V_f_2,V_z_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__other) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4350 (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,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,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)))) # label(fact_inj__on__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4351 (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]. 6.44/6.45 4352 (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),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),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),V_b_2))))) # label(fact_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4353 (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]. 6.44/6.45 4354 (all V_b_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),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_b_2)) <-> V_a_2 = V_b_2)) # label(fact_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4355 (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]. 6.44/6.45 4356 (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]. 6.44/6.45 4357 (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]. 6.44/6.45 4358 (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]. 6.44/6.45 4359 (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]. 6.44/6.45 4360 (all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n))) # label(fact_int__mult) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4361 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4362 (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]. 6.44/6.45 4363 (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_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_k_2,V_l_2)))) # label(fact_greaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4364 (all V_x all V_y all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = V_x)))) # label(fact_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4365 (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))) <-> (exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)) & hBOOL(hAPP(hAPP(c_member(tc_prod(T_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))))))) # label(fact_trancl__power) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4366 (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]. 6.44/6.45 4367 (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]. 6.44/6.45 4368 (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]. 6.44/6.45 4369 (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_x_2 = V_y_2))) # label(fact_injD) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4370 (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]. 6.44/6.45 4371 (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]. 6.44/6.45 4372 (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]. 6.44/6.45 4373 (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]. 6.44/6.45 4374 (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)) <-> (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_INF2__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4375 (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]. 6.44/6.45 4376 (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]. 6.44/6.45 4377 (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]. 6.44/6.45 4378 (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]. 6.44/6.45 4379 (all V_n all V_a all T_a (class_Rings_Oring__1(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)) = 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)))) # label(fact_power__minus__even) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4380 (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]. 6.44/6.45 4381 (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),V_xs),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_i)) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),V_i))) # label(fact_nth__drop) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4382 (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]. 6.44/6.45 4383 (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_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),V_n))) # label(fact_power__mult) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4384 (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]. 6.44/6.45 4385 (all V_u_2 all V_l_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_u_2)),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))) # label(fact_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4386 (all V_xs_2 all V_Pa_2 all T_c c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = 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)) # label(fact_takeWhile__eq__take) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4387 (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]. 6.44/6.45 4388 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist_ONil(T_c) != c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) -> -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]. 6.44/6.45 4389 (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 (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),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(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]. 6.44/6.45 4390 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != 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(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]. 6.44/6.45 4391 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Oall__interval__nat(V_Pa_2,V_i_2,V_j_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)))))) # label(fact_all__interval__nat__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4392 (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]. 6.44/6.45 4393 (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(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_power__minus1__even) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4394 (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]. 6.44/6.45 4395 (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_OInf__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__inf__class_Oinf(T_c)),V_a_2),V_A_2)))) # label(fact_Inf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4396 (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]. 6.44/6.45 4397 (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]. 6.44/6.45 4398 (all V_com2_2 all V_com1_2 all V_fun_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))) = c_Com_Ocom_Ocom__size(c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2))) # label(fact_com_Osize_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4399 (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]. 6.44/6.45 4400 (all V_c_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_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]. 6.44/6.45 4401 (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]. 6.44/6.45 4402 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Int_Osucc(V_k)) # label(fact_succ__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4403 (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__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4404 (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]. 6.44/6.45 4405 (all V_n_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | V_x_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)),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]. 6.44/6.45 4406 (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__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2))))) # label(fact_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4407 (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(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))) & (-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)))))))))) # label(fact_Random_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4408 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_B_2)) -> V_t_2 != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xa),B_x))))) -> V_t_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),V_D_2))))) # label(fact_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4409 (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]. 6.44/6.45 4410 (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]. 6.44/6.45 4411 (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)) & V_y_2 = hAPP(hAPP(V_f_2,V_a_2),B_y_H))))))) # label(fact_fun__left__comm_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4412 (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)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)),V_A_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__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4413 (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]. 6.44/6.45 4414 (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]. 6.44/6.45 4415 (all V_n all V_m 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) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n)) # label(fact_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4416 (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]. 6.44/6.45 4417 (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_ys_2 = V_xs_2 <-> hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2)))) # label(fact_inj__map__eq__map) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4418 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord_Omax(T_c,c_Orderings_Oord__class_Oless__eq(T_c)) = c_Orderings_Oord__class_Omax(T_c))) # label(fact_max__ord__max) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4419 (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]. 6.44/6.45 4420 (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]. 6.44/6.45 4421 (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]. 6.44/6.45 4422 (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]. 6.44/6.45 4423 (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]. 6.44/6.45 4424 (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]. 6.44/6.45 4425 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4426 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)))) # label(fact_min__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4427 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))) # label(fact_diff__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4428 (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]. 6.44/6.45 4429 (all V_xs_2 all V_Pa_2 all T_c V_xs_2 = 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))) # label(fact_takeWhile__dropWhile__id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4430 (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]. 6.44/6.45 4431 (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]. 6.44/6.45 4432 (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_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]. 6.44/6.45 4433 (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]. 6.44/6.45 4434 (all V_a all V_b all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),V_a)) -> (V_a != V_b -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_b),V_a)))))) # label(fact_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4435 (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__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]. 6.44/6.45 4436 (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]. 6.44/6.45 4437 (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]. 6.44/6.45 4438 (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]. 6.44/6.45 4439 (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_e,tc_HOL_Obool)) != V_A_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)) = 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]. 6.44/6.45 4440 (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]. 6.44/6.45 4441 (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]. 6.44/6.45 4442 (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]. 6.44/6.45 4443 (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]. 6.44/6.45 4444 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c 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)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4445 (all V_y all V_x all T_a (class_Orderings_Olinorder(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) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4446 (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]. 6.44/6.45 4447 (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]. 6.44/6.45 4448 (all 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)))) = hAPP(c_Set_OCollect(tc_Nat_Onat),V_Pa_2)) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4449 (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]. 6.44/6.45 4450 (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]. 6.44/6.45 4451 (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]. 6.44/6.45 4452 (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]. 6.44/6.45 4453 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4454 (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_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_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_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__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4455 (all V_r_2 all T_c c_Transitive__Closure_Otrancl(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) = hAPP(c_Relation_Oconverse(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2))) # label(fact_trancl__converse) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4456 (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]. 6.44/6.45 4457 (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)),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)),V_k)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H))),V_k))))) # label(fact_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4458 (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]. 6.44/6.45 4459 (all V_B_2 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_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2) <-> V_A_2 = V_B_2)) # label(fact_Compl__eq__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4460 (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]. 6.44/6.45 4461 (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)),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_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4462 (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]. 6.44/6.45 4463 (all V_n all T_a all V_xs (c_List_Olist_ONil(T_a) != 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))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs))) # label(fact_take__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4464 (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_H = V_qr))) # label(fact_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4465 (all V_x c_Int_Osucc(c_Int_Opred(V_x)) = V_x) # label(fact_succ__pred) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4466 (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 -> ((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]. 6.44/6.45 4467 (all T_c c_Relation_Osym(T_c,c_Relation_OId(T_c))) # label(fact_sym__Id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4468 (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))) & (c_Groups_Ozero__class_Ozero(T_a) = V_x -> c_Groups_Ozero__class_Ozero(T_a) = c_Groups_Osgn__class_Osgn(T_a,V_x)))) # label(fact_sgn__if) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4469 (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]. 6.44/6.45 4470 (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]. 6.44/6.45 4471 (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)) & (V_i != 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_i)) -> c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = c_Groups_Oone__class_Oone(tc_Int_Oint)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i)) -> 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))))) # label(fact_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4472 (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))) <-> 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_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4473 (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_Onth(tc_List_Olist(T_c)),c_List_Otranspose(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)))),V_xs_2)))) # label(fact_nth__transpose) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4474 (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]. 6.44/6.45 4475 (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]. 6.44/6.45 4476 (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]. 6.44/6.45 4477 (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]. 6.44/6.45 4478 (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]. 6.44/6.45 4479 (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]. 6.44/6.45 4480 (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)) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_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_B_2))) # label(fact_INT__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4481 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),V_n)) # label(fact_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4482 (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]. 6.44/6.45 4483 (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]. 6.44/6.45 4484 (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]. 6.44/6.45 4485 (all V_n all V_i c_Code__Numeral_Onat__of__aux(V_i,V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_i)),V_n)) # label(fact_nat__of__aux__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4486 (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]. 6.44/6.45 4487 (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]. 6.44/6.45 4488 (all V_n all V_i hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_i)),V_n) = c_Int_Onat__aux(V_i,V_n)) # label(fact_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4489 (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]. 6.44/6.45 4490 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4491 (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]. 6.44/6.45 4492 (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]. 6.44/6.45 4493 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) -> (V_a_2 = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2) <-> (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)) & (V_c_2 = c_Groups_Ozero__class_Ozero(T_c) -> c_Groups_Ozero__class_Ozero(T_c) = V_a_2)))) # label(fact_eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4494 (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]. 6.44/6.45 4495 (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]. 6.44/6.45 4496 (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),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))) = 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)))) # label(fact_setsum__shift__bounds__cl__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4497 (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]. 6.44/6.45 4498 (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]. 6.44/6.45 4499 (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]. 6.44/6.45 4500 (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]. 6.44/6.45 4501 (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(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_d),V_B_2) <-> (exists B_f c_Fun_Obij__betw(T_c,T_d,B_f,V_A_2,V_B_2)))))) # label(fact_BIJ) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4502 (all V_q all T_a hAPP(hAPP(c_Nitpick_Ofrac(T_a),hAPP(c_Nitpick_Odenom(T_a),V_q)),hAPP(c_Nitpick_Onum(T_a),V_q)) = c_Nitpick_Oinverse__frac(T_a,V_q)) # label(fact_Nitpick_Oinverse__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4503 (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]. 6.44/6.45 4504 (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]. 6.44/6.45 4505 (all V_g_2 all V_f_2 all T_d all T_c 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)) = hAPP(c_Finite__Set_Ofold__image(T_c,T_d,V_f_2),V_g_2)) # label(fact_fold__image__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4506 (all T_d all T_c 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))) = c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4507 (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]. 6.44/6.45 4508 (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]. 6.44/6.45 4509 (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]. 6.44/6.45 4510 (all V_code__numeral_H c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) # label(fact_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4511 (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_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_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)) = 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))))))) # label(fact_fold__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4512 (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]. 6.44/6.45 4513 (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]. 6.44/6.45 4514 (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_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)) = 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)) # label(fact_scomp__scomp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4515 (all V_n hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_min__0R) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4516 (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]. 6.44/6.45 4517 (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]. 6.44/6.45 4518 (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]. 6.44/6.45 4519 (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]. 6.44/6.45 4520 (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_A_2) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__partition2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4521 (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_A_2 = V_B_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4522 (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]. 6.44/6.45 4523 (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]. 6.44/6.45 4524 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(hAPP(c_List_Olist_OCons(T_c),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_Olist_OCons(T_c),V_x_2),V_xs_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)) = c_List_Olist_ONil(T_c)))) # label(fact_takeWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4525 (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]. 6.44/6.45 4526 (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(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))) = 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))))) # label(fact_map__pair__surj__on) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4527 (all V_R_2 all V_Qa_2 all V_Pa_2 all T_e all T_c all T_d hAPP(hAPP(hAPP(c_COMBB(T_d,T_c,T_e),V_Pa_2),V_Qa_2),V_R_2) = hAPP(V_Pa_2,hAPP(V_Qa_2,V_R_2))) # label(help_c__COMBB__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4528 (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 (hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) = V_k & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)))))) # label(fact_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.45 4529 (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]. 6.44/6.46 4530 (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),V_xs_2) = hAPP(c_List_OremoveAll(T_c,V_x_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))) # label(fact_removeAll__filter__not) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4531 (all V_size_2 all V_nrandom_2 all V_x_2 all T_c hAPP(hAPP(hAPP(c_Random__Sequence_Osingle(T_c),V_x_2),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)),hAPP(c_DSequence_Osingle(T_c),V_x_2))) # label(fact_Random__Sequence_Osingle__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4532 (all V_y all V_x all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_y),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_y)) -> V_x = V_y))))) # label(fact_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4533 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_i_2),V_j_2) = hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_j_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_atLeastLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4534 (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]. 6.44/6.46 4535 (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) = 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)) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) = 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)))) # label(fact_sublist__singleton) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4536 (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]. 6.44/6.46 4537 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))))) # label(fact_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4538 (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)) -> 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),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_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))))) # label(fact_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4539 (all V_f_2 all T_d all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> 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]. 6.44/6.46 4540 (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),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),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),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),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]. 6.44/6.46 4541 (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]. 6.44/6.46 4542 (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]. 6.44/6.46 4543 (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]. 6.44/6.46 4544 (all V_A_2 all V_B_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)) -> (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]. 6.44/6.46 4545 (all V_x hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_x)) = V_x) # label(fact_nat__of__inverse) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4546 (all T_a c_List_Olist_ONil(T_a) = c_List_Oremdups(T_a,c_List_Olist_ONil(T_a))) # label(fact_remdups_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4547 (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]. 6.44/6.46 4548 (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,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)))) <-> hBOOL(hAPP(V_Pa_2,B_x)))) -> ((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]. 6.44/6.46 4549 (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_n),V_m)),V_n))) # label(fact_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4550 (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]. 6.44/6.46 4551 (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]. 6.44/6.46 4552 (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]. 6.44/6.46 4553 (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]. 6.44/6.46 4554 (all V_fs_2 all T_c 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)) = c_List_Omeasures(T_c,V_fs_2)) # label(fact_measures__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4555 (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__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)),V_C_2))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4556 (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]. 6.44/6.46 4557 (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]. 6.44/6.46 4558 (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]. 6.44/6.46 4559 (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_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__is__surj) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4560 (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]. 6.44/6.46 4561 (all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OPls) = V_k) # label(fact_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4562 (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]. 6.44/6.46 4563 (all V_B_2 all V_A_2 all V_R_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_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)) = 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))) # label(fact_Image__Un) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4564 (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),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),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),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(T_c),V_a_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)))))) # label(fact_less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4565 (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]. 6.44/6.46 4566 (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]. 6.44/6.46 4567 (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]. 6.44/6.46 4568 (all V_lazy__sequence_2 all V_a_2 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_OInsert(T_d,V_a_2,V_lazy__sequence_2)) = 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))) # label(fact_lazy__sequence_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4569 (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]. 6.44/6.46 4570 (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]. 6.44/6.46 4571 (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]. 6.44/6.46 4572 (all V_l_2 all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Divides_Odivmod__int(V_k_2,V_l_2)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_k_2 -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l_2 -> hAPP(c_Product__Type_Oapsnd(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint,hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),hAPP(hAPP(hAPP(c_If(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_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))))) # label(fact_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4573 (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]. 6.44/6.46 4574 (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]. 6.44/6.46 4575 (all V_A_2 all T_c 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))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)) # label(fact_Compl__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4576 (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]. 6.44/6.46 4577 (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]. 6.44/6.46 4578 (all V_d_2 all T_c all V_Pa_2 ((-(exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & (all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_x = B_y)))) -> c_FunDef_OTHE__default(T_c,V_d_2,V_Pa_2) = V_d_2) & ((exists B_x ((all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_y = B_x)) & hBOOL(hAPP(V_Pa_2,B_x)))) -> c_FunDef_OTHE__default(T_c,V_d_2,V_Pa_2) = c_HOL_OThe(T_c,V_Pa_2)))) # label(fact_THE__default__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4579 (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]. 6.44/6.46 4580 (all V_u_2 all 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),V_l_2)) # label(fact_card__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4581 (all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_a) = V_a)) # label(fact_min__max_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4582 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_y_2) <-> V_y_2 = V_x_2))) # label(fact_lessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4583 (all V_n all V_k all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_k))) # label(fact_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4584 (all V_z_2 all V_w_2 all T_c (class_Int_Oring__char__0(T_c) -> (V_w_2 = V_z_2 <-> 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__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4585 (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]. 6.44/6.46 4586 (all V_B_2 all V_A_2 all T_c 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)) = 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))) # label(fact_Pow__Int__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4587 (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]. 6.44/6.46 4588 (all V_A_2 all V_y_2 all V_x_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4589 (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_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)) = 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),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)))))))) # label(fact_setsum__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4590 (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_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__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__eq__Union) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4591 (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_e,T_c,hAPP(hAPP(c_COMBB(T_d,T_c,T_e),V_f_2),V_g_2)),V_A_2) = 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))) # label(fact_image__image) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4592 (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)) -> 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__absorb2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4593 (all V_z (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> V_z = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)))) # label(fact_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4594 (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]. 6.44/6.46 4595 (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]. 6.44/6.46 4596 (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]. 6.44/6.46 4597 (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]. 6.44/6.46 4598 (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]. 6.44/6.46 4599 (all V_x all V_k all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> (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))) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) != V_k -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x))))) # label(fact_last__list__update) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4600 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Pa_2) | hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)))) # label(help_c__fimplies__1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4601 (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_OBit1(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]. 6.44/6.46 4602 (all V_Pa_2 all T_d all T_c 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))) = 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))) # label(fact_The__split) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4603 (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]. 6.44/6.46 4604 (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_Olist_ONil(T_c)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_insort__key_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4605 (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)) -> 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)) = V_a_2)))) # label(fact_min__max_Osup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4606 (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]. 6.44/6.46 4607 (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]. 6.44/6.46 4608 (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_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_a),V_b)),V_b)))) # label(fact_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4609 (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]. 6.44/6.46 4610 (all V_B_H_2 all V_A_H_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_A_H_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)) -> V_B_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))))) # label(fact_image__inv__into__cancel) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4611 (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]. 6.44/6.46 4612 (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]. 6.44/6.46 4613 (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_X_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_X_2)))) # label(fact_inv__image__comp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4614 (all V_u_2 all 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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),hAPP(c_Nat_OSuc,V_l_2))) # label(fact_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4615 (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]. 6.44/6.46 4616 (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]. 6.44/6.46 4617 (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]. 6.44/6.46 4618 (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]. 6.44/6.46 4619 (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]. 6.44/6.46 4620 (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]. 6.44/6.46 4621 (all V_n_2 (V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4622 (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]. 6.44/6.46 4623 (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_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)))) # label(fact_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4624 (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]. 6.44/6.46 4625 (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]. 6.44/6.46 4626 (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]. 6.44/6.46 4627 (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]. 6.44/6.46 4628 (all V_q all V_p all T_b all T_a (hAPP(c_Product__Type_Ofst(T_a,T_b),V_q) = hAPP(c_Product__Type_Ofst(T_a,T_b),V_p) -> (hAPP(c_Product__Type_Osnd(T_a,T_b),V_p) = hAPP(c_Product__Type_Osnd(T_a,T_b),V_q) -> V_p = V_q))) # label(fact_prod__eqI) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4629 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield__inverse__zero(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Oone__class_Oone(T_a)),V_x)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_x))) # label(fact_minus1__divide) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4630 (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]. 6.44/6.46 4631 (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]. 6.44/6.46 4632 (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]. 6.44/6.46 4633 (all V_z all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),V_z))) # label(fact_minus__number__of__mult) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4634 (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]. 6.44/6.46 4635 (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_inj__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4636 (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]. 6.44/6.46 4637 (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]. 6.44/6.46 4638 (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]. 6.44/6.46 4639 (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]. 6.44/6.46 4640 (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]. 6.44/6.46 4641 (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]. 6.44/6.46 4642 (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]. 6.44/6.46 4643 (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]. 6.44/6.46 4644 (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]. 6.44/6.46 4645 (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]. 6.44/6.46 4646 (all V_code__numeral_2 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_Code__Numeral_OSuc__code__numeral(V_code__numeral_2)) = hAPP(V_f2_2,V_code__numeral_2)) # label(fact_code__numeral_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4647 (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),V_a),c_Groups_Ozero__class_Ozero(T_a))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_b),c_Groups_Ozero__class_Ozero(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(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]. 6.44/6.46 4648 (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]. 6.44/6.46 4649 (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]. 6.44/6.46 4650 (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]. 6.44/6.46 4651 (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_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 -> hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)) = hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4652 (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]. 6.44/6.46 4653 (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]. 6.44/6.46 4654 (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]. 6.44/6.46 4655 (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)))) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) -> 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]. 6.44/6.46 4656 (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_Omin(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_min__less__iff__disj) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4657 (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]. 6.44/6.46 4658 (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]. 6.44/6.46 4659 (all V_x_2 all V_n_2 all T_c (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> c_List_Oreplicate(T_c,V_n_2,V_x_2) = c_List_Olist_ONil(T_c))) # label(fact_replicate__empty) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4660 (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]. 6.44/6.46 4661 (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)) -> (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)) = 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]. 6.44/6.46 4662 (all V_B_2 all V_A_2 all T_e all V_r_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)),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_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Image__UN) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4663 (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]. 6.44/6.46 4664 (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)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)))) <-> 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_bot__empty__eq2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4665 (all V_R_2 all V_f_2 all T_c 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))) = c_Wellfounded_Omlex__prod(T_c,V_f_2,V_R_2)) # label(fact_mlex__prod__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4666 (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]. 6.44/6.46 4667 (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]. 6.44/6.46 4668 (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]. 6.44/6.46 4669 (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]. 6.44/6.46 4670 (all V_y_2 all V_x_2 all V_Pa_2 (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)))) <-> (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(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))))))) # label(fact_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4671 (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) <-> 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__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4672 (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_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)))))) # label(fact_listrel__iff__zip) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4673 (all V_i_2 all V_M_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_M_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2))))) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))))) # label(fact_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4674 (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]. 6.44/6.46 4675 (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]. 6.44/6.46 4676 (all V_m_2 all V_n_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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_n_2),V_m_2)))) # label(fact_less__eq__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4677 (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]. 6.44/6.46 4678 (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_Un__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4679 (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(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)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Groups_Otimes__class_Otimes(T_c)),V_a_2),V_A_2))))) # label(fact_fold1__eq__fold) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4680 (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]. 6.44/6.46 4681 (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)) -> 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)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_c_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2)))))) # label(fact_image__minus__const__atLeastLessThan__nat) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4682 (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) -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = V_A_2)) # label(fact_bij__betw__empty1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4683 (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_d)),V_b_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_a_2) -> ((all B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_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]. 6.44/6.46 4684 (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]. 6.44/6.46 4685 (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]. 6.44/6.46 4686 (all V_n hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_n) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4687 (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]. 6.44/6.46 4688 (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)) -> (V_j_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2) = V_k_2))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4689 (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]. 6.44/6.46 4690 (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]. 6.44/6.46 4691 (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]. 6.44/6.46 4692 (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),hAPP(c_Groups_Ouminus__class_Ouminus(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_Oplus__class_Oplus(T_a),V_a),V_b)))) # label(fact_minus__add__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4693 (all V_w_2 hAPP(hAPP(c_HOL_OLet(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2)),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_COMBI(tc_Nat_Onat))) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) # label(fact_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4694 (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]. 6.44/6.46 4695 (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]. 6.44/6.46 4696 (all V_f_2 all T_c all T_d c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c) = c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_d))) # label(fact__07) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4697 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_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)))))) # label(fact_power2__abs) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4698 (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_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))) & (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))))) # label(fact_INT__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4699 (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_e),V_x_2) = 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))) # label(fact_fst__apsnd) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4700 (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]. 6.44/6.46 4701 (all V_A_2 all V_a_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_finite__insert) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4702 (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]. 6.44/6.46 4703 (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_OdropWhile(T_c,V_Pa_2,V_xs_2) = c_List_Olist_ONil(T_c))) # label(fact_dropWhile__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4704 (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_Fun_Oid(T_c),c_Fun_Oid(T_d))) # label(fact_map__pair_Oid) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4705 (all V_x_2 all V_w_2 all T_c (class_Int_Onumber(T_c) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = V_x_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]. 6.44/6.46 4706 (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]. 6.44/6.46 4707 (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]. 6.44/6.46 4708 (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)),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__int_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4709 (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]. 6.44/6.46 4710 (all V_a all T_a (class_Groups_Oab__group__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_a))) # label(fact_ab__left__minus) # label(axiom) # label(non_clause). [assumption]. 6.44/6.46 4711 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_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_c)),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)) | 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_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)),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))))) # label(fact_Cons__in__lex) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4712 (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))) <-> (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))))))) # label(fact_in__set__conv__decomp__first) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4713 (all V_l_2 c_List_Oembed__list(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)) # label(fact_embed__list__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4714 (all V_m_2 all V_n_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_n_2,V_m_2) = hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_n_2,hAPP(c_Nat_OSuc,V_m_2)))) # label(fact_atLeastAtMost__upt) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4715 (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]. 6.44/6.47 4716 (all V_y_2 all V_x_2 all T_c (class_Orderings_Opreorder(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)) & -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2))))) # label(fact_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4717 (all V_t_2 all V_B_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_t_2))))))) # label(fact_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4718 (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_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)) = 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)))) # label(fact_Int__insert__right__if1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4719 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b))) # label(fact_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4720 (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]. 6.44/6.47 4721 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y))))) # label(fact_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4722 (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]. 6.44/6.47 4723 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4724 (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]. 6.44/6.47 4725 (all V_n 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(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_n)))) # label(fact_abs__power__minus__one) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4726 (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]. 6.44/6.47 4727 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a),V_b))) # label(fact_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4728 (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]. 6.44/6.47 4729 (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]. 6.44/6.47 4730 (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]. 6.44/6.47 4731 (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]. 6.44/6.47 4732 (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]. 6.44/6.47 4733 (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_xs_2)))) | 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,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]. 6.44/6.47 4734 (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_i) != hAPP(V_r_2,B_j) -> 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]. 6.44/6.47 4735 (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]. 6.44/6.47 4736 (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]. 6.44/6.47 4737 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),V_n) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)))) # label(fact_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4738 (all V_m_2 all V_x_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_x_2 <-> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4739 (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]. 6.44/6.47 4740 (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]. 6.44/6.47 4741 (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]. 6.44/6.47 4742 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x))) # label(fact_inf__top__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4743 (all V_m all V_n hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m)))) # label(fact_div__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4744 (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]. 6.44/6.47 4745 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Obot__class_Obot(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x))) # label(fact_compl__inf__bot) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4746 (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]. 6.44/6.47 4747 (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]. 6.44/6.47 4748 (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]. 6.44/6.47 4749 (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]. 6.44/6.47 4750 (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,hAPP(hAPP(V_f_2,V_a_2),V_x_2)),V_xs_2) = 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))) # label(fact_foldl__Cons) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4751 (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]. 6.44/6.47 4752 (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]. 6.44/6.47 4753 (all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = 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)))) # label(fact_Compl__UNIV__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4754 (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]. 6.44/6.47 4755 (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]. 6.44/6.47 4756 (all V_K_2 all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Oordered__ab__semigroup__add(T_d) & class_Rings_Osemiring__1(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]. 6.44/6.47 4757 (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]. 6.44/6.47 4758 (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]. 6.44/6.47 4759 (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]. 6.44/6.47 4760 (all V_ys_2 all V_y_2 all V_xs_2 all T_d all T_c 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) = 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))) # label(fact_zip__Cons) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4761 (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]. 6.44/6.47 4762 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),V_m)) -> V_n != V_m)) # label(fact_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4763 (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]. 6.44/6.47 4764 (all V_com2_H all V_com1_H c_Com_Ocom_OSKIP != c_Com_Ocom_OSemi(V_com1_H,V_com2_H)) # label(fact_com_Osimps_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4765 (all V_n_2 all T_c all V_xs_2 ((c_List_Olist_ONil(tc_List_Olist(T_c)) = V_xs_2 -> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> ((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]. 6.44/6.47 4766 (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))) -> V_n_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) = V_n_2)) # label(fact_lexn__length) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4767 (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]. 6.44/6.47 4768 (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]. 6.44/6.47 4769 (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]. 6.44/6.47 4770 (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]. 6.44/6.47 4771 (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)),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_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4772 (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_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)) = 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))) & (-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),V_B_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))))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4773 (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]. 6.44/6.47 4774 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> (V_n != hAPP(c_Nat_OSuc,V_m) -> 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]. 6.44/6.47 4775 (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]. 6.44/6.47 4776 (all V_n_2 (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)))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4777 (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]. 6.44/6.47 4778 (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))) -> hAPP(hAPP(c_List_Oappend(T_c),c_List_Oremove1(T_c,V_x_2,V_xs_2)),V_ys_2) = c_List_Oremove1(T_c,V_x_2,hAPP(hAPP(c_List_Oappend(T_c),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))) -> 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))))) # label(fact_remove1__append) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4779 (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]. 6.44/6.47 4780 (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]. 6.44/6.47 4781 (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]. 6.44/6.47 4782 (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_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))) = 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))))) # label(fact_setsum__head__Suc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4783 (all V_A_2 all V_l_2 all V_x_2 all T_c 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)))) = c_List_Osublist(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_l_2),V_A_2)) # label(fact_sublist__Cons) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4784 (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]. 6.44/6.47 4785 (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]. 6.44/6.47 4786 (all V_k hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k) = V_k) # label(fact_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4787 (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]. 6.44/6.47 4788 (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]. 6.44/6.47 4789 (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]. 6.44/6.47 4790 (all V_A_2 all T_c hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = 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)) # label(fact_card__eq__setsum) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4791 (all V_n_2 all V_k_2 all V_m_2 (V_m_2 = V_n_2 <-> 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_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4792 (all V_com_H_2 all V_fun_H_2 all V_com2_2 all V_com1_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OSemi(V_com1_2,V_com2_2)) # label(fact_com_Osimps_I46_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4793 (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)),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)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_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))))) # label(fact_Int__insert__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4794 (all V_list_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)),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))) = c_List_Olist_Olist__size(T_c,V_fa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_a_2),V_list_2))) # label(fact_list_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4795 (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]. 6.44/6.47 4796 (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]. 6.44/6.47 4797 (all V_b_2 all V_a_2 all V_c_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_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) | c_Groups_Ozero__class_Ozero(T_c) = V_c_2))) # label(fact_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4798 (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]. 6.44/6.47 4799 (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]. 6.44/6.47 4800 (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]. 6.44/6.47 4801 (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]. 6.44/6.47 4802 (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),V_z_2),hAPP(hAPP(c_Set_Oinsert(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)))) # label(fact_fun__left__comm__idem_Ofold__insert__idem2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4803 (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]. 6.44/6.47 4804 (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]. 6.44/6.47 4805 (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]. 6.44/6.47 4806 (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_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_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__subset) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4807 (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]. 6.44/6.47 4808 (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) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_r_2),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) # label(fact_r__comp__rtrancl__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4809 (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]. 6.44/6.47 4810 (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]. 6.44/6.47 4811 (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]. 6.44/6.47 4812 (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]. 6.44/6.47 4813 (all V_ys_2 all V_x_2 all V_xs_2 all T_c (V_ys_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))) <-> c_List_Obutlast(T_c,V_ys_2) = V_xs_2 & c_List_Olast(T_c,V_ys_2) = V_x_2 & V_ys_2 != c_List_Olist_ONil(T_c))) # label(fact_snoc__eq__iff__butlast) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4814 (all V_s_2 all V_r_2 all T_e all T_c all T_d c_Predicate_Oconversep(T_d,T_c,c_Predicate_Opred__comp(T_d,T_e,T_c,V_r_2,V_s_2)) = 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))) # label(fact_converse__pred__comp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4815 (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]. 6.44/6.47 4816 (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]. 6.44/6.47 4817 (all V_ys_2 all V_xs_2 all T_d all T_c 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)) = 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))))))) # label(fact_set__zip) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4818 (all V_x all T_a all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(c_List_Otl(T_a),c_List_Oreplicate(T_a,V_n,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_x))) # label(fact_tl__replicate) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4819 (all V_xs_2 all V_n_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_2)),V_n_2)) <-> c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2))) # label(fact_drop__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4820 (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]. 6.44/6.47 4821 (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]. 6.44/6.47 4822 (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]. 6.44/6.47 4823 (all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4824 (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]. 6.44/6.47 4825 (all V_w_2 all V_x_2 all V_z_2 all V_y_2 all T_c (class_Fields_Ofield(T_c) -> (V_y_2 != c_Groups_Ozero__class_Ozero(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_z_2 -> (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_w_2),V_z_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_x_2),V_y_2)))))) # label(fact_frac__eq__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4826 (all V_B_2 all V_x_2 all V_A_2 all T_c ((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))) & (-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__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_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_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))))) # label(fact_psubset__insert__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4827 (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]. 6.44/6.47 4828 (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]. 6.44/6.47 4829 (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]. 6.44/6.47 4830 (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]. 6.44/6.47 4831 (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))) -> (V_xs_2 = c_List_Olist_ONil(tc_List_Olist(T_c)) -> 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]. 6.44/6.47 4832 (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_x_2),V_v_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2))) # label(fact_intrel__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4833 (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]. 6.44/6.47 4834 (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_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)) = 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)))))) # label(fact_dropWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4835 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (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)))) = 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)))) <-> V_a_2 = V_d_2 & V_b_2 = V_c_2 | V_c_2 = V_a_2 & V_d_2 = V_b_2)) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4836 (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]. 6.44/6.47 4837 (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]. 6.44/6.47 4838 (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]. 6.44/6.47 4839 (all V_x_2 all T_e all T_d all T_c V_x_2 = 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))) # label(fact_scomp__Pair) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4840 (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]. 6.44/6.47 4841 (all V_x all V_y all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_sup__absorb1) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4842 (all V_n_2 all T_c c_List_Orotate(T_c,V_n_2) = hAPP(hAPP(c_Nat_Ocompow(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_n_2),c_List_Orotate1(T_c))) # label(fact_rotate__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4843 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (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)) <-> c_List_Olinorder__class_Osorted(T_c,V_xs_2)))) # label(fact_sorted__insort) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4844 (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]. 6.44/6.47 4845 (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]. 6.44/6.47 4846 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Qa_2) | -hBOOL(V_Pa_2) | -hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)))) # label(help_c__fimplies__3) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4847 (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]. 6.44/6.47 4848 (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]. 6.44/6.47 4849 (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)) -> (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,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)))) -> V_x_2 = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_folding__one_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4850 (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),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_n),V_x)),V_ys) = 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_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)))) # label(fact_list__update__append) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4851 (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]. 6.44/6.47 4852 (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]. 6.44/6.47 4853 (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(V_k_2,B_x) = 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_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]. 6.44/6.47 4854 (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)),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)),V_g_2))) # label(fact_folding__idem_Oidem__left__comp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4855 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4856 (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) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4857 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__bot(T_c) -> (c_Orderings_Obot__class_Obot(T_c) = V_y_2 & c_Orderings_Obot__class_Obot(T_c) = V_x_2 <-> c_Orderings_Obot__class_Obot(T_c) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2)))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4858 (all V_b all V_a 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)) = 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))) # label(fact_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4859 (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]. 6.44/6.47 4860 (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]. 6.44/6.47 4861 (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]. 6.44/6.47 4862 (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)) -> (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) <-> 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_eq__equiv__class__iff2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4863 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OgreaterThan(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_OlessThan(T_c),V_x_2)))) # label(fact_image__uminus__lessThan) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4864 (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_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_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)))))))))) # label(fact_setsum__diff1_H) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4865 (all V_p_2 all V_g_2 all V_f_2 all T_d all T_c c_Product__Type_Oprod_Oprod__size(T_c,T_d,V_f_2,V_g_2,V_p_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_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)))) # label(fact_prod__size__simp) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4866 (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]. 6.44/6.47 4867 (all V_a_2 all T_c all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 -> -hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4868 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4869 (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]. 6.44/6.47 4870 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))))) # label(fact_diff__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4871 (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]. 6.44/6.47 4872 (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]. 6.44/6.47 4873 (all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> ((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))))))) <-> c_List_Olinorder__class_Osorted(T_c,V_xs_2)))) # label(fact_sorted__equals__nth__mono) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4874 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2) = 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))) # label(fact_length__map) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4875 (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]. 6.44/6.47 4876 (all V_S_2 all T_c c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_S_2) = 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)))) # label(fact_Inter__def) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4877 (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]. 6.44/6.47 4878 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_p),V_q)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),V_q))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4879 (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]. 6.44/6.47 4880 (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]. 6.44/6.47 4881 (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]. 6.44/6.47 4882 (all V_y_2 all V_x_2 all V_r_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_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(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)))) # label(fact_reflcl__set__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4883 (all V_xs_2 all V_x_2 all 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)) <-> 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]. 6.44/6.47 4884 (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(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4885 (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_OSup__fin(T_c,c_Orderings_Oord__class_Omax(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_Omax(T_c)),V_a_2),V_A_2)))) # label(fact_min__max_OSup__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4886 (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_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__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)))) # label(fact_setsum__reindex) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4887 (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]. 6.44/6.47 4888 (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]. 6.44/6.47 4889 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)))) # label(fact_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4890 (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(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_f),V_x_2)) = 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))) # label(fact_fst__map__pair) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4891 (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]. 6.44/6.47 4892 (all V_x all V_xs all T_a V_xs = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a))))) # label(fact_butlast__snoc) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4893 (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]. 6.44/6.47 4894 (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_OMax(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_OMin(T_c,V_S_2)))))) # label(fact_minus__Min__eq__Max) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4895 (all V_n all T_a (class_Rings_Osemiring__0(T_a) & class_Power_Opower(T_a) -> (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> 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)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_n) = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_power__0__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4896 (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(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(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)))) # label(fact_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.44/6.47 4897 (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)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H))),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k))) & (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))))) # label(fact_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 6.44/6.48 4898 (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]. 6.44/6.48 4899 (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),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(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),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(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(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))))) # label(fact_divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.44/6.48 4900 (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]. 6.44/6.48 4901 (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]. 6.44/6.48 4902 (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]. 6.44/6.48 4903 (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]. 6.44/6.48 4904 (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]. 6.44/6.48 4905 (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_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2)))) # label(fact_greaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.44/6.48 4906 (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]. 8.68/8.66 4907 (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]. 8.68/8.66 8.68/8.66 ============================== end of process non-clausal formulas === 8.68/8.66 8.68/8.66 ============================== PROCESS INITIAL CLAUSES =============== 8.68/8.66 8.68/8.66 ============================== PREDICATE ELIMINATION ================= 8.68/8.66 4908 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 8.68/8.66 4909 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f1(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,F),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,F,A),D)),hAPP(hAPP(c_Product__Type_OSigma(B,F),C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(F,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(F,A),tc_fun(tc_fun(F,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(F,A)),D)),E)),C) # label(fact_setsum__Sigma) # label(axiom). [clausify(4)]. 8.68/8.66 4910 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f1(F,E,D,C,B,A)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,D),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,D,A),F)),hAPP(hAPP(c_Product__Type_OSigma(B,D),C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(tc_fun(D,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A)),F)),E)),C) # label(fact_setsum__Sigma) # label(axiom). [clausify(4)]. 8.68/8.66 4911 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Int_Oint),B),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),C) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom). [clausify(18)]. 8.68/8.66 4912 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__disjoint) # label(axiom). [clausify(76)]. 8.68/8.66 4913 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,C)),hAPP(c_Nat_OSuc,D))) # label(fact_setsum__shift__bounds__Suc__ivl) # label(axiom). [clausify(96)]. 8.68/8.67 4914 -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__mult__class_Osetprod(tc_Int_Oint,A),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,tc_Nat_Onat),C),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),B)) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom). [clausify(112)]. 8.68/8.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f1(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Nat_Onat),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Nat_Onat),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Nat_Onat)),C)),D)),B). [resolve(4908,a,4909,a)]. 8.68/8.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f1(E,D,C,B,A,tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Nat_Onat),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat)),E)),D)),B). [resolve(4908,a,4910,a)]. 8.68/8.67 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),B). [resolve(4908,a,4911,b)]. 8.68/8.67 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(4908,a,4912,a)]. 8.68/8.67 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(4908,a,4913,a)]. 8.68/8.67 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4908,a,4914,a)]. 8.68/8.67 4915 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add_Ocomm__neutral) # label(axiom). [clausify(243)]. 8.68/8.67 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(4915,a,4908,a)]. 8.68/8.71 4916 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_member(B),C),hAPP(c_List_Oset(B),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),E),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,C)),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),E),c_List_Oremove1(B,C,D)))) # label(fact_listsum__map__remove1) # label(axiom). [clausify(366)]. 8.68/8.71 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),D),c_List_Oremove1(A,B,C)))). [resolve(4916,a,4908,a)]. 8.68/8.71 4917 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f80(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__right) # label(axiom). [clausify(588)]. 8.68/8.71 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),f80(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(4917,a,4908,a)]. 8.68/8.71 4918 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f80(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__right) # label(axiom). [clausify(588)]. 8.68/8.71 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f80(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(4918,a,4908,a)]. 8.68/8.71 4919 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum_Oinfinite) # label(axiom). [clausify(610)]. 8.68/8.71 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B). [resolve(4919,a,4908,a)]. 8.68/8.71 4920 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_COMBI(A)),hAPP(c_List_Oset(A),B)) # label(fact_distinct__listsum__conv__Setsum) # label(axiom). [clausify(634)]. 8.68/8.71 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_COMBI(tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A)). [resolve(4920,a,4908,a)]. 8.68/8.71 4921 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),A,D),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),c_Set_OCollect(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C)))),hAPP(c_COMBC(B,D,tc_HOL_Obool),hAPP(hAPP(c_COMBB(D,tc_fun(D,tc_HOL_Obool),B),c_fequal),F)))))),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) # label(fact_setsum__image__gen) # label(axiom). [clausify(727)]. 8.76/8.71 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B). [resolve(4921,a,4908,a)]. 8.76/8.71 4922 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,C),hAPP(c_Nat_OSuc,D))) # label(fact_setsum__shift__bounds__cl__Suc__ivl) # label(axiom). [clausify(732)]. 8.76/8.71 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(4922,a,4908,a)]. 8.76/8.71 4923 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),C))) # label(fact_setsum__head__upt__Suc) # label(axiom). [clausify(736)]. 8.76/8.71 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(4923,a,4908,a)]. 8.76/8.71 4924 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),C)),D)),E) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)) # label(fact_setsum__addf) # label(axiom). [clausify(775)]. 8.76/8.71 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)). [resolve(4924,a,4908,a)]. 8.76/8.72 4925 class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 8.76/8.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f1(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Code__Numeral_Ocode__numeral)),C)),D)),B). [resolve(4925,a,4909,a)]. 8.76/8.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f1(E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Code__Numeral_Ocode__numeral),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral)),E)),D)),B). [resolve(4925,a,4910,a)]. 8.76/8.72 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B). [resolve(4925,a,4911,b)]. 8.76/8.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(4925,a,4912,a)]. 8.76/8.72 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(4925,a,4913,a)]. 8.76/8.72 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4925,a,4914,a)]. 8.76/8.73 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(4925,a,4915,a)]. 8.76/8.73 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),D),c_List_Oremove1(A,B,C)))). [resolve(4925,a,4916,a)]. 8.76/8.73 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),f80(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(4925,a,4917,a)]. 8.76/8.73 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f80(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(4925,a,4918,a)]. 8.76/8.73 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(4925,a,4919,a)]. 8.76/8.73 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_COMBI(tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A)). [resolve(4925,a,4920,a)]. 8.76/8.73 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B). [resolve(4925,a,4921,a)]. 8.76/8.73 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(4925,a,4922,a)]. 8.76/8.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(4925,a,4923,a)]. 8.76/8.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)). [resolve(4925,a,4924,a)]. 8.76/8.75 4926 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f111(E,D,C,B,A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__zero) # label(axiom). [clausify(905)]. 8.76/8.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f111(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(4926,a,4908,a)]. 8.76/8.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f111(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(4926,a,4925,a)]. 8.76/8.75 4927 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f111(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__zero) # label(axiom). [clausify(905)]. 8.76/8.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f111(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(4927,a,4908,a)]. 8.83/8.79 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f111(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(4927,a,4925,a)]. 8.83/8.79 4928 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum__insert) # label(axiom). [clausify(914)]. 8.83/8.79 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)). [resolve(4928,a,4908,a)]. 8.83/8.79 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)). [resolve(4928,a,4925,a)]. 8.83/8.79 4929 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),C))),hAPP(B,C)) # label(fact_setsum__lessThan__Suc) # label(axiom). [clausify(944)]. 8.83/8.79 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(4929,a,4908,a)]. 8.83/8.79 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(4929,a,4925,a)]. 8.83/8.79 4930 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Nitpick_Osetsum_H(B,A,D,C) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),c_Hilbert__Choice_OEps(tc_List_Olist(B),hAPP(hAPP(c_COMBS(tc_List_Olist(B),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(B)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(B),tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(B)),c_fequal),c_List_Oset(B))),C))),c_List_Odistinct(B))))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom). [clausify(1186)]. 8.83/8.80 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Nat_Onat,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(4930,a,4908,a)]. 8.83/8.80 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Code__Numeral_Ocode__numeral,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(4930,a,4925,a)]. 8.83/8.80 4931 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Nitpick_Osetsum_H(B,A,D,C) = c_Groups_Ozero__class_Ozero(A) # label(fact_Nitpick_Osetsum_H__def) # label(axiom). [clausify(1186)]. 8.83/8.80 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Nat_Onat,C,B) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(4931,a,4908,a)]. 8.83/8.80 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Code__Numeral_Ocode__numeral,C,B) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(4931,a,4925,a)]. 8.83/8.80 4932 -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)),f153(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(1200)]. 8.83/8.80 Derived: -class_Groups_Ocomm__monoid__mult(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)),f153(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(4932,b,4908,a)]. 8.83/8.80 Derived: -class_Groups_Ocomm__monoid__mult(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)),f153(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(4932,b,4925,a)]. 8.83/8.80 4933 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f153(D,E,C,B,A)) != hAPP(E,f153(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(1200)]. 8.83/8.80 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f153(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f153(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(4933,b,4908,a)]. 8.83/8.81 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f153(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f153(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(4933,b,4925,a)]. 8.83/8.81 4934 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(B),C),f155(V6,V7,E,F,C,D,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1229)]. 8.83/8.81 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f155(F,V6,D,E,B,C,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4934,a,4908,a)]. 8.83/8.81 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f155(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4934,a,4925,a)]. 8.83/8.81 4935 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(B),f155(V6,V7,E,F,C,D,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1229)]. 8.83/8.81 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f155(F,V6,D,E,B,C,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4935,a,4908,a)]. 8.83/8.81 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f155(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4935,a,4925,a)]. 8.83/8.81 4936 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hAPP(V6,f155(V6,V7,E,F,C,D,B,A)) != hAPP(V7,f155(V6,V7,E,F,C,D,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(1229)]. 8.92/8.86 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f155(F,V6,D,E,B,C,A,tc_Nat_Onat)) != hAPP(V6,f155(F,V6,D,E,B,C,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4936,a,4908,a)]. 8.92/8.86 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f155(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(V6,f155(F,V6,D,E,B,C,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4936,a,4925,a)]. 8.92/8.86 4937 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,A),B),c_List_Oupt(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(C,D))) # label(fact_interv__listsum__conv__setsum__set__nat) # label(axiom). [clausify(1280)]. 8.92/8.86 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Nat_Onat),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(4937,a,4908,a)]. 8.92/8.86 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(4937,a,4925,a)]. 8.92/8.86 4938 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_List_Odistinct(B),C)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(c_List_Oset(B),C)) # label(fact_listsum__distinct__conv__setsum__set) # label(axiom). [clausify(1418)]. 8.92/8.86 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(c_List_Oset(A),B)). [resolve(4938,a,4908,a)]. 8.92/8.86 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_List_Oset(A),B)). [resolve(4938,a,4925,a)]. 8.92/8.86 4939 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Int_Oint),B),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),B),C) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom). [clausify(1479)]. 8.92/8.86 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Nat_Onat),A),B). [resolve(4939,a,4908,a)]. 8.92/8.86 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),B). [resolve(4939,a,4925,a)]. 8.99/8.94 4940 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D)) # label(fact_setsum__restrict__set) # label(axiom). [clausify(1480)]. 8.99/8.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(4940,a,4908,a)]. 8.99/8.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(4940,a,4925,a)]. 8.99/8.94 4941 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum_Oremove) # label(axiom). [clausify(1617)]. 8.99/8.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4941,a,4908,a)]. 8.99/8.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4941,a,4925,a)]. 8.99/8.94 4942 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,tc_fun(C,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(C,A),tc_fun(tc_fun(C,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A)),hAPP(c_COMBC(C,B,A),D))),E)),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),hAPP(hAPP(c_COMBC(C,tc_fun(B,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(B,A),tc_fun(tc_fun(B,tc_HOL_Obool),A),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A)),D)),F)),E) # label(fact_setsum__commute) # label(axiom). [clausify(1835)]. 8.99/8.95 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Nat_Onat),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat)),hAPP(c_COMBC(B,A,tc_Nat_Onat),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Nat_Onat),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat)),C)),E)),D). [resolve(4942,a,4908,a)]. 8.99/8.95 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_COMBC(B,A,tc_Code__Numeral_Ocode__numeral),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral)),C)),E)),D). [resolve(4942,a,4925,a)]. 8.99/8.95 4943 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -c_Fun_Oinj__on(B,B,D,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(B,A,B,E),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(c_Set_Oimage(B,B,D),C)) # label(fact_setsum_Oreindex) # label(axiom). [clausify(1840)]. 8.99/8.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(A,tc_Nat_Onat,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(4943,a,4908,a)]. 8.99/8.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(A,tc_Code__Numeral_Ocode__numeral,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(4943,a,4925,a)]. 8.99/8.95 4944 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_member(B),f224(C,D,E,B,A)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),E))) # label(fact_setsum__mono) # label(axiom). [clausify(1852)]. 8.99/8.95 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f224(B,C,D,A,tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),D))). [resolve(4944,a,4908,a)]. 8.99/8.95 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f224(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(4944,a,4925,a)]. 8.99/8.96 4945 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f224(C,B,D,E,A))),hAPP(C,f224(C,B,D,E,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),C),D))) # label(fact_setsum__mono) # label(axiom). [clausify(1852)]. 8.99/8.96 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f224(B,A,C,D,tc_Nat_Onat))),hAPP(B,f224(B,A,C,D,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(D,tc_Nat_Onat),A),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),B),C))). [resolve(4945,a,4908,a)]. 8.99/8.96 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,f224(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),hAPP(B,f224(B,A,C,D,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(D,tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(4945,a,4925,a)]. 8.99/8.96 4946 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f227(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__left) # label(axiom). [clausify(1863)]. 8.99/8.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f227(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(4946,a,4908,a)]. 8.99/8.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f227(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(4946,a,4925,a)]. 8.99/8.96 4947 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f227(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D) # label(fact_setsum__mono__zero__left) # label(axiom). [clausify(1863)]. 8.99/8.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f227(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C). [resolve(4947,a,4908,a)]. 9.07/9.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f227(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C). [resolve(4947,a,4925,a)]. 9.07/9.07 4948 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(c_List_Orev(A),B)) = hAPP(c_List_Omonoid__add__class_Olistsum(A),B) # label(fact_listsum__rev) # label(axiom). [clausify(1992)]. 9.07/9.07 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(c_List_Orev(tc_Nat_Onat),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A). [resolve(4948,a,4908,a)]. 9.07/9.07 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A). [resolve(4948,a,4925,a)]. 9.07/9.07 4949 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(A) | -c_Nat__Transfer_Onat__set(B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,tc_Nat_Onat),C),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),B)) # label(fact_transfer__int__nat__sum__prod_I1_J) # label(axiom). [clausify(2060)]. 9.07/9.07 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4949,a,4908,a)]. 9.07/9.07 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4949,a,4925,a)]. 9.07/9.07 4950 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Oplus__class_Oplus(A)),D),c_Groups_Ozero__class_Ozero(A)),C) # label(fact_setsum_OF__eq) # label(axiom). [clausify(2222)]. 9.07/9.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B). [resolve(4950,a,4908,a)]. 9.07/9.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),C),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B). [resolve(4950,a,4925,a)]. 9.07/9.07 4951 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum_OF__eq) # label(axiom). [clausify(2222)]. 9.15/9.11 4952 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Int_Oint,A),B),c_List_Oupto(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),B),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(C,D))) # label(fact_setsum__set__upto__conv__listsum__int) # label(axiom). [clausify(2228)]. 9.15/9.11 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(4952,a,4908,a)]. 9.15/9.11 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(4952,a,4925,a)]. 9.15/9.11 4953 -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)),f280(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(2302)]. 9.15/9.11 Derived: -class_Groups_Ocomm__monoid__mult(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)),f280(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(4953,b,4908,a)]. 9.15/9.11 Derived: -class_Groups_Ocomm__monoid__mult(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)),f280(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(4953,b,4925,a)]. 9.15/9.11 4954 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f280(D,E,C,B,A)) != hAPP(E,f280(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(2302)]. 9.15/9.11 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f280(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f280(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(4954,b,4908,a)]. 9.15/9.11 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f280(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f280(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(4954,b,4925,a)]. 9.15/9.11 4955 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum__empty) # label(axiom). [clausify(2363)]. 9.15/9.15 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(4955,a,4908,a)]. 9.15/9.15 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(4955,a,4925,a)]. 9.15/9.15 4956 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Int_Oint,A),B),c_List_Oupto(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,A),B),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(C,D))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom). [clausify(2424)]. 9.15/9.15 4957 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(tc_fun(D,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A)),F)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),c_Set_OCollect(D)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),hAPP(c_COMBS(D,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),D),c_fconj),hAPP(hAPP(c_COMBC(D,tc_fun(D,tc_HOL_Obool),tc_HOL_Obool),c_member(D)),E)))),V6)))),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),hAPP(hAPP(c_COMBS(D,tc_fun(B,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(B,A),tc_fun(tc_fun(B,tc_HOL_Obool),A),D),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A)),hAPP(c_COMBC(B,D,A),F))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),c_Set_OCollect(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C)))),hAPP(c_COMBC(B,D,tc_HOL_Obool),V6))))),E) # label(fact_setsum__setsum__restrict) # label(axiom). [clausify(2458)]. 9.15/9.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Nat_Onat),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat)),hAPP(c_COMBC(A,C,tc_Nat_Onat),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D). [resolve(4957,a,4908,a)]. 9.15/9.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_COMBC(A,C,tc_Code__Numeral_Ocode__numeral),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D). [resolve(4957,a,4925,a)]. 9.23/9.17 4958 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),E),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D))) # label(fact_setsum__add__nat__ivl) # label(axiom). [clausify(2481)]. 9.23/9.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(4958,a,4908,a)]. 9.23/9.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(4958,a,4925,a)]. 9.23/9.17 4959 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,D),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),E),C))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom). [clausify(2530)]. 9.23/9.19 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(4959,a,4908,a)]. 9.23/9.19 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(4959,a,4925,a)]. 9.23/9.19 4960 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f307(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(2561)]. 9.23/9.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f307(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4960,a,4908,a)]. 9.23/9.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f307(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4960,a,4925,a)]. 9.23/9.19 4961 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f308(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(2561)]. 9.23/9.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f308(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4961,a,4908,a)]. 9.23/9.19 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f308(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4961,a,4925,a)]. 9.23/9.19 4962 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | f308(D,E,F,C,B,A) != f307(D,E,F,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(2561)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f308(C,D,E,B,A,tc_Nat_Onat) != f307(C,D,E,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4962,a,4908,a)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f308(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) != f307(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4962,a,4925,a)]. 9.25/9.20 4963 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f308(E,F,D,C,B,A)) = hAPP(D,f307(E,F,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(E,A,B,F),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),F),hAPP(c_Set_Oimage(B,E,D),C)) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(2561)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f308(D,E,C,B,A,tc_Nat_Onat)) = hAPP(C,f307(D,E,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(D,tc_Nat_Onat,A,E),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,D,C),B)). [resolve(4963,a,4908,a)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f308(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) = hAPP(C,f307(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(D,tc_Code__Numeral_Ocode__numeral,A,E),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,D,C),B)). [resolve(4963,a,4925,a)]. 9.25/9.20 4964 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,hAPP(E,f307(F,D,E,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(F,A,B,D),E)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(F,A),D),hAPP(c_Set_Oimage(B,F,E),C)) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(2561)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,hAPP(D,f307(E,C,D,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(E,tc_Nat_Onat,A,C),D)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Nat_Onat),C),hAPP(c_Set_Oimage(A,E,D),B)). [resolve(4964,a,4908,a)]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,hAPP(D,f307(E,C,D,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(E,tc_Code__Numeral_Ocode__numeral,A,C),D)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Set_Oimage(A,E,D),B)). [resolve(4964,a,4925,a)]. 9.25/9.20 4965 class_Groups_Ocomm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 9.25/9.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f1(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,E),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,E,tc_Int_Oint),C)),hAPP(hAPP(c_Product__Type_OSigma(A,E),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(E,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(E,tc_Int_Oint),tc_fun(tc_fun(E,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Int_Oint)),C)),D)),B). [resolve(4965,a,4909,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f1(E,D,C,B,A,tc_Int_Oint)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,C),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,C,tc_Int_Oint),E)),hAPP(hAPP(c_Product__Type_OSigma(A,C),B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint)),E)),D)),B). [resolve(4965,a,4910,a)]. 9.25/9.21 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),B). [resolve(4965,a,4911,b)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(4965,a,4912,a)]. 9.25/9.21 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),c_Nat_OSuc)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,B)),hAPP(c_Nat_OSuc,C))). [resolve(4965,a,4913,a)]. 9.25/9.21 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4965,a,4914,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,B)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),D),c_List_Oremove1(A,B,C)))). [resolve(4965,a,4916,a)]. 9.25/9.21 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),f80(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(4965,a,4917,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f80(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(4965,a,4918,a)]. 9.25/9.21 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B). [resolve(4965,a,4919,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),A)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),c_COMBI(tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A)). [resolve(4965,a,4920,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_Int_Oint,C),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),hAPP(hAPP(c_COMBB(C,tc_fun(C,tc_HOL_Obool),A),c_fequal),E)))))),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B). [resolve(4965,a,4921,a)]. 9.25/9.21 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),c_Nat_OSuc)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),hAPP(c_Nat_OSuc,C))). [resolve(4965,a,4922,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,A)),B))). [resolve(4965,a,4923,a)]. 9.25/9.21 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)). [resolve(4965,a,4924,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f111(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(4965,a,4926,a)]. 9.25/9.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f111(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(4965,a,4927,a)]. 9.28/9.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)). [resolve(4965,a,4928,a)]. 9.28/9.22 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B))),hAPP(A,B)). [resolve(4965,a,4929,a)]. 9.28/9.22 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Int_Oint,C,B) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),c_Hilbert__Choice_OEps(tc_List_Olist(A),hAPP(hAPP(c_COMBS(tc_List_Olist(A),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(A)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(A),tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(A)),c_fequal),c_List_Oset(A))),B))),c_List_Odistinct(A))))). [resolve(4965,a,4930,a)]. 9.28/9.22 Derived: hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Nitpick_Osetsum_H(A,tc_Int_Oint,C,B) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(4965,a,4931,a)]. 9.28/9.22 Derived: -class_Groups_Ocomm__monoid__mult(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)),f153(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(4965,a,4932,b)]. 9.28/9.22 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f153(C,D,B,A,tc_Int_Oint)) != hAPP(D,f153(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(4965,a,4933,b)]. 9.28/9.22 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f155(F,V6,D,E,B,C,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4965,a,4934,a)]. 9.28/9.22 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f155(F,V6,D,E,B,C,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4965,a,4935,a)]. 9.28/9.22 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f155(F,V6,D,E,B,C,A,tc_Int_Oint)) != hAPP(V6,f155(F,V6,D,E,B,C,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)). [resolve(4965,a,4936,a)]. 9.28/9.22 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,tc_Int_Oint),A),c_List_Oupt(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(B,C))). [resolve(4965,a,4937,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(c_List_Oset(A),B)). [resolve(4965,a,4938,a)]. 9.28/9.23 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Int_Oint),A),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,tc_Int_Oint),A),B). [resolve(4965,a,4939,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C)). [resolve(4965,a,4940,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4965,a,4941,a)]. 9.28/9.23 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Int_Oint),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint)),hAPP(c_COMBC(B,A,tc_Int_Oint),C))),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),hAPP(hAPP(c_COMBC(B,tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Int_Oint),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint)),C)),E)),D). [resolve(4965,a,4942,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,A,C,B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(A,tc_Int_Oint,A,D),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,A,C),B)). [resolve(4965,a,4943,a)]. 9.28/9.23 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f224(B,C,D,A,tc_Int_Oint)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),D))). [resolve(4965,a,4944,a)]. 9.28/9.23 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f224(B,A,C,D,tc_Int_Oint))),hAPP(B,f224(B,A,C,D,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(D,tc_Int_Oint),A),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),B),C))). [resolve(4965,a,4945,a)]. 9.28/9.23 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),f227(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(4965,a,4946,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f227(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C). [resolve(4965,a,4947,a)]. 9.28/9.23 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(c_List_Orev(tc_Int_Oint),A)) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A). [resolve(4965,a,4948,a)]. 9.28/9.23 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | -c_Nat__Transfer_Onat__set(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),B),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Nat_Onat),B),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),A)). [resolve(4965,a,4949,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Oplus__class_Oplus(tc_Int_Oint)),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B). [resolve(4965,a,4950,a)]. 9.28/9.23 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Int_Oint),A),c_List_Oupto(B,C))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(B,C))). [resolve(4965,a,4952,a)]. 9.28/9.23 Derived: -class_Groups_Ocomm__monoid__mult(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)),f280(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(4965,a,4953,b)]. 9.28/9.23 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f280(C,D,B,A,tc_Int_Oint)) != hAPP(D,f280(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(4965,a,4954,b)]. 9.28/9.23 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool))). [resolve(4965,a,4955,a)]. 9.28/9.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Int_Oint),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint)),hAPP(c_COMBC(A,C,tc_Int_Oint),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D). [resolve(4965,a,4957,a)]. 9.28/9.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(4965,a,4958,a)]. 9.28/9.27 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(4965,a,4959,a)]. 9.28/9.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f307(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4965,a,4960,a)]. 9.28/9.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f308(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4965,a,4961,a)]. 9.28/9.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f308(C,D,E,B,A,tc_Int_Oint) != f307(C,D,E,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)). [resolve(4965,a,4962,a)]. 9.28/9.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f308(D,E,C,B,A,tc_Int_Oint)) = hAPP(C,f307(D,E,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(D,tc_Int_Oint,A,E),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,D,C),B)). [resolve(4965,a,4963,a)]. 9.28/9.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,hAPP(D,f307(E,C,D,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(E,tc_Int_Oint,A,C),D)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Int_Oint),C),hAPP(c_Set_Oimage(A,E,D),B)). [resolve(4965,a,4964,a)]. 9.28/9.28 4966 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),C)),D)),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))) # label(fact_listsum__addf) # label(axiom). [clausify(2706)]. 9.28/9.28 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))). [resolve(4966,a,4908,a)]. 9.36/9.31 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),D))). [resolve(4966,a,4925,a)]. 9.36/9.31 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))). [resolve(4966,a,4965,a)]. 9.36/9.31 4967 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Oplus__class_Oplus(A)),D),c_Groups_Ozero__class_Ozero(A)),C) # label(fact_setsum_Oeq__fold) # label(axiom). [clausify(2746)]. 9.36/9.31 4968 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(c_List_Onth(A),B)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) # label(fact_listsum__setsum__nth) # label(axiom). [clausify(2761)]. 9.36/9.31 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_List_Onth(tc_Nat_Onat),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),A))). [resolve(4968,a,4908,a)]. 9.36/9.31 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))). [resolve(4968,a,4925,a)]. 9.36/9.31 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(c_List_Onth(tc_Int_Oint),A)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),A))). [resolve(4968,a,4965,a)]. 9.36/9.31 4969 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E)))) # label(fact_setsum__ub__add__nat) # label(axiom). [clausify(2791)]. 9.36/9.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(4969,a,4908,a)]. 9.36/9.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(4969,a,4925,a)]. 9.36/9.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),D)))). [resolve(4969,a,4965,a)]. 9.36/9.34 4970 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(hAPP(c_Set_Oinsert(B),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,E)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),E),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum_Oinsert__remove) # label(axiom). [clausify(2893)]. 9.36/9.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4970,a,4908,a)]. 9.43/9.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4970,a,4925,a)]. 9.43/9.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(hAPP(c_Set_Oinsert(A),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),D),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(4970,a,4965,a)]. 9.43/9.40 4971 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_add__0) # label(axiom). [clausify(2895)]. 9.43/9.40 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A) = A. [resolve(4971,a,4908,a)]. 9.43/9.40 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(4971,a,4925,a)]. 9.43/9.40 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = A. [resolve(4971,a,4965,a)]. 9.43/9.40 4972 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,A,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_Fun_Oid(A)),hAPP(c_Set_Oimage(B,A,C),D)) # label(fact_setsum__reindex__id) # label(axiom). [clausify(2928)]. 9.43/9.40 Derived: -c_Fun_Oinj__on(A,tc_Nat_Onat,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_Fun_Oid(tc_Nat_Onat)),hAPP(c_Set_Oimage(A,tc_Nat_Onat,B),C)). [resolve(4972,a,4908,a)]. 9.43/9.40 Derived: -c_Fun_Oinj__on(A,tc_Code__Numeral_Ocode__numeral,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_Fun_Oid(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Set_Oimage(A,tc_Code__Numeral_Ocode__numeral,B),C)). [resolve(4972,a,4925,a)]. 9.43/9.40 Derived: -c_Fun_Oinj__on(A,tc_Int_Oint,B,C) | 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(tc_Int_Oint,tc_Int_Oint),c_Fun_Oid(tc_Int_Oint)),hAPP(c_Set_Oimage(A,tc_Int_Oint,B),C)). [resolve(4972,a,4965,a)]. 9.43/9.40 4973 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_member(B),f361(E,D,C,B,A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono3) # label(axiom). [clausify(3015)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f361(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(4973,b,4908,a)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f361(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(4973,b,4925,a)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f361(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(4973,b,4965,a)]. 9.43/9.40 4974 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(E,f361(E,D,C,B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C))) # label(fact_setsum__mono3) # label(axiom). [clausify(3015)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(D,f361(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(4974,b,4908,a)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(D,f361(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(4974,b,4925,a)]. 9.43/9.40 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(D,f361(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(4974,b,4965,a)]. 9.55/9.51 4975 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool))) # label(fact_setsum_Oempty) # label(axiom). [clausify(3086)]. 9.55/9.51 4976 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hBOOL(hAPP(hAPP(c_member(B),f390(E,D,C,B,A)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum_Ounion__inter__neutral) # label(axiom). [clausify(3281)]. 9.55/9.51 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f390(D,C,B,A,tc_Nat_Onat)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(4976,a,4908,a)]. 9.55/9.51 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f390(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(4976,a,4925,a)]. 9.55/9.51 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f390(D,C,B,A,tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(4976,a,4965,a)]. 9.55/9.51 4977 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | c_Groups_Ozero__class_Ozero(A) != hAPP(E,f390(E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum_Ounion__inter__neutral) # label(axiom). [clausify(3281)]. 9.55/9.51 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f390(D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(4977,a,4908,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(D,f390(D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(4977,a,4925,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(D,f390(D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(4977,a,4965,a)]. 9.58/9.55 4978 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),D)),E)),F)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),hAPP(c_Set_OCollect(B),D)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),F),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(B,tc_HOL_Obool)),hAPP(c_Set_OCollect(B),D))))) # label(fact_setsum__cases) # label(axiom). [clausify(3334)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(4978,a,4908,a)]. 9.58/9.55 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(4978,a,4925,a)]. 9.58/9.60 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),C)),D)),E)),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Set_OCollect(A),C)))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_Set_OCollect(A),C))))). [resolve(4978,a,4965,a)]. 9.58/9.60 4979 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_COMBK(A,B),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__0) # label(axiom). [clausify(3346)]. 9.58/9.60 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(4979,a,4908,a)]. 9.58/9.60 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(4979,a,4925,a)]. 9.58/9.60 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(4979,a,4965,a)]. 9.58/9.60 4980 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),C))),hAPP(B,hAPP(c_Nat_OSuc,C))) # label(fact_setsum__atMost__Suc) # label(axiom). [clausify(3448)]. 9.58/9.60 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(4980,a,4908,a)]. 9.58/9.60 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(4980,a,4925,a)]. 9.58/9.60 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B))),hAPP(A,hAPP(c_Nat_OSuc,B))). [resolve(4980,a,4965,a)]. 9.58/9.60 4981 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),B),c_Groups_Oplus__class_Oplus(A)),D)),E)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum_Odistrib) # label(axiom). [clausify(3467)]. 9.67/9.62 4982 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(c_fequal,D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(E,D) # label(fact_setsum__delta_H) # label(axiom). [clausify(3483)]. 9.67/9.62 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(4982,a,4908,a)]. 9.67/9.62 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(4982,a,4925,a)]. 9.67/9.62 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(D,C). [resolve(4982,a,4965,a)]. 9.67/9.62 4983 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(c_fequal,D))),E)),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__delta_H) # label(axiom). [clausify(3483)]. 9.67/9.62 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(4983,a,4908,a)]. 9.67/9.62 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(4983,a,4925,a)]. 9.85/9.80 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(c_fequal,C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(4983,a,4965,a)]. 9.85/9.80 4984 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hBOOL(hAPP(hAPP(c_member(B),f419(V6,V7,F,E,D,C,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(3506)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f419(F,V6,E,D,C,B,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E). [resolve(4984,a,4908,a)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f419(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E). [resolve(4984,a,4925,a)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f419(F,V6,E,D,C,B,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E). [resolve(4984,a,4965,a)]. 9.85/9.80 4985 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(c_Set_Oimage(B,C,D),E) != F | hAPP(V6,hAPP(D,f419(V6,V7,F,E,D,C,B,A))) != hAPP(V7,f419(V6,V7,F,E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(3506)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f419(F,V6,E,D,C,B,A,tc_Nat_Onat))) != hAPP(V6,f419(F,V6,E,D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E). [resolve(4985,a,4908,a)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f419(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(V6,f419(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E). [resolve(4985,a,4925,a)]. 9.85/9.80 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f419(F,V6,E,D,C,B,A,tc_Int_Oint))) != hAPP(V6,f419(F,V6,E,D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E). [resolve(4985,a,4965,a)]. 9.85/9.80 4986 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),C) # label(fact_setsum__infinite) # label(axiom). [clausify(3554)]. 9.85/9.80 4987 -class_Groups_Ocomm__monoid__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),C)) # label(fact_setsum__shift__lb__Suc0__0__upt) # label(axiom). [clausify(3868)]. 9.86/9.82 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(4987,a,4908,a)]. 9.86/9.82 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(4987,a,4925,a)]. 9.86/9.82 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(4987,a,4965,a)]. 9.86/9.82 4988 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f467(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.82 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4988,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4988,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4988,a,4965,a)]. 9.86/9.83 4989 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f468(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4989,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4989,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4989,a,4965,a)]. 9.86/9.83 4990 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | f468(D,C,B,A) != f467(D,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | f468(C,B,A,tc_Nat_Onat) != f467(C,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4990,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | f468(C,B,A,tc_Code__Numeral_Ocode__numeral) != f467(C,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4990,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | f468(C,B,A,tc_Int_Oint) != f467(C,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4990,a,4965,a)]. 9.86/9.83 4991 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f469(D,C,B,A)),f467(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Nat_Onat)),f467(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4991,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Code__Numeral_Ocode__numeral)),f467(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4991,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Int_Oint)),f467(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4991,a,4965,a)]. 9.86/9.83 4992 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f469(D,C,B,A)),f468(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Nat_Onat)),f468(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4992,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Code__Numeral_Ocode__numeral)),f468(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4992,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Int_Oint)),f468(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4992,a,4965,a)]. 9.86/9.83 4993 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f466(D,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f469(D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f469(C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4993,a,4908,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f469(C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4993,a,4925,a)]. 9.86/9.83 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f466(C,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f469(C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4993,a,4965,a)]. 9.86/9.84 4994 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f467(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4994,a,4908,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4994,a,4925,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f467(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4994,a,4965,a)]. 9.86/9.84 4995 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f468(D,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4995,a,4908,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4995,a,4925,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f468(C,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4995,a,4965,a)]. 9.86/9.84 4996 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | f468(D,C,B,A) != f467(D,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | f468(C,B,A,tc_Nat_Onat) != f467(C,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4996,a,4908,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | f468(C,B,A,tc_Code__Numeral_Ocode__numeral) != f467(C,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4996,a,4925,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | f468(C,B,A,tc_Int_Oint) != f467(C,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4996,a,4965,a)]. 9.86/9.84 4997 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f469(D,C,B,A)),f467(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Nat_Onat)),f467(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4997,a,4908,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Code__Numeral_Ocode__numeral)),f467(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4997,a,4925,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Int_Oint)),f467(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4997,a,4965,a)]. 9.86/9.84 4998 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f469(D,C,B,A)),f468(D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Nat_Onat)),f468(C,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4998,a,4908,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Code__Numeral_Ocode__numeral)),f468(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4998,a,4925,a)]. 9.86/9.84 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f469(C,B,A,tc_Int_Oint)),f468(C,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4998,a,4965,a)]. 9.86/9.84 4999 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(B,tc_HOL_Obool)),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),f466(D,C,B,A))) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f469(D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(B,tc_HOL_Obool),A),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D)),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),c_Complete__Lattice_OSup__class_OSup(tc_fun(B,tc_HOL_Obool),C)) # label(fact_setsum__UNION__zero) # label(axiom). [clausify(3914)]. 9.86/9.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Nat_Onat))) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f469(C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4999,a,4908,a)]. 9.86/9.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Code__Numeral_Ocode__numeral))) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f469(C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4999,a,4925,a)]. 9.86/9.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f466(C,B,A,tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f469(C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C)),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_fun(A,tc_HOL_Obool),B)). [resolve(4999,a,4965,a)]. 9.86/9.85 5000 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),D)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,tc_HOL_Obool)),C),D))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,tc_HOL_Obool)),C),D))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),D)) # label(fact_setsum__Un__Int) # label(axiom). [clausify(3927)]. 9.86/9.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),C)). [resolve(5000,a,4908,a)]. 9.86/9.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),C)). [resolve(5000,a,4925,a)]. 9.86/9.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,tc_HOL_Obool)),B),C))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),C)). [resolve(5000,a,4965,a)]. 9.86/9.86 5001 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,B,tc_HOL_Obool),c_fequal),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(E,D) # label(fact_setsum__delta) # label(axiom). [clausify(3941)]. 9.86/9.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(D,C). [resolve(5001,a,4908,a)]. 9.86/9.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(D,C). [resolve(5001,a,4925,a)]. 9.86/9.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(D,C). [resolve(5001,a,4965,a)]. 9.86/9.86 5002 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,B,tc_HOL_Obool),c_fequal),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) # label(fact_setsum__delta) # label(axiom). [clausify(3941)]. 9.86/9.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B). [resolve(5002,a,4908,a)]. 9.97/9.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B). [resolve(5002,a,4925,a)]. 9.97/9.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,A,tc_HOL_Obool),c_fequal),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B). [resolve(5002,a,4965,a)]. 9.97/9.91 5003 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Set_Oinsert(B),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C)) # label(fact_setsum_Oinsert) # label(axiom). [clausify(3953)]. 9.97/9.91 5004 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,A,A),hAPP(hAPP(c_COMBS(B,A,tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),B),c_If(A)),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D))),E)),c_Groups_Ozero__class_Ozero(A))),C) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(c_Set_OCollect(B),hAPP(hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C))),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),D)))) # label(fact_setsum__restrict__set_H) # label(axiom). [clausify(4047)]. 9.97/9.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),A),c_If(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5004,a,4908,a)]. 9.97/9.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),A),c_If(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5004,a,4925,a)]. 9.97/9.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),A),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C))),D)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),C)))). [resolve(5004,a,4965,a)]. 9.97/9.97 5005 -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),f490(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(4145)]. 9.97/9.97 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f490(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(5005,a,4908,a)]. 9.97/9.97 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f490(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(5005,a,4925,a)]. 9.97/9.97 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f490(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(5005,a,4965,a)]. 9.97/9.97 5006 -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,f490(E,D,C,B,A))),hAPP(E,f490(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(4145)]. 10.06/10.01 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(C,f490(D,C,B,A,tc_Nat_Onat))),hAPP(D,f490(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(5006,a,4908,a)]. 10.06/10.01 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(C,f490(D,C,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f490(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(5006,a,4925,a)]. 10.06/10.01 Derived: -class_Groups_Oordered__cancel__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(C,f490(D,C,B,A,tc_Int_Oint))),hAPP(D,f490(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(5006,a,4965,a)]. 10.06/10.01 5007 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(B,C),A),hAPP(c_Product__Type_Oprod_Oprod__case(B,C,A),D)),hAPP(hAPP(c_Product__Type_OSigma(B,C),E),hAPP(c_COMBK(tc_fun(C,tc_HOL_Obool),B),F))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBC(B,tc_fun(C,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(C,A),tc_fun(tc_fun(C,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A)),D)),F)),E) # label(fact_setsum__cartesian__product) # label(axiom). [clausify(4217)]. 10.06/10.01 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Nat_Onat),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Nat_Onat),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat)),C)),E)),D). [resolve(5007,a,4908,a)]. 10.06/10.01 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Code__Numeral_Ocode__numeral),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Code__Numeral_Ocode__numeral),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral)),C)),E)),D). [resolve(5007,a,4925,a)]. 10.06/10.01 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(A,B),tc_Int_Oint),hAPP(c_Product__Type_Oprod_Oprod__case(A,B,tc_Int_Oint),C)),hAPP(hAPP(c_Product__Type_OSigma(A,B),D),hAPP(c_COMBK(tc_fun(B,tc_HOL_Obool),A),E))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBC(A,tc_fun(B,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(B,tc_Int_Oint),tc_fun(tc_fun(B,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint)),C)),E)),D). [resolve(5007,a,4965,a)]. 10.18/10.20 5008 -class_Groups_Ocomm__monoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,A),B),c_List_Oupt(C,D))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(C,D))) # label(fact_setsum__set__upt__conv__listsum__nat) # label(axiom). [clausify(4297)]. 10.18/10.20 5009 -class_Groups_Ocomm__monoid__add(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,tc_Nat_Onat),B),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,D,E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),B),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),E),C))) # label(fact_setsum__shift__bounds__cl__nat__ivl) # label(axiom). [clausify(4496)]. 10.18/10.20 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5009,a,4908,a)]. 10.18/10.20 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5009,a,4925,a)]. 10.18/10.20 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),A),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),D),B))). [resolve(5009,a,4965,a)]. 10.18/10.20 5010 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),hAPP(c_Nat_OSuc,B))) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(4538)]. 10.18/10.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5010,a,4908,a)]. 10.28/10.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5010,a,4925,a)]. 10.28/10.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))). [resolve(5010,a,4965,a)]. 10.28/10.26 5011 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),B))),hAPP(D,B)) # label(fact_setsum__op__ivl__Suc) # label(axiom). [clausify(4538)]. 10.28/10.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5011,a,4908,a)]. 10.28/10.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5011,a,4925,a)]. 10.28/10.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),A))),hAPP(C,A)). [resolve(5011,a,4965,a)]. 10.28/10.26 5012 -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,B)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,B))),hAPP(D,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(4622)]. 10.28/10.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5012,a,4908,a)]. 10.28/10.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5012,a,4925,a)]. 10.28/10.27 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,A))),hAPP(C,hAPP(c_Nat_OSuc,A))). [resolve(5012,a,4965,a)]. 10.28/10.27 5013 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,B)),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,C,hAPP(c_Nat_OSuc,B))) # label(fact_setsum__cl__ivl__Suc) # label(axiom). [clausify(4622)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5013,a,4908,a)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5013,a,4925,a)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,hAPP(c_Nat_OSuc,A))). [resolve(5013,a,4965,a)]. 10.28/10.27 5014 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,B,C))) # label(fact_setsum__head) # label(axiom). [clausify(4632)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5014,a,4908,a)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5014,a,4925,a)]. 10.28/10.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B))). [resolve(5014,a,4965,a)]. 11.14/11.08 5015 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,B,C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(D,B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,A),D),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,B),C))) # label(fact_setsum__head__Suc) # label(axiom). [clausify(4782)]. 11.14/11.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5015,a,4908,a)]. 11.14/11.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5015,a,4925,a)]. 11.14/11.08 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,A,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(C,A)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),C),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,A),B))). [resolve(5015,a,4965,a)]. 11.14/11.08 5016 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(E,D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),E),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(B,tc_HOL_Obool)),C),hAPP(hAPP(c_Set_Oinsert(B),D),c_Orderings_Obot__class_Obot(tc_fun(B,tc_HOL_Obool)))))) # label(fact_setsum__diff1_H) # label(axiom). [clausify(4864)]. 11.14/11.08 5017 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(C,A,B,F),D)),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),F),hAPP(c_Set_Oimage(B,C,D),E)) # label(fact_setsum__reindex) # label(axiom). [clausify(4886)]. 11.14/11.08 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,E),C)),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,B,C),D)). [resolve(5017,a,4908,a)]. 11.14/11.08 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,E),C)),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,B,C),D)). [resolve(5017,a,4925,a)]. 11.14/11.08 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,E),C)),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,B,C),D)). [resolve(5017,a,4965,a)]. 11.14/11.08 5018 class_Orderings_Olinorder(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Orderings_Olinorder) # label(axiom). [assumption]. 11.14/11.08 5019 -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(6)]. 11.14/11.08 5020 -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(6)]. 11.14/11.08 5021 -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(11)]. 11.14/11.08 5022 -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(33)]. 11.14/11.08 5023 -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(47)]. 11.14/11.08 5024 -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(48)]. 11.14/11.08 5025 -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_less__max__iff__disj) # label(axiom). [clausify(93)]. 11.14/11.08 5026 -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_less__max__iff__disj) # label(axiom). [clausify(93)]. 11.14/11.08 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),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_less__max__iff__disj) # label(axiom). [clausify(93)]. 11.14/11.08 5028 -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(100)]. 11.14/11.08 5029 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f18(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(116)]. 11.14/11.08 5030 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f18(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(116)]. 11.14/11.08 5031 -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(134)]. 11.14/11.08 5032 -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(142)]. 11.14/11.08 5033 -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(150)]. 11.14/11.08 5034 -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(163)]. 11.14/11.08 5035 -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(191)]. 11.14/11.08 5036 -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(197)]. 11.14/11.08 5037 -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(216)]. 11.14/11.08 5038 -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(216)]. 11.14/11.08 5039 -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(239)]. 11.14/11.08 5040 -class_Orderings_Olinorder(A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)) = hAPP(c_SetInterval_Oord__class_OatMost(A),B) # label(fact_Compl__greaterThan) # label(axiom). [clausify(255)]. 11.14/11.08 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 | 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(271)]. 11.14/11.08 5042 -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(304)]. 11.14/11.08 5043 -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(308)]. 11.14/11.08 5044 -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(358)]. 11.14/11.08 5045 -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(358)]. 11.14/11.08 5046 -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(379)]. 11.14/11.08 5047 -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(395)]. 11.14/11.08 5048 -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(395)]. 11.14/11.08 5049 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__max_Ole__inf__iff) # label(axiom). [clausify(395)]. 11.14/11.08 5050 -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(406)]. 11.14/11.08 5051 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f56(C,B,A))),hAPP(B,f57(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f56(C,B,A)),f57(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(408)]. 11.14/11.08 5052 -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(432)]. 11.14/11.08 5053 -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(446)]. 11.14/11.08 5054 -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(457)]. 11.14/11.08 5055 -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(457)]. 11.14/11.08 5056 -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(481)]. 11.14/11.08 5057 -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(517)]. 11.14/11.08 5058 -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(533)]. 11.14/11.08 5059 -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(548)]. 11.14/11.08 5060 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f73(B,A)),f74(B,A))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(558)]. 11.14/11.08 5061 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f74(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(558)]. 11.14/11.08 5062 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f73(B,A))),hAPP(hAPP(c_List_Onth(A),B),f74(B,A)))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(558)]. 11.14/11.08 5063 -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(566)]. 11.14/11.08 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_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(568)]. 11.14/11.08 5065 -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),f76(C,B,A)),B)) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(568)]. 11.14/11.08 5066 -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),f76(C,B,A))) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(568)]. 11.14/11.08 5067 -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(591)]. 11.14/11.08 5068 -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(594)]. 11.14/11.08 5069 -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(604)]. 11.14/11.08 5070 -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(606)]. 11.14/11.08 5071 -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(617)]. 11.14/11.08 5072 -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(627)]. 11.14/11.08 5073 -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_linorder__not__le) # label(axiom). [clausify(648)]. 11.14/11.08 5074 -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_linorder__not__le) # label(axiom). [clausify(648)]. 11.14/11.08 5075 -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(651)]. 11.14/11.08 5076 -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(682)]. 11.14/11.08 5077 -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(705)]. 11.14/11.08 5078 -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(742)]. 11.14/11.08 5079 -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(763)]. 11.14/11.08 5080 -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(814)]. 11.14/11.08 5081 -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(838)]. 11.14/11.08 5082 -class_Orderings_Olinorder(A) | -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),D),E))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),E)) # label(fact_sorted__insort__key) # label(axiom). [clausify(840)]. 11.14/11.08 5083 -class_Orderings_Olinorder(A) | 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),D),E))) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Omap(B,A),C),E)) # label(fact_sorted__insort__key) # label(axiom). [clausify(840)]. 11.14/11.08 5084 -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(852)]. 11.14/11.08 5085 -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(876)]. 11.14/11.08 5086 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f112(C,B,A))),hAPP(B,f113(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f112(C,B,A)),f113(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(906)]. 11.14/11.08 5087 -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(931)]. 11.14/11.08 5088 -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),f118(C,B,A)),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))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(948)]. 11.14/11.08 5089 -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),f118(C,B,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(948)]. 11.14/11.08 5090 -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),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(948)]. 11.14/11.08 5091 -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(952)]. 11.14/11.08 5092 -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_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(955)]. 11.14/11.08 5093 -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(974)]. 11.14/11.08 5094 -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(1039)]. 11.14/11.08 5095 -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(1042)]. 11.14/11.08 5096 -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(1042)]. 11.14/11.08 5097 -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(1042)]. 11.14/11.08 5098 -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(1048)]. 11.14/11.08 5099 -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(1076)]. 11.14/11.08 5100 -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(1083)]. 11.14/11.08 5101 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f134(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(1086)]. 11.14/11.08 5102 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f134(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(1086)]. 11.14/11.08 5103 -class_Orderings_Olinorder(A) | -class_Lattices_Osemilattice__inf(A) | c_Orderings_Oord__class_Omin(A) = c_Lattices_Osemilattice__inf__class_Oinf(A) # label(fact_inf__min) # label(axiom). [clausify(1088)]. 11.14/11.08 5104 -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(1101)]. 11.14/11.08 5105 -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(1114)]. 11.16/11.09 5106 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_min__max_Osup_Oidem) # label(axiom). [clausify(1189)]. 11.16/11.09 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(5018,a,5019,a)]. 11.16/11.09 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(5018,a,5020,a)]. 11.16/11.09 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(5018,a,5021,a)]. 11.16/11.09 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(5018,a,5022,a)]. 11.16/11.09 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(5018,a,5023,a)]. 11.16/11.09 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(5018,a,5024,a)]. 11.16/11.10 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),B),C))). [resolve(5018,a,5025,a)]. 11.16/11.10 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(5018,a,5026,a)]. 11.16/11.10 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),C)) | -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))). [resolve(5018,a,5027,a)]. 11.16/11.10 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(5018,a,5028,a)]. 11.16/11.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f18(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(5018,a,5029,a)]. 11.16/11.10 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),f18(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(5018,a,5030,a)]. 11.16/11.10 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(5018,a,5031,a)]. 11.16/11.10 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(5018,a,5032,a)]. 11.16/11.10 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(5018,a,5033,a)]. 11.16/11.10 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(5018,a,5034,a)]. 11.16/11.10 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(5018,a,5035,a)]. 11.16/11.11 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(5018,a,5036,a)]. 11.16/11.11 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(5018,a,5037,a)]. 11.16/11.11 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(5018,a,5038,a)]. 11.16/11.11 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(5018,a,5039,a)]. 11.16/11.11 Derived: 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)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A). [resolve(5018,a,5040,a)]. 11.16/11.11 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(5018,a,5041,a)]. 11.16/11.11 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(5018,a,5042,a)]. 11.16/11.11 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(5018,a,5043,a)]. 11.16/11.11 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(5018,a,5044,a)]. 11.16/11.11 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(5018,a,5045,a)]. 11.16/11.11 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(5018,a,5046,a)]. 11.19/11.12 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(5018,a,5047,a)]. 11.19/11.12 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(5018,a,5048,a)]. 11.19/11.12 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5018,a,5049,a)]. 11.19/11.12 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(5018,a,5050,a)]. 11.19/11.12 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f56(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f57(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f56(B,A,tc_Code__Numeral_Ocode__numeral)),f57(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(5018,a,5051,a)]. 11.19/11.12 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),c_List_Oremove1(A,D,C))). [resolve(5018,a,5052,a)]. 11.19/11.12 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(5018,a,5053,a)]. 11.19/11.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | B != A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5018,a,5054,a)]. 11.19/11.12 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | B = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5018,a,5055,a)]. 11.19/11.12 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(5018,a,5056,a)]. 11.20/11.13 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(5018,a,5057,a)]. 11.20/11.13 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(5018,a,5058,a)]. 11.20/11.13 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(5018,a,5059,a)]. 11.20/11.13 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f73(A,tc_Code__Numeral_Ocode__numeral)),f74(A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5018,a,5060,a)]. 11.20/11.13 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f74(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(5018,a,5061,a)]. 11.20/11.13 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),f73(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f74(A,tc_Code__Numeral_Ocode__numeral)))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5018,a,5062,a)]. 11.20/11.13 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),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(5018,a,5063,a)]. 11.20/11.13 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(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(5018,a,5064,a)]. 11.20/11.13 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(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),f76(B,A,tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5018,a,5065,a)]. 11.20/11.13 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(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),f76(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5018,a,5066,a)]. 11.20/11.14 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(5018,a,5067,a)]. 11.20/11.14 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_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(5018,a,5068,a)]. 11.20/11.14 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(5018,a,5069,a)]. 11.20/11.14 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(5018,a,5070,a)]. 11.20/11.14 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_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(5018,a,5071,a)]. 11.20/11.14 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(5018,a,5072,a)]. 11.20/11.14 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),A)). [resolve(5018,a,5073,a)]. 11.20/11.14 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),A)). [resolve(5018,a,5074,a)]. 11.20/11.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B)). [resolve(5018,a,5076,a)]. 11.20/11.14 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(5018,a,5077,a)]. 11.20/11.15 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(5018,a,5078,a)]. 11.20/11.15 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 | -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(5018,a,5079,a)]. 11.20/11.15 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(5018,a,5080,a)]. 11.20/11.15 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(5018,a,5081,a)]. 11.20/11.15 Derived: -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),C),D))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5018,a,5082,a)]. 11.20/11.15 Derived: 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),C),D))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5018,a,5083,a)]. 11.20/11.15 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),A),B) = B. [resolve(5018,a,5084,a)]. 11.20/11.15 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(5018,a,5085,a)]. 11.20/11.15 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f112(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f113(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f112(B,A,tc_Code__Numeral_Ocode__numeral)),f113(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(5018,a,5086,a)]. 11.20/11.17 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(5018,a,5087,a)]. 11.20/11.17 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),f118(B,A,tc_Code__Numeral_Ocode__numeral)),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))). [resolve(5018,a,5088,a)]. 11.20/11.17 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),f118(B,A,tc_Code__Numeral_Ocode__numeral))) | 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))). [resolve(5018,a,5089,a)]. 11.20/11.17 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))). [resolve(5018,a,5090,a)]. 11.20/11.17 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(5018,a,5091,a)]. 11.20/11.17 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_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(5018,a,5092,a)]. 11.20/11.17 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(5018,a,5093,a)]. 11.20/11.17 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(5018,a,5094,a)]. 11.20/11.17 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(5018,a,5097,a)]. 11.20/11.17 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(5018,a,5098,a)]. 11.20/11.19 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(5018,a,5099,a)]. 11.20/11.19 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f134(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(5018,a,5101,a)]. 11.20/11.19 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),f134(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(5018,a,5102,a)]. 11.20/11.19 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(5018,a,5103,a)]. 11.20/11.19 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(5018,a,5104,a)]. 11.20/11.19 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(5018,a,5105,a)]. 11.20/11.19 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5018,a,5106,a)]. 11.20/11.19 5107 -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(1201)]. 11.20/11.19 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(5107,a,5018,a)]. 11.20/11.19 5108 -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(1203)]. 11.20/11.19 5109 -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),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),D),C))) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1256)]. 11.20/11.19 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),C)) | 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),C),B))). [resolve(5109,a,5018,a)]. 11.28/11.20 5110 -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_Omin(A),D),C))) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1256)]. 11.28/11.20 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_Omin(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5110,a,5018,a)]. 11.28/11.20 5111 -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_Omin(A),C),D))) # label(fact_min__less__iff__conj) # label(axiom). [clausify(1256)]. 11.28/11.20 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_Omin(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5111,a,5018,a)]. 11.28/11.20 5112 -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(1302)]. 11.28/11.20 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(5112,a,5018,a)]. 11.28/11.20 5113 -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(1305)]. 11.28/11.20 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(5113,a,5018,a)]. 11.28/11.20 5114 -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(1305)]. 11.28/11.20 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(5114,a,5018,a)]. 11.28/11.20 5115 -class_Orderings_Olinorder(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))) | -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(1307)]. 11.28/11.20 5116 -class_Orderings_Olinorder(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))) | 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(1307)]. 11.28/11.20 5117 -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(1309)]. 11.28/11.21 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(5117,a,5018,a)]. 11.28/11.21 5118 -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(1309)]. 11.28/11.21 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(5118,a,5018,a)]. 11.28/11.21 5119 -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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) | D = B # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1315)]. 11.28/11.21 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)) | 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) | C = A. [resolve(5119,a,5018,a)]. 11.28/11.21 5120 -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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) | E = C # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1315)]. 11.28/11.21 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)) | 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) | D = B. [resolve(5120,a,5018,a)]. 11.28/11.22 5121 -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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),C) | D != B | E != C # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1315)]. 11.28/11.22 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)) | 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) | C != A | D != B. [resolve(5121,a,5018,a)]. 11.28/11.22 5122 -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(1321)]. 11.28/11.22 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(5122,a,5018,a)]. 11.28/11.22 5123 -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,C) # label(fact_sorted__Cons) # label(axiom). [clausify(1325)]. 11.28/11.22 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5123,a,5018,a)]. 11.28/11.22 5124 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),C)) | -hBOOL(hAPP(hAPP(c_member(A),D),hAPP(c_List_Oset(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_sorted__Cons) # label(axiom). [clausify(1325)]. 11.28/11.22 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5124,a,5018,a)]. 11.28/11.22 5125 -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,C) | hBOOL(hAPP(hAPP(c_member(A),f162(C,B,A)),hAPP(c_List_Oset(A),C))) # label(fact_sorted__Cons) # label(axiom). [clausify(1325)]. 11.28/11.22 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f162(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))). [resolve(5125,a,5018,a)]. 11.28/11.22 5126 -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,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f162(C,B,A))) # label(fact_sorted__Cons) # label(axiom). [clausify(1325)]. 11.28/11.22 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) | -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),f162(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5126,a,5018,a)]. 11.28/11.22 5127 -class_Orderings_Olinorder(A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)) = hAPP(c_SetInterval_Oord__class_OatLeast(A),B) # label(fact_Compl__lessThan) # label(axiom). [clausify(1354)]. 11.28/11.25 Derived: 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)) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A). [resolve(5127,a,5018,a)]. 11.28/11.25 5128 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(B,C)),D)) | -hBOOL(hAPP(hAPP(c_member(E),C),F)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(E,A),F),B)),D)) # label(fact_INF__less__iff) # label(axiom). [clausify(1389)]. 11.28/11.25 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(A,B)),C)) | -hBOOL(hAPP(hAPP(c_member(D),B),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Code__Numeral_Ocode__numeral),E),A)),C)). [resolve(5128,a,5018,a)]. 11.28/11.25 5129 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(B,f170(C,B,D,E,A))),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(E,A),D),B)),C)) # label(fact_INF__less__iff) # label(axiom). [clausify(1389)]. 11.28/11.25 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(A,f170(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Code__Numeral_Ocode__numeral),C),A)),B)). [resolve(5129,a,5018,a)]. 11.28/11.25 5130 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_member(B),f170(C,D,E,B,A)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(B,A),E),D)),C)) # label(fact_INF__less__iff) # label(axiom). [clausify(1389)]. 11.28/11.25 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f170(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),D)) | -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),D),C)),B)). [resolve(5130,a,5018,a)]. 11.28/11.25 5131 -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(1416)]. 11.28/11.25 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(5131,a,5018,a)]. 11.28/11.25 5132 -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(1425)]. 11.28/11.25 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_Omin(A),D),B)),C)) # label(fact_min__max_Oless__infI2) # label(axiom). [clausify(1428)]. 11.28/11.26 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(5133,a,5018,a)]. 11.28/11.26 5134 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,f178(B,A)) | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f178(A,tc_Code__Numeral_Ocode__numeral)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5134,a,5018,a)]. 11.28/11.26 5135 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),f178(B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f179(B,A)),C)) | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 Derived: -c_List_Olinorder__class_Osorted(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),f178(A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f179(A,tc_Code__Numeral_Ocode__numeral)),B)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5135,a,5018,a)]. 11.28/11.26 5136 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hAPP(hAPP(c_List_Olist_OCons(A),f179(B,A)),f178(B,A)) = B | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),f179(A,tc_Code__Numeral_Ocode__numeral)),f178(A,tc_Code__Numeral_Ocode__numeral)) = A | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5136,a,5018,a)]. 11.28/11.26 5137 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,C) | hBOOL(hAPP(hAPP(c_member(A),f180(B,A,C,D)),hAPP(c_List_Oset(A),C))) | hAPP(hAPP(c_List_Olist_OCons(A),D),C) != B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 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),f180(A,tc_Code__Numeral_Ocode__numeral,B,C)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),C),B) != A. [resolve(5137,a,5018,a)]. 11.28/11.26 5138 -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),D),f180(B,A,C,D))) | hAPP(hAPP(c_List_Olist_OCons(A),D),C) != B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 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),C),f180(A,tc_Code__Numeral_Ocode__numeral,B,C))) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),C),B) != A. [resolve(5138,a,5018,a)]. 11.28/11.26 5139 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | c_List_Olist_ONil(A) != B # label(fact_sorted_Osimps) # label(axiom). [clausify(1430)]. 11.28/11.26 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) != A. [resolve(5139,a,5018,a)]. 11.28/11.26 5140 -class_Orderings_Olinorder(A) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_lessThan__subset__iff) # label(axiom). [clausify(1439)]. 11.34/11.30 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_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__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5140,a,5018,a)]. 11.34/11.30 5141 -class_Orderings_Olinorder(A) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_lessThan__subset__iff) # label(axiom). [clausify(1439)]. 11.34/11.30 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_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__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5141,a,5018,a)]. 11.34/11.30 5142 -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(1501)]. 11.34/11.30 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(5142,a,5018,a)]. 11.34/11.30 5143 -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(1505)]. 11.34/11.30 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(5143,a,5018,a)]. 11.34/11.30 5144 -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(1523)]. 11.34/11.30 5145 -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(1578)]. 11.34/11.30 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(5145,a,5018,a)]. 11.34/11.30 5146 -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(1579)]. 11.34/11.30 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(5146,a,5018,a)]. 11.34/11.38 5147 -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(1592)]. 11.34/11.38 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(5147,a,5018,a)]. 11.34/11.38 5148 -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(1602)]. 11.34/11.38 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_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(5148,a,5018,a)]. 11.34/11.38 5149 -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(1619)]. 11.34/11.38 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(5149,a,5018,a)]. 11.34/11.38 5150 -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(1690)]. 11.34/11.38 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(5150,a,5018,a)]. 11.34/11.38 5151 -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(1789)]. 11.34/11.38 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)) = 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_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(5151,a,5018,a)]. 11.45/11.39 5152 -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(1793)]. 11.45/11.39 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5152,a,5018,a)]. 11.45/11.39 5153 -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(1793)]. 11.45/11.39 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5153,a,5018,a)]. 11.45/11.39 5154 -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(1798)]. 11.45/11.39 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)) | 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(5154,a,5018,a)]. 11.45/11.39 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(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(1821)]. 11.45/11.39 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(5155,a,5018,a)]. 11.45/11.40 5156 -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(1834)]. 11.45/11.40 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(5156,a,5018,a)]. 11.45/11.40 5157 -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(1836)]. 11.45/11.40 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(5157,a,5018,a)]. 11.45/11.40 5158 -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(1853)]. 11.45/11.40 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(5158,a,5018,a)]. 11.45/11.40 5159 -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(1856)]. 11.45/11.40 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 | 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(5159,a,5018,a)]. 11.51/11.45 5160 -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(1857)]. 11.51/11.45 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(5160,a,5018,a)]. 11.51/11.45 5161 -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(1870)]. 11.51/11.45 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(5161,a,5018,a)]. 11.51/11.45 5162 -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(1887)]. 11.51/11.45 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(5162,a,5018,a)]. 11.51/11.45 5163 -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(1928)]. 11.51/11.45 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(5163,a,5018,a)]. 11.51/11.45 5164 -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(1969)]. 11.51/11.45 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(5164,a,5018,a)]. 11.51/11.45 5165 -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(2004)]. 11.51/11.45 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(5165,a,5018,a)]. 11.51/11.47 5166 -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(2009)]. 11.51/11.47 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_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(5166,a,5018,a)]. 11.51/11.47 5167 -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(2010)]. 11.51/11.47 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(5167,a,5018,a)]. 11.51/11.47 5168 -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(2014)]. 11.51/11.47 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(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(5168,a,5018,a)]. 11.51/11.47 5169 class_Orderings_Olinorder(tc_Int_Oint) # label(arity_Int__Oint__Orderings_Olinorder) # label(axiom). [assumption]. 11.51/11.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != 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(5169,a,5019,a)]. 11.51/11.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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(5169,a,5020,a)]. 11.51/11.47 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(5169,a,5021,a)]. 11.51/11.47 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(5169,a,5022,a)]. 11.51/11.48 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(5169,a,5023,a)]. 11.51/11.48 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(5169,a,5024,a)]. 11.51/11.48 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),B),C))). [resolve(5169,a,5025,a)]. 11.51/11.48 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(5169,a,5026,a)]. 11.51/11.48 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),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))). [resolve(5169,a,5027,a)]. 11.51/11.48 Derived: c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5169,a,5028,a)]. 11.51/11.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f18(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(5169,a,5029,a)]. 11.51/11.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f18(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(5169,a,5030,a)]. 11.51/11.48 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(5169,a,5031,a)]. 11.51/11.48 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(5169,a,5032,a)]. 11.51/11.48 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(5169,a,5033,a)]. 11.51/11.48 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(5169,a,5034,a)]. 11.51/11.49 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(5169,a,5035,a)]. 11.51/11.49 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(5169,a,5036,a)]. 11.51/11.49 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(5169,a,5037,a)]. 11.51/11.49 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(5169,a,5038,a)]. 11.51/11.49 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(5169,a,5039,a)]. 11.51/11.49 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A). [resolve(5169,a,5040,a)]. 11.51/11.49 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(5169,a,5041,a)]. 11.51/11.49 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(5169,a,5042,a)]. 11.51/11.49 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(5169,a,5043,a)]. 11.51/11.49 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B) | A = B. [resolve(5169,a,5044,a)]. 11.51/11.49 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B) | A != B. [resolve(5169,a,5045,a)]. 11.51/11.49 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(5169,a,5046,a)]. 11.51/11.49 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(5169,a,5047,a)]. 11.51/11.49 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(5169,a,5048,a)]. 11.51/11.49 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5169,a,5049,a)]. 11.51/11.50 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(5169,a,5050,a)]. 11.51/11.50 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f56(B,A,tc_Int_Oint))),hAPP(A,f57(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f56(B,A,tc_Int_Oint)),f57(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(5169,a,5051,a)]. 11.51/11.50 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(5169,a,5052,a)]. 11.51/11.50 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(5169,a,5053,a)]. 11.51/11.50 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(5169,a,5056,a)]. 11.51/11.50 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(5169,a,5057,a)]. 11.51/11.50 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(5169,a,5058,a)]. 11.51/11.50 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(5169,a,5059,a)]. 11.51/11.50 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f73(A,tc_Int_Oint)),f74(A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5169,a,5060,a)]. 11.51/11.50 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f74(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(5169,a,5061,a)]. 11.51/11.50 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f73(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f74(A,tc_Int_Oint)))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5169,a,5062,a)]. 11.51/11.50 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),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(5169,a,5063,a)]. 11.51/11.51 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(5169,a,5064,a)]. 11.51/11.51 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),f76(B,A,tc_Int_Oint)),A)). [resolve(5169,a,5065,a)]. 11.51/11.51 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),f76(B,A,tc_Int_Oint))). [resolve(5169,a,5066,a)]. 11.51/11.51 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)). [resolve(5169,a,5067,a)]. 11.51/11.51 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_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(5169,a,5068,a)]. 11.51/11.51 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(5169,a,5069,a)]. 11.51/11.51 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(5169,a,5070,a)]. 11.51/11.51 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),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(5169,a,5071,a)]. 11.51/11.51 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(5169,a,5072,a)]. 11.51/11.51 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),A)). [resolve(5169,a,5073,a)]. 11.51/11.51 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),A)). [resolve(5169,a,5074,a)]. 11.51/11.51 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(5169,a,5076,a)]. 11.51/11.51 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(5169,a,5077,a)]. 11.51/11.51 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(5169,a,5078,a)]. 11.51/11.52 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(5169,a,5079,a)]. 11.51/11.52 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(5169,a,5080,a)]. 11.51/11.52 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(5169,a,5081,a)]. 11.51/11.52 Derived: -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),C),D))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)). [resolve(5169,a,5082,a)]. 11.51/11.52 Derived: 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),C),D))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)). [resolve(5169,a,5083,a)]. 11.51/11.52 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),A),B) = B. [resolve(5169,a,5084,a)]. 11.51/11.52 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(5169,a,5085,a)]. 11.51/11.52 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f112(B,A,tc_Int_Oint))),hAPP(A,f113(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f112(B,A,tc_Int_Oint)),f113(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(5169,a,5086,a)]. 11.51/11.52 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(5169,a,5087,a)]. 11.51/11.52 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),f118(B,A,tc_Int_Oint)),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))). [resolve(5169,a,5088,a)]. 11.51/11.52 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),f118(B,A,tc_Int_Oint))) | 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))). [resolve(5169,a,5089,a)]. 11.51/11.52 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))). [resolve(5169,a,5090,a)]. 11.61/11.54 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(5169,a,5091,a)]. 11.61/11.54 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_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(5169,a,5092,a)]. 11.61/11.54 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(5169,a,5093,a)]. 11.61/11.54 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(5169,a,5094,a)]. 11.61/11.54 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(5169,a,5097,a)]. 11.61/11.54 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(5169,a,5098,a)]. 11.61/11.54 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(5169,a,5099,a)]. 11.61/11.54 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f134(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(5169,a,5101,a)]. 11.61/11.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f134(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(5169,a,5102,a)]. 11.61/11.54 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(5169,a,5103,a)]. 11.61/11.54 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(5169,a,5104,a)]. 11.61/11.54 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(5169,a,5105,a)]. 11.61/11.54 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),A) = A. [resolve(5169,a,5106,a)]. 11.61/11.54 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(5169,a,5107,a)]. 11.61/11.55 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),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),B))). [resolve(5169,a,5109,a)]. 11.61/11.55 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_Omin(tc_Int_Oint),C),B))). [resolve(5169,a,5110,a)]. 11.61/11.55 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_Omin(tc_Int_Oint),B),C))). [resolve(5169,a,5111,a)]. 11.61/11.55 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(5169,a,5112,a)]. 11.61/11.55 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(5169,a,5113,a)]. 11.61/11.55 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(5169,a,5114,a)]. 11.61/11.55 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(5169,a,5117,a)]. 11.61/11.55 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(5169,a,5118,a)]. 11.61/11.55 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) | C = A. [resolve(5169,a,5119,a)]. 11.61/11.55 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) | D = B. [resolve(5169,a,5120,a)]. 11.61/11.55 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) | C != A | D != B. [resolve(5169,a,5121,a)]. 11.61/11.55 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(5169,a,5122,a)]. 11.61/11.55 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5169,a,5123,a)]. 11.61/11.56 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),hAPP(c_List_Oset(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(5169,a,5124,a)]. 11.61/11.56 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f162(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),B))). [resolve(5169,a,5125,a)]. 11.61/11.56 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f162(B,A,tc_Int_Oint))). [resolve(5169,a,5126,a)]. 11.61/11.56 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A). [resolve(5169,a,5127,a)]. 11.61/11.56 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(A,B)),C)) | -hBOOL(hAPP(hAPP(c_member(D),B),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Int_Oint),E),A)),C)). [resolve(5169,a,5128,a)]. 11.61/11.56 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(A,f170(B,A,C,D,tc_Int_Oint))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Int_Oint),C),A)),B)). [resolve(5169,a,5129,a)]. 11.61/11.56 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f170(B,C,D,A,tc_Int_Oint)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Int_Oint),D),C)),B)). [resolve(5169,a,5130,a)]. 11.61/11.56 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(5169,a,5131,a)]. 11.61/11.56 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(5169,a,5133,a)]. 11.61/11.56 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,f178(A,tc_Int_Oint)) | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5169,a,5134,a)]. 11.61/11.56 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),hAPP(c_List_Oset(tc_Int_Oint),f178(A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f179(A,tc_Int_Oint)),B)) | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5169,a,5135,a)]. 11.61/11.56 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),f179(A,tc_Int_Oint)),f178(A,tc_Int_Oint)) = A | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5169,a,5136,a)]. 11.61/11.56 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),f180(A,tc_Int_Oint,B,C)),hAPP(c_List_Oset(tc_Int_Oint),B))) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),C),B) != A. [resolve(5169,a,5137,a)]. 11.61/11.56 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),C),f180(A,tc_Int_Oint,B,C))) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),C),B) != A. [resolve(5169,a,5138,a)]. 11.61/11.56 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olist_ONil(tc_Int_Oint) != A. [resolve(5169,a,5139,a)]. 11.61/11.58 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5169,a,5140,a)]. 11.61/11.58 Derived: 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5169,a,5141,a)]. 11.61/11.58 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(5169,a,5142,a)]. 11.61/11.58 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(5169,a,5143,a)]. 11.61/11.58 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(5169,a,5145,a)]. 11.61/11.58 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(5169,a,5146,a)]. 11.61/11.58 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(5169,a,5147,a)]. 11.61/11.58 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(5169,a,5148,a)]. 11.61/11.58 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(5169,a,5149,a)]. 11.61/11.58 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(5169,a,5150,a)]. 11.61/11.58 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_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(5169,a,5151,a)]. 11.61/11.58 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | A = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(5169,a,5153,a)]. 11.61/11.58 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(5169,a,5154,a)]. 11.61/11.58 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(5169,a,5155,a)]. 11.61/11.58 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(5169,a,5156,a)]. 11.61/11.58 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(5169,a,5157,a)]. 11.61/11.58 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(5169,a,5158,a)]. 11.67/11.61 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(5169,a,5159,a)]. 11.67/11.61 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(5169,a,5160,a)]. 11.67/11.61 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(5169,a,5161,a)]. 11.67/11.61 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(5169,a,5162,a)]. 11.67/11.61 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(5169,a,5163,a)]. 11.67/11.61 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(5169,a,5164,a)]. 11.67/11.61 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(5169,a,5165,a)]. 11.67/11.61 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(5169,a,5166,a)]. 11.67/11.61 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(5169,a,5167,a)]. 11.67/11.61 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(5169,a,5168,a)]. 11.67/11.61 5170 -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(2070)]. 11.67/11.61 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(5170,a,5018,a)]. 11.67/11.65 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(5170,a,5169,a)]. 11.67/11.65 5171 -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_linorder__not__less) # label(axiom). [clausify(2079)]. 11.67/11.65 5172 -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_linorder__not__less) # label(axiom). [clausify(2079)]. 11.67/11.65 5173 -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(2124)]. 11.67/11.65 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(5173,a,5018,a)]. 11.67/11.65 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(5173,a,5169,a)]. 11.67/11.65 5174 -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(2127)]. 11.67/11.65 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(5174,a,5018,a)]. 11.67/11.65 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(5174,a,5169,a)]. 11.67/11.65 5175 -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(2136)]. 11.67/11.65 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(5175,a,5018,a)]. 11.67/11.65 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(5175,a,5169,a)]. 11.67/11.65 5176 -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(2155)]. 11.67/11.65 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(5176,a,5018,a)]. 11.75/11.67 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(5176,a,5169,a)]. 11.75/11.67 5177 -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(2168)]. 11.75/11.67 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D). [resolve(5177,a,5018,a)]. 11.75/11.67 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D). [resolve(5177,a,5169,a)]. 11.75/11.67 5178 -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),f262(B,A)),f263(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f262(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f263(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(2188)]. 11.75/11.67 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),f262(A,tc_Code__Numeral_Ocode__numeral)),f263(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f262(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f263(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(5178,a,5018,a)]. 11.75/11.67 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),f262(A,tc_Int_Oint)),f263(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f262(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f263(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(5178,a,5169,a)]. 11.78/11.71 5179 -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(2213)]. 11.78/11.71 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(5179,a,5018,a)]. 11.78/11.71 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(5179,a,5169,a)]. 11.78/11.71 5180 -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(2223)]. 11.78/11.71 5181 -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(2244)]. 11.78/11.71 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(5181,a,5018,a)]. 11.78/11.71 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(5181,a,5169,a)]. 11.78/11.71 5182 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f277(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(2282)]. 11.78/11.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f277(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(5182,a,5018,a)]. 11.78/11.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f277(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(5182,a,5169,a)]. 11.78/11.72 5183 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f277(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(2282)]. 11.78/11.72 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),f277(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(5183,a,5018,a)]. 11.78/11.72 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f277(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(5183,a,5169,a)]. 11.78/11.72 5184 -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(2303)]. 11.78/11.72 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(5184,a,5018,a)]. 11.78/11.72 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(5184,a,5169,a)]. 11.78/11.72 5185 -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(2304)]. 11.78/11.73 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(5185,a,5018,a)]. 11.78/11.73 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(5185,a,5169,a)]. 11.78/11.73 5186 -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(2304)]. 11.78/11.73 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(5186,a,5018,a)]. 11.78/11.73 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(5186,a,5169,a)]. 11.78/11.73 5187 -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(2320)]. 11.82/11.77 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(5187,a,5018,a)]. 11.82/11.77 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(5187,a,5169,a)]. 11.82/11.77 5188 -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_Ole__iff__sup) # label(axiom). [clausify(2331)]. 11.82/11.77 5189 -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_Ole__iff__sup) # label(axiom). [clausify(2331)]. 11.82/11.77 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),A),B) != B. [resolve(5189,a,5018,a)]. 11.82/11.77 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),A),B) != B. [resolve(5189,a,5169,a)]. 11.82/11.77 5190 -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(2339)]. 11.82/11.77 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(5190,a,5018,a)]. 11.82/11.77 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(5190,a,5169,a)]. 11.82/11.77 5191 -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(2342)]. 11.82/11.77 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(5191,a,5018,a)]. 11.82/11.77 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(5191,a,5169,a)]. 11.82/11.77 5192 -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(2394)]. 11.91/11.83 5193 -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(2396)]. 11.91/11.83 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(5193,a,5018,a)]. 11.91/11.83 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(5193,a,5169,a)]. 11.91/11.83 5194 -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(2415)]. 11.91/11.83 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(5194,a,5018,a)]. 11.91/11.83 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(5194,a,5169,a)]. 11.91/11.83 5195 -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(2443)]. 11.91/11.83 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(5195,a,5018,a)]. 11.91/11.83 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(5195,a,5169,a)]. 11.91/11.83 5196 -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(2451)]. 11.91/11.83 5197 -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(2457)]. 11.91/11.83 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(5197,a,5018,a)]. 11.91/11.83 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(5197,a,5169,a)]. 11.91/11.83 5198 -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(2524)]. 11.91/11.83 5199 -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(2527)]. 11.91/11.83 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(5199,a,5018,a)]. 11.91/11.88 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(5199,a,5169,a)]. 11.91/11.88 5200 -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(2556)]. 11.91/11.88 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(5200,a,5018,a)]. 11.91/11.88 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(5200,a,5169,a)]. 11.91/11.88 5201 -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(2563)]. 11.91/11.88 5202 -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(2565)]. 11.91/11.88 5203 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f316(B,C,A)),hAPP(c_List_Oset(A),B))) | -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(2614)]. 11.91/11.88 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f316(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,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(5203,a,5018,a)]. 11.91/11.88 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f316(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | -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(5203,a,5169,a)]. 11.91/11.88 5204 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f316(C,B,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_Oequations_I2_J) # label(axiom). [clausify(2614)]. 11.91/11.88 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),f316(B,A,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(5204,a,5018,a)]. 11.91/11.88 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f316(B,A,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(5204,a,5169,a)]. 11.91/11.88 5205 -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(2614)]. 11.97/11.90 5206 -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(2614)]. 11.97/11.90 5207 -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(2617)]. 11.97/11.90 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(5207,a,5018,a)]. 11.97/11.90 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(5207,a,5169,a)]. 11.97/11.90 5208 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Complete__Lattice_OSup__class_OSup(A,C))) | hBOOL(hAPP(hAPP(c_member(A),f327(C,B,A)),C)) # label(fact_less__Sup__iff) # label(axiom). [clausify(2648)]. 11.97/11.90 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),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f327(B,A,tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5208,b,5018,a)]. 11.97/11.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f327(B,A,tc_Int_Oint)),B)). [resolve(5208,b,5169,a)]. 11.97/11.90 5209 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Complete__Lattice_OSup__class_OSup(A,C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),f327(C,B,A))) # label(fact_less__Sup__iff) # label(axiom). [clausify(2648)]. 11.97/11.90 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),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),f327(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5209,b,5018,a)]. 11.97/11.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),f327(B,A,tc_Int_Oint))). [resolve(5209,b,5169,a)]. 11.97/11.90 5210 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Complete__Lattice_OSup__class_OSup(A,C))) | -hBOOL(hAPP(hAPP(c_member(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_less__Sup__iff) # label(axiom). [clausify(2648)]. 11.97/11.90 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),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5210,b,5018,a)]. 11.97/11.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5210,b,5169,a)]. 11.97/11.90 5211 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_sorted__many__eq) # label(axiom). [clausify(2652)]. 11.97/11.92 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5211,a,5018,a)]. 11.97/11.92 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5211,a,5169,a)]. 11.97/11.92 5212 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),D)) # label(fact_sorted__many__eq) # label(axiom). [clausify(2652)]. 11.97/11.92 5213 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),B),hAPP(hAPP(c_List_Olist_OCons(A),C),D))) | -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)) # label(fact_sorted__many__eq) # label(axiom). [clausify(2652)]. 11.97/11.92 5214 -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(2665)]. 11.97/11.92 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(5214,a,5018,a)]. 11.97/11.92 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(5214,a,5169,a)]. 11.97/11.92 5215 -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(2665)]. 11.97/11.92 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(5215,a,5018,a)]. 11.97/11.92 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(5215,a,5169,a)]. 11.97/11.92 5216 -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(2670)]. 11.97/11.92 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(5216,a,5018,a)]. 11.97/11.92 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(5216,a,5169,a)]. 11.97/11.92 5217 -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(2677)]. 11.97/11.92 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(5217,a,5018,a)]. 11.97/11.92 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(5217,a,5169,a)]. 11.97/11.94 5218 -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(2689)]. 11.97/11.94 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(5218,a,5018,a)]. 11.97/11.94 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(5218,a,5169,a)]. 11.97/11.94 5219 -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(2696)]. 11.97/11.94 5220 -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(2717)]. 12.05/11.98 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(5220,a,5018,a)]. 12.05/11.98 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(5220,a,5169,a)]. 12.05/11.98 5221 -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(2740)]. 12.05/11.98 5222 -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(2775)]. 12.05/11.98 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(5222,a,5018,a)]. 12.05/11.98 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(5222,a,5169,a)]. 12.05/11.98 5223 -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(2784)]. 12.05/11.98 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(5223,a,5018,a)]. 12.05/11.98 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(5223,a,5169,a)]. 12.05/11.98 5224 -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(2785)]. 12.05/11.98 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(5224,a,5018,a)]. 12.06/11.98 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(5224,a,5169,a)]. 12.06/11.98 5225 -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(2785)]. 12.06/11.98 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(5225,a,5018,a)]. 12.06/11.98 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(5225,a,5169,a)]. 12.06/11.98 5226 -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(2790)]. 12.06/11.98 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(5226,a,5018,a)]. 12.06/11.98 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(5226,a,5169,a)]. 12.06/11.98 5227 -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(2794)]. 12.06/11.98 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(5227,a,5018,a)]. 12.06/12.00 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(5227,a,5169,a)]. 12.06/12.00 5228 -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(2798)]. 12.06/12.00 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(5228,a,5018,a)]. 12.06/12.00 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(5228,a,5169,a)]. 12.06/12.00 5229 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_Min_Oidem) # label(axiom). [clausify(2803)]. 12.06/12.00 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5229,a,5018,a)]. 12.06/12.00 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),A) = A. [resolve(5229,a,5169,a)]. 12.06/12.00 5230 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(B,f340(C,B,A))),hAPP(B,f341(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),f340(C,B,A)),f341(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(2824)]. 12.06/12.00 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(A,f340(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f341(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f340(B,A,tc_Code__Numeral_Ocode__numeral)),f341(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(5230,a,5018,a)]. 12.06/12.00 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(A,f340(B,A,tc_Int_Oint))),hAPP(A,f341(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),f340(B,A,tc_Int_Oint)),f341(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(5230,a,5169,a)]. 12.06/12.00 5231 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_Oequations_I1_J) # label(axiom). [clausify(2826)]. 12.06/12.01 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)). [resolve(5231,a,5018,a)]. 12.06/12.01 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olist_ONil(tc_Int_Oint)). [resolve(5231,a,5169,a)]. 12.06/12.01 5232 -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(2831)]. 12.06/12.01 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(5232,a,5018,a)]. 12.06/12.01 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(5232,a,5169,a)]. 12.06/12.01 5233 -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(2831)]. 12.06/12.01 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(5233,a,5018,a)]. 12.06/12.01 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(5233,a,5169,a)]. 12.12/12.03 5234 -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(2840)]. 12.12/12.03 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(5234,a,5018,a)]. 12.12/12.03 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(5234,a,5169,a)]. 12.12/12.03 5235 -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(2848)]. 12.12/12.03 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(5235,a,5018,a)]. 12.12/12.03 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(5235,a,5169,a)]. 12.12/12.03 5236 -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(2855)]. 12.12/12.03 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(5236,a,5018,a)]. 12.12/12.03 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(5236,a,5169,a)]. 12.12/12.03 5237 -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(2884)]. 12.12/12.04 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(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(2886)]. 12.12/12.04 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(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(5238,a,5018,a)]. 12.12/12.04 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(5238,a,5169,a)]. 12.12/12.04 5239 -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(2896)]. 12.12/12.04 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(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(5239,a,5018,a)]. 12.12/12.09 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(5239,a,5169,a)]. 12.12/12.09 5240 -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(2912)]. 12.12/12.09 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(5240,a,5018,a)]. 12.12/12.09 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(5240,a,5169,a)]. 12.12/12.09 5241 -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(2945)]. 12.12/12.09 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(5241,a,5018,a)]. 12.12/12.09 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),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B)). [resolve(5241,a,5169,a)]. 12.12/12.09 5242 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f351(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_Oappend(A),C),B)) # label(fact_sorted__append) # label(axiom). [clausify(2980)]. 12.12/12.09 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f351(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_Oappend(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5242,a,5018,a)]. 12.12/12.09 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f351(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_Oappend(tc_Int_Oint),B),A)). [resolve(5242,a,5169,a)]. 12.12/12.09 5243 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f352(B,C,A)),hAPP(c_List_Oset(A),B))) | -c_List_Olinorder__class_Osorted(A,C) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),C),B)) # label(fact_sorted__append) # label(axiom). [clausify(2980)]. 12.12/12.09 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f352(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,B) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5243,a,5018,a)]. 12.12/12.09 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f352(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),B),A)). [resolve(5243,a,5169,a)]. 12.12/12.10 5244 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f351(B,C,A)),f352(B,C,A))) | -c_List_Olinorder__class_Osorted(A,C) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),C),B)) # label(fact_sorted__append) # label(axiom). [clausify(2980)]. 12.12/12.10 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),f351(A,B,tc_Code__Numeral_Ocode__numeral)),f352(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_Oappend(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5244,a,5018,a)]. 12.12/12.10 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f351(A,B,tc_Int_Oint)),f352(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_Oappend(tc_Int_Oint),B),A)). [resolve(5244,a,5169,a)]. 12.12/12.10 5245 -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(2980)]. 12.12/12.10 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(5245,a,5018,a)]. 12.12/12.10 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(5245,a,5169,a)]. 12.12/12.10 5246 -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(2980)]. 12.12/12.10 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(5246,a,5018,a)]. 12.12/12.10 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(5246,a,5169,a)]. 12.12/12.10 5247 -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(2980)]. 12.12/12.10 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(5247,a,5018,a)]. 12.12/12.10 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(5247,a,5169,a)]. 12.12/12.10 5248 -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(2981)]. 12.20/12.12 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(5248,a,5018,a)]. 12.20/12.12 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(5248,a,5169,a)]. 12.20/12.12 5249 -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),f353(B,A)),f354(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f353(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f354(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(2984)]. 12.20/12.12 5250 -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(2987)]. 12.20/12.12 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(5250,a,5018,a)]. 12.20/12.12 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(5250,a,5169,a)]. 12.20/12.12 5251 -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(2993)]. 12.20/12.12 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),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(5251,a,5018,a)]. 12.20/12.12 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(5251,a,5169,a)]. 12.20/12.12 5252 -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(3021)]. 12.20/12.12 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(5252,a,5018,a)]. 12.20/12.12 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(5252,a,5169,a)]. 12.20/12.14 5253 -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(3027)]. 12.20/12.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5253,a,5018,a)]. 12.20/12.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B))). [resolve(5253,a,5169,a)]. 12.20/12.14 5254 -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(3027)]. 12.20/12.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5254,a,5018,a)]. 12.20/12.14 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))). [resolve(5254,a,5169,a)]. 12.20/12.14 5255 -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),D),C))) # label(fact_le__max__iff__disj) # label(axiom). [clausify(3027)]. 12.20/12.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),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),C),B))). [resolve(5255,a,5018,a)]. 12.20/12.14 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),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),C),B))). [resolve(5255,a,5169,a)]. 12.20/12.14 5256 -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(3036)]. 12.20/12.14 5257 -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(3052)]. 12.20/12.14 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(5257,a,5018,a)]. 12.20/12.14 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(5257,a,5169,a)]. 12.20/12.14 5258 -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(3053)]. 12.20/12.16 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(5258,b,5018,a)]. 12.20/12.16 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(5258,b,5169,a)]. 12.20/12.16 5259 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f368(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(3053)]. 12.20/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f368(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(5259,b,5018,a)]. 12.20/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f368(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(5259,b,5169,a)]. 12.20/12.16 5260 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f368(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(3053)]. 12.20/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f368(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(5260,b,5018,a)]. 12.20/12.16 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),f368(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(5260,b,5169,a)]. 12.20/12.16 5261 -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(3062)]. 12.20/12.16 5262 -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(3064)]. 12.20/12.16 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(5262,a,5018,a)]. 12.20/12.16 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(5262,a,5169,a)]. 12.26/12.21 5263 -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(3068)]. 12.26/12.21 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(5263,a,5018,a)]. 12.26/12.21 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(5263,a,5169,a)]. 12.26/12.21 5264 -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(3084)]. 12.26/12.21 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(5264,a,5018,a)]. 12.26/12.21 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(5264,a,5169,a)]. 12.26/12.21 5265 -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(3101)]. 12.26/12.21 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(5265,a,5018,a)]. 12.26/12.21 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)). [resolve(5265,a,5169,a)]. 12.26/12.21 5266 -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(3139)]. 12.26/12.21 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(5266,a,5018,a)]. 12.26/12.21 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(5266,a,5169,a)]. 12.26/12.21 5267 -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(3149)]. 12.26/12.23 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(5267,a,5018,a)]. 12.26/12.23 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(5267,a,5169,a)]. 12.26/12.23 5268 -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(3172)]. 12.26/12.23 5269 -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(3177)]. 12.26/12.23 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(5269,a,5018,a)]. 12.26/12.23 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(5269,a,5169,a)]. 12.26/12.23 5270 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 12.26/12.23 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(5270,a,5019,a)]. 12.26/12.23 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(5270,a,5020,a)]. 12.26/12.23 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(5270,a,5021,a)]. 12.26/12.23 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(5270,a,5022,a)]. 12.26/12.23 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(5270,a,5023,a)]. 12.26/12.24 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(5270,a,5024,a)]. 12.26/12.24 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),B),C))). [resolve(5270,a,5025,a)]. 12.26/12.24 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(5270,a,5026,a)]. 12.26/12.24 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),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C))). [resolve(5270,a,5027,a)]. 12.26/12.24 Derived: c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5270,a,5028,a)]. 12.26/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f18(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(5270,a,5029,a)]. 12.26/12.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f18(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(5270,a,5030,a)]. 12.26/12.24 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(5270,a,5031,a)]. 12.26/12.24 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(5270,a,5032,a)]. 12.26/12.24 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(5270,a,5033,a)]. 12.26/12.24 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(5270,a,5034,a)]. 12.26/12.24 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(5270,a,5035,a)]. 12.26/12.25 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(5270,a,5036,a)]. 12.26/12.25 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(5270,a,5037,a)]. 12.26/12.25 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(5270,a,5038,a)]. 12.26/12.25 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(5270,a,5039,a)]. 12.26/12.25 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A). [resolve(5270,a,5040,a)]. 12.26/12.25 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(5270,a,5041,a)]. 12.26/12.25 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(5270,a,5042,a)]. 12.26/12.25 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(5270,a,5043,a)]. 12.26/12.25 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B) | A = B. [resolve(5270,a,5044,a)]. 12.26/12.25 Derived: hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B) | A != B. [resolve(5270,a,5045,a)]. 12.26/12.25 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(5270,a,5046,a)]. 12.26/12.25 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(5270,a,5047,a)]. 12.26/12.25 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(5270,a,5048,a)]. 12.26/12.25 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5270,a,5049,a)]. 12.26/12.25 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(5270,a,5050,a)]. 12.26/12.26 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f56(B,A,tc_Nat_Onat))),hAPP(A,f57(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f56(B,A,tc_Nat_Onat)),f57(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(5270,a,5051,a)]. 12.26/12.26 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(5270,a,5052,a)]. 12.26/12.26 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(5270,a,5053,a)]. 12.26/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | B = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5270,a,5055,a)]. 12.26/12.26 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(5270,a,5056,a)]. 12.26/12.26 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(5270,a,5057,a)]. 12.26/12.26 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(5270,a,5058,a)]. 12.26/12.26 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(5270,a,5059,a)]. 12.26/12.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f73(A,tc_Nat_Onat)),f74(A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5270,a,5060,a)]. 12.26/12.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f74(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(5270,a,5061,a)]. 12.26/12.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f73(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f74(A,tc_Nat_Onat)))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5270,a,5062,a)]. 12.26/12.26 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),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(5270,a,5063,a)]. 12.26/12.26 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(5270,a,5064,a)]. 12.36/12.26 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),f76(B,A,tc_Nat_Onat)),A)). [resolve(5270,a,5065,a)]. 12.36/12.26 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),f76(B,A,tc_Nat_Onat))). [resolve(5270,a,5066,a)]. 12.36/12.26 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)). [resolve(5270,a,5067,a)]. 12.36/12.26 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_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(5270,a,5068,a)]. 12.36/12.26 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(5270,a,5069,a)]. 12.36/12.27 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(5270,a,5070,a)]. 12.36/12.27 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),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(5270,a,5071,a)]. 12.36/12.27 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(5270,a,5072,a)]. 12.36/12.27 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),A)). [resolve(5270,a,5073,a)]. 12.36/12.27 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),A)). [resolve(5270,a,5074,a)]. 12.36/12.27 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(5270,a,5076,a)]. 12.36/12.27 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(5270,a,5077,a)]. 12.36/12.27 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(5270,a,5078,a)]. 12.36/12.27 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_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(5270,a,5079,a)]. 12.36/12.27 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(5270,a,5080,a)]. 12.36/12.27 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(5270,a,5081,a)]. 12.36/12.27 Derived: -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),C),D))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)). [resolve(5270,a,5082,a)]. 12.36/12.27 Derived: 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),C),D))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)). [resolve(5270,a,5083,a)]. 12.36/12.27 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),A),B) = B. [resolve(5270,a,5084,a)]. 12.36/12.27 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(5270,a,5085,a)]. 12.36/12.27 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f112(B,A,tc_Nat_Onat))),hAPP(A,f113(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f112(B,A,tc_Nat_Onat)),f113(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(5270,a,5086,a)]. 12.36/12.27 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(5270,a,5087,a)]. 12.36/12.27 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(hAPP(c_member(tc_Nat_Onat),f118(B,A,tc_Nat_Onat)),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))). [resolve(5270,a,5088,a)]. 12.36/12.27 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(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f118(B,A,tc_Nat_Onat))) | 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))). [resolve(5270,a,5089,a)]. 12.36/12.27 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(hAPP(c_member(tc_Nat_Onat),B),A)) | 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),C),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))). [resolve(5270,a,5090,a)]. 12.38/12.28 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(5270,a,5091,a)]. 12.38/12.28 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_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(5270,a,5092,a)]. 12.38/12.28 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(5270,a,5093,a)]. 12.38/12.28 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(5270,a,5094,a)]. 12.38/12.28 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(5270,a,5097,a)]. 12.38/12.28 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(5270,a,5098,a)]. 12.38/12.28 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(5270,a,5099,a)]. 12.38/12.28 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f134(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(5270,a,5101,a)]. 12.38/12.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f134(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(5270,a,5102,a)]. 12.38/12.28 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(5270,a,5103,a)]. 12.38/12.28 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(5270,a,5104,a)]. 12.38/12.28 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(5270,a,5105,a)]. 12.38/12.28 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),A) = A. [resolve(5270,a,5106,a)]. 12.38/12.28 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(5270,a,5107,a)]. 12.38/12.28 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),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),C),B))). [resolve(5270,a,5109,a)]. 12.38/12.29 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_Omin(tc_Nat_Onat),C),B))). [resolve(5270,a,5110,a)]. 12.38/12.29 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_Omin(tc_Nat_Onat),B),C))). [resolve(5270,a,5111,a)]. 12.38/12.29 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(5270,a,5112,a)]. 12.38/12.29 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(5270,a,5113,a)]. 12.38/12.29 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(5270,a,5114,a)]. 12.38/12.29 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(5270,a,5117,a)]. 12.38/12.29 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(5270,a,5118,a)]. 12.38/12.29 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B) | C = A. [resolve(5270,a,5119,a)]. 12.38/12.29 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B) | D = B. [resolve(5270,a,5120,a)]. 12.38/12.29 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)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B) | C != A | D != B. [resolve(5270,a,5121,a)]. 12.38/12.29 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(5270,a,5122,a)]. 12.38/12.29 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5270,a,5123,a)]. 12.38/12.29 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),hAPP(c_List_Oset(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5270,a,5124,a)]. 12.38/12.30 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f162(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),B))). [resolve(5270,a,5125,a)]. 12.38/12.30 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f162(B,A,tc_Nat_Onat))). [resolve(5270,a,5126,a)]. 12.38/12.30 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A). [resolve(5270,a,5127,a)]. 12.38/12.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(A,B)),C)) | -hBOOL(hAPP(hAPP(c_member(D),B),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Nat_Onat),E),A)),C)). [resolve(5270,a,5128,a)]. 12.38/12.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(A,f170(B,A,C,D,tc_Nat_Onat))),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(D,tc_Nat_Onat),C),A)),B)). [resolve(5270,a,5129,a)]. 12.38/12.30 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f170(B,C,D,A,tc_Nat_Onat)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Nat_Onat),D),C)),B)). [resolve(5270,a,5130,a)]. 12.38/12.30 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(5270,a,5131,a)]. 12.38/12.30 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(5270,a,5133,a)]. 12.38/12.30 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f178(A,tc_Nat_Onat)) | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5270,a,5134,a)]. 12.38/12.30 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),hAPP(c_List_Oset(tc_Nat_Onat),f178(A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f179(A,tc_Nat_Onat)),B)) | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5270,a,5135,a)]. 12.38/12.30 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),f179(A,tc_Nat_Onat)),f178(A,tc_Nat_Onat)) = A | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5270,a,5136,a)]. 12.38/12.30 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),f180(A,tc_Nat_Onat,B,C)),hAPP(c_List_Oset(tc_Nat_Onat),B))) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),C),B) != A. [resolve(5270,a,5137,a)]. 12.38/12.30 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),C),f180(A,tc_Nat_Onat,B,C))) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),C),B) != A. [resolve(5270,a,5138,a)]. 12.38/12.30 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olist_ONil(tc_Nat_Onat) != A. [resolve(5270,a,5139,a)]. 12.38/12.30 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5270,a,5140,a)]. 12.38/12.31 Derived: 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5270,a,5141,a)]. 12.38/12.31 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(5270,a,5142,a)]. 12.38/12.31 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(5270,a,5143,a)]. 12.38/12.31 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(5270,a,5145,a)]. 12.38/12.31 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(5270,a,5146,a)]. 12.38/12.31 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(5270,a,5147,a)]. 12.38/12.31 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(5270,a,5148,a)]. 12.38/12.31 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(5270,a,5149,a)]. 12.38/12.31 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(5270,a,5150,a)]. 12.38/12.31 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_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(5270,a,5151,a)]. 12.38/12.31 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(5270,a,5154,a)]. 12.38/12.31 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(5270,a,5155,a)]. 12.38/12.31 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(5270,a,5156,a)]. 12.38/12.31 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(5270,a,5157,a)]. 12.38/12.31 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(5270,a,5158,a)]. 12.38/12.31 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(5270,a,5159,a)]. 12.42/12.32 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(5270,a,5160,a)]. 12.42/12.32 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(5270,a,5161,a)]. 12.42/12.32 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(5270,a,5162,a)]. 12.42/12.32 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(5270,a,5163,a)]. 12.42/12.32 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(5270,a,5164,a)]. 12.42/12.32 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(5270,a,5165,a)]. 12.42/12.32 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(5270,a,5166,a)]. 12.42/12.32 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(5270,a,5167,a)]. 12.42/12.32 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(5270,a,5168,a)]. 12.42/12.32 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(5270,a,5170,a)]. 12.42/12.32 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(5270,a,5173,a)]. 12.42/12.32 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(5270,a,5174,a)]. 12.42/12.32 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(5270,a,5175,a)]. 12.42/12.32 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(5270,a,5176,a)]. 12.42/12.33 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D). [resolve(5270,a,5177,a)]. 12.42/12.33 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),f262(A,tc_Nat_Onat)),f263(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f262(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f263(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(5270,a,5178,a)]. 12.42/12.33 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(5270,a,5179,a)]. 12.42/12.33 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(5270,a,5181,a)]. 12.42/12.33 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f277(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(5270,a,5182,a)]. 12.42/12.33 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f277(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(5270,a,5183,a)]. 12.42/12.33 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(5270,a,5184,a)]. 12.42/12.33 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(5270,a,5185,a)]. 12.43/12.34 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(5270,a,5186,a)]. 12.43/12.34 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(5270,a,5187,a)]. 12.43/12.34 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),A),B) != B. [resolve(5270,a,5189,a)]. 12.43/12.34 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(5270,a,5190,a)]. 12.43/12.34 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(5270,a,5191,a)]. 12.43/12.34 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(5270,a,5193,a)]. 12.43/12.34 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(5270,a,5194,a)]. 12.43/12.34 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(5270,a,5195,a)]. 12.43/12.34 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(5270,a,5197,a)]. 12.43/12.34 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(5270,a,5199,a)]. 12.43/12.34 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(5270,a,5200,a)]. 12.43/12.34 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f316(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | -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(5270,a,5203,a)]. 12.43/12.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f316(B,A,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(5270,a,5204,a)]. 12.43/12.34 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(5270,a,5207,a)]. 12.43/12.34 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f327(B,A,tc_Nat_Onat)),B)). [resolve(5270,a,5208,b)]. 12.43/12.34 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),f327(B,A,tc_Nat_Onat))). [resolve(5270,a,5209,b)]. 12.43/12.34 Derived: -class_Complete__Lattice_Ocomplete__lattice(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))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5270,a,5210,b)]. 12.43/12.34 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5270,a,5211,a)]. 12.43/12.34 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(5270,a,5214,a)]. 12.43/12.34 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(5270,a,5215,a)]. 12.43/12.34 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(5270,a,5216,a)]. 12.43/12.34 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(5270,a,5217,a)]. 12.43/12.34 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(5270,a,5218,a)]. 12.43/12.34 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(5270,a,5220,a)]. 12.43/12.35 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(5270,a,5222,a)]. 12.43/12.35 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(5270,a,5223,a)]. 12.43/12.35 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5270,a,5224,a)]. 12.43/12.35 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != 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(5270,a,5225,a)]. 12.43/12.35 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(5270,a,5226,a)]. 12.43/12.35 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(5270,a,5227,a)]. 12.43/12.35 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(5270,a,5228,a)]. 12.43/12.35 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),A) = A. [resolve(5270,a,5229,a)]. 12.43/12.35 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(A,f340(B,A,tc_Nat_Onat))),hAPP(A,f341(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),f340(B,A,tc_Nat_Onat)),f341(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(5270,a,5230,a)]. 12.43/12.35 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olist_ONil(tc_Nat_Onat)). [resolve(5270,a,5231,a)]. 12.43/12.35 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(5270,a,5232,a)]. 12.43/12.36 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(5270,a,5233,a)]. 12.43/12.36 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(5270,a,5234,a)]. 12.43/12.36 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(5270,a,5235,a)]. 12.43/12.36 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(5270,a,5236,a)]. 12.43/12.36 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(5270,a,5238,a)]. 12.43/12.36 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(5270,a,5239,a)]. 12.43/12.36 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(5270,a,5240,a)]. 12.43/12.36 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(5270,a,5241,a)]. 12.43/12.36 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f351(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_Oappend(tc_Nat_Onat),B),A)). [resolve(5270,a,5242,a)]. 12.43/12.36 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f352(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),B),A)). [resolve(5270,a,5243,a)]. 12.43/12.37 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f351(A,B,tc_Nat_Onat)),f352(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_Oappend(tc_Nat_Onat),B),A)). [resolve(5270,a,5244,a)]. 12.43/12.37 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(5270,a,5245,a)]. 12.43/12.37 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(5270,a,5246,a)]. 12.43/12.37 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(5270,a,5247,a)]. 12.43/12.37 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(5270,a,5248,a)]. 12.43/12.37 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(5270,a,5250,a)]. 12.43/12.37 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(5270,a,5251,a)]. 12.43/12.37 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(5270,a,5252,a)]. 12.43/12.37 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(5270,a,5253,a)]. 12.43/12.37 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(5270,a,5254,a)]. 12.43/12.37 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),C),B))). [resolve(5270,a,5255,a)]. 12.43/12.37 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(5270,a,5257,a)]. 12.43/12.37 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(5270,a,5258,b)]. 12.43/12.37 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f368(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(5270,a,5259,b)]. 12.43/12.38 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f368(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(5270,a,5260,b)]. 12.43/12.38 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(5270,a,5262,a)]. 12.43/12.38 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(5270,a,5263,a)]. 12.43/12.38 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(5270,a,5264,a)]. 12.43/12.38 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)). [resolve(5270,a,5265,a)]. 12.43/12.38 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(5270,a,5266,a)]. 12.43/12.38 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(5270,a,5267,a)]. 12.43/12.38 5271 -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(3211)]. 12.43/12.38 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(5271,a,5018,a)]. 12.43/12.38 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(5271,a,5169,a)]. 12.43/12.38 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(5271,a,5270,a)]. 12.43/12.38 5272 -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(3211)]. 12.43/12.38 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(5272,a,5018,a)]. 12.43/12.38 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(5272,a,5169,a)]. 12.43/12.38 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(5272,a,5270,a)]. 12.43/12.38 5273 -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(3211)]. 12.43/12.40 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(5273,a,5018,a)]. 12.43/12.40 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(5273,a,5169,a)]. 12.43/12.40 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(5273,a,5270,a)]. 12.43/12.40 5274 -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(3234)]. 12.43/12.40 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(5274,a,5018,a)]. 12.43/12.40 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(5274,a,5169,a)]. 12.43/12.40 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(5274,a,5270,a)]. 12.43/12.40 5275 -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(3234)]. 12.43/12.40 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(5275,a,5018,a)]. 12.43/12.40 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(5275,a,5169,a)]. 12.43/12.40 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(5275,a,5270,a)]. 12.52/12.42 5276 -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(3241)]. 12.52/12.42 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(5276,a,5018,a)]. 12.52/12.42 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(5276,a,5169,a)]. 12.52/12.42 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(5276,a,5270,a)]. 12.52/12.42 5277 -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(3241)]. 12.52/12.42 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(5277,a,5018,a)]. 12.52/12.42 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(5277,a,5169,a)]. 12.52/12.42 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(5277,a,5270,a)]. 12.52/12.42 5278 -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(3241)]. 12.52/12.42 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(5278,a,5018,a)]. 12.52/12.42 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(5278,a,5169,a)]. 12.52/12.42 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(5278,a,5270,a)]. 12.52/12.42 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 | 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(3251)]. 12.52/12.42 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 | 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(5279,a,5018,a)]. 12.52/12.42 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(5279,a,5169,a)]. 12.52/12.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5279,a,5270,a)]. 12.52/12.44 5280 -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(3262)]. 12.52/12.44 5281 -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(3262)]. 12.52/12.44 5282 -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(3262)]. 12.52/12.44 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),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(5282,a,5018,a)]. 12.52/12.44 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(5282,a,5169,a)]. 12.52/12.44 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(5282,a,5270,a)]. 12.52/12.44 5283 -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(3269)]. 12.52/12.44 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_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(5283,a,5018,a)]. 12.52/12.46 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(5283,a,5169,a)]. 12.52/12.46 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(5283,a,5270,a)]. 12.52/12.46 5284 -class_Orderings_Olinorder(A) | -class_Lattices_Osemilattice__sup(A) | c_Orderings_Oord__class_Omax(A) = c_Lattices_Osemilattice__sup__class_Osup(A) # label(fact_sup__max) # label(axiom). [clausify(3272)]. 12.52/12.46 Derived: -class_Lattices_Osemilattice__sup(tc_Code__Numeral_Ocode__numeral) | c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral) = c_Lattices_Osemilattice__sup__class_Osup(tc_Code__Numeral_Ocode__numeral). [resolve(5284,a,5018,a)]. 12.52/12.46 Derived: -class_Lattices_Osemilattice__sup(tc_Int_Oint) | c_Orderings_Oord__class_Omax(tc_Int_Oint) = c_Lattices_Osemilattice__sup__class_Osup(tc_Int_Oint). [resolve(5284,a,5169,a)]. 12.52/12.46 Derived: -class_Lattices_Osemilattice__sup(tc_Nat_Onat) | c_Orderings_Oord__class_Omax(tc_Nat_Onat) = c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat). [resolve(5284,a,5270,a)]. 12.52/12.46 5285 -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(3292)]. 12.52/12.46 5286 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(3295)]. 12.52/12.46 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_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5286,a,5018,a)]. 12.52/12.47 Derived: -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),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5286,a,5169,a)]. 12.52/12.47 Derived: -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),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),A)). [resolve(5286,a,5270,a)]. 12.52/12.47 5287 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),B)) # label(fact_greaterThan__subset__iff) # label(axiom). [clausify(3295)]. 12.52/12.47 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_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5287,a,5018,a)]. 12.52/12.47 Derived: 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),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5287,a,5169,a)]. 12.52/12.47 Derived: 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),A)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),A)). [resolve(5287,a,5270,a)]. 12.52/12.47 5288 -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(3309)]. 12.52/12.47 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)) = 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(5288,a,5018,a)]. 12.52/12.47 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_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(5288,a,5169,a)]. 12.60/12.50 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(5288,a,5270,a)]. 12.60/12.50 5289 -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(3319)]. 12.60/12.50 5290 -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(3355)]. 12.60/12.50 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(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(5290,a,5018,a)]. 12.60/12.50 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(5290,a,5169,a)]. 12.60/12.52 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(5290,a,5270,a)]. 12.60/12.52 5291 -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(3372)]. 12.60/12.52 5292 -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(3378)]. 12.60/12.53 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(5292,a,5018,a)]. 12.60/12.53 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(5292,a,5169,a)]. 12.60/12.53 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(5292,a,5270,a)]. 12.60/12.53 5293 -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(3379)]. 12.60/12.53 5294 -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(A),C),D)) | 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(3390)]. 12.60/12.53 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(tc_Code__Numeral_Ocode__numeral),B),C)) | 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(5294,a,5018,a)]. 12.60/12.53 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(tc_Int_Oint),B),C)) | 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(5294,a,5169,a)]. 12.60/12.53 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(tc_Nat_Onat),B),C)) | 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(5294,a,5270,a)]. 12.60/12.53 5295 -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),f409(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(3390)]. 12.60/12.53 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),f409(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(5295,a,5018,a)]. 12.60/12.53 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),f409(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(5295,a,5169,a)]. 12.60/12.54 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),f409(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(5295,a,5270,a)]. 12.60/12.54 5296 -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),f409(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(3390)]. 12.60/12.54 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_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f409(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(5296,a,5018,a)]. 12.60/12.54 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),f409(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(5296,a,5169,a)]. 12.60/12.54 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),f409(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(5296,a,5270,a)]. 12.60/12.54 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 | -hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f410(B,A)),f411(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f410(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f411(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(3397)]. 12.60/12.54 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),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f410(A,tc_Code__Numeral_Ocode__numeral)),f411(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f410(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f411(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(5297,a,5018,a)]. 12.60/12.54 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),f410(A,tc_Int_Oint)),f411(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f410(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f411(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(5297,a,5169,a)]. 12.60/12.57 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),f410(A,tc_Nat_Onat)),f411(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f410(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f411(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(5297,a,5270,a)]. 12.60/12.57 5298 -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(3433)]. 12.60/12.57 5299 -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(3438)]. 12.60/12.57 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(5299,a,5018,a)]. 12.60/12.57 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(5299,a,5169,a)]. 12.60/12.57 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(5299,a,5270,a)]. 12.60/12.57 5300 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(3443)]. 12.60/12.57 Derived: -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),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5300,a,5018,a)]. 12.60/12.57 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5300,a,5169,a)]. 12.60/12.57 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5300,a,5270,a)]. 12.60/12.57 5301 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(3443)]. 12.60/12.57 Derived: -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),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5301,a,5018,a)]. 12.70/12.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(5301,a,5169,a)]. 12.70/12.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5301,a,5270,a)]. 12.70/12.59 5302 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(3443)]. 12.70/12.59 Derived: 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),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),B),C)). [resolve(5302,a,5018,a)]. 12.70/12.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),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),B),C)). [resolve(5302,a,5169,a)]. 12.70/12.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),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),B),C)). [resolve(5302,a,5270,a)]. 12.70/12.59 5303 -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(3451)]. 12.70/12.59 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(5303,a,5018,a)]. 12.70/12.59 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(5303,a,5169,a)]. 12.70/12.59 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(5303,a,5270,a)]. 12.70/12.59 5304 -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(3475)]. 12.70/12.59 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(5304,a,5018,a)]. 12.70/12.59 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(5304,a,5169,a)]. 12.73/12.64 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(5304,a,5270,a)]. 12.73/12.64 5305 -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(3494)]. 12.73/12.64 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(5305,a,5018,a)]. 12.73/12.64 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(5305,a,5169,a)]. 12.73/12.64 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(5305,a,5270,a)]. 12.73/12.64 5306 -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(3494)]. 12.73/12.64 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(5306,a,5018,a)]. 12.73/12.64 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(5306,a,5169,a)]. 12.73/12.64 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(5306,a,5270,a)]. 12.73/12.64 5307 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(A),B)) # label(fact_Compl__atLeast) # label(axiom). [clausify(3512)]. 12.73/12.64 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(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_OatLeast(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5307,a,5018,a)]. 12.73/12.64 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A)). [resolve(5307,a,5169,a)]. 12.73/12.64 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A)). [resolve(5307,a,5270,a)]. 12.73/12.64 5308 -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_linorder__le__less__linear) # label(axiom). [clausify(3517)]. 12.73/12.64 5309 -class_Orderings_Olinorder(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),D),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C) # label(fact_min__max_Oinf__sup__distrib2) # label(axiom). [clausify(3547)]. 12.73/12.64 Derived: 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),C),B)) = 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),C)),B). [resolve(5309,a,5018,a)]. 12.73/12.64 Derived: 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),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),B). [resolve(5309,a,5169,a)]. 12.73/12.68 Derived: 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),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)),B). [resolve(5309,a,5270,a)]. 12.73/12.68 5310 -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(3561)]. 12.73/12.68 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(5310,a,5018,a)]. 12.73/12.68 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(5310,a,5169,a)]. 12.73/12.68 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(5310,a,5270,a)]. 12.73/12.68 5311 -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(3605)]. 12.73/12.68 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(5311,a,5018,a)]. 12.73/12.68 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(5311,a,5169,a)]. 12.73/12.68 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(5311,a,5270,a)]. 12.73/12.68 5312 -class_Orderings_Olinorder(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_Oinf__sup__distrib1) # label(axiom). [clausify(3615)]. 12.73/12.68 Derived: 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(5312,a,5018,a)]. 12.81/12.70 Derived: 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(5312,a,5169,a)]. 12.81/12.70 Derived: 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(5312,a,5270,a)]. 12.81/12.70 5313 -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(3637)]. 12.81/12.70 5314 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(B),f429(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(3638)]. 12.81/12.70 Derived: hBOOL(hAPP(hAPP(c_member(A),f429(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(5314,a,5018,a)]. 12.81/12.70 Derived: hBOOL(hAPP(hAPP(c_member(A),f429(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(5314,a,5169,a)]. 12.81/12.70 Derived: hBOOL(hAPP(hAPP(c_member(A),f429(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(5314,a,5270,a)]. 12.81/12.70 5315 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,C)),hAPP(B,f429(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(3638)]. 12.81/12.70 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(A,f429(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(5315,a,5018,a)]. 12.81/12.70 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,B)),hAPP(A,f429(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(5315,a,5169,a)]. 12.81/12.70 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,B)),hAPP(A,f429(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(5315,a,5270,a)]. 12.81/12.70 5316 -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(3661)]. 12.81/12.70 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(5316,a,5018,a)]. 12.81/12.70 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(5316,a,5169,a)]. 12.81/12.70 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(5316,a,5270,a)]. 12.81/12.71 5317 -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(3668)]. 12.81/12.71 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(5317,a,5018,a)]. 12.81/12.71 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(5317,a,5169,a)]. 12.81/12.71 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(5317,a,5270,a)]. 12.81/12.71 5318 -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(3669)]. 12.81/12.71 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(5318,a,5018,a)]. 12.81/12.71 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(5318,a,5169,a)]. 12.81/12.71 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(5318,a,5270,a)]. 12.81/12.71 5319 -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(3674)]. 12.81/12.71 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(5319,a,5018,a)]. 12.81/12.77 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(5319,a,5169,a)]. 12.81/12.77 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(5319,a,5270,a)]. 12.81/12.77 5320 -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(3675)]. 12.81/12.77 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(5320,a,5018,a)]. 12.81/12.77 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(5320,a,5169,a)]. 12.81/12.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5320,a,5270,a)]. 12.81/12.77 5321 -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(3750)]. 12.81/12.77 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(5321,a,5018,a)]. 12.81/12.77 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(5321,a,5169,a)]. 12.81/12.77 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(5321,a,5270,a)]. 12.81/12.77 5322 -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(3752)]. 12.89/12.79 5323 -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(3754)]. 12.89/12.79 5324 -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(3758)]. 12.89/12.79 5325 -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(3758)]. 12.89/12.79 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5325,a,5018,a)]. 12.89/12.79 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5325,a,5169,a)]. 12.89/12.79 5326 -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(3758)]. 12.89/12.79 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5326,a,5018,a)]. 12.89/12.79 Derived: A != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(5326,a,5169,a)]. 12.89/12.79 5327 -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(3775)]. 12.89/12.79 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(5327,a,5018,a)]. 12.89/12.79 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(5327,a,5169,a)]. 12.89/12.79 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(5327,a,5270,a)]. 12.89/12.79 5328 -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(3785)]. 12.89/12.79 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(5328,a,5018,a)]. 12.89/12.79 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(5328,a,5169,a)]. 12.89/12.81 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(5328,a,5270,a)]. 12.89/12.81 5329 -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(3795)]. 12.89/12.81 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(5329,a,5018,a)]. 12.89/12.81 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(5329,a,5169,a)]. 12.89/12.81 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(5329,a,5270,a)]. 12.89/12.81 5330 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(c_List_Oset(A),f447(B,A)) = B # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(3804)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),f447(A,tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5330,a,5018,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(c_List_Oset(tc_Int_Oint),f447(A,tc_Int_Oint)) = A. [resolve(5330,a,5169,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(c_List_Oset(tc_Nat_Onat),f447(A,tc_Nat_Onat)) = A. [resolve(5330,a,5270,a)]. 12.89/12.81 5331 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted(A,f447(B,A)) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(3804)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f447(A,tc_Code__Numeral_Ocode__numeral)). [resolve(5331,a,5018,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,f447(A,tc_Int_Oint)). [resolve(5331,a,5169,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f447(A,tc_Nat_Onat)). [resolve(5331,a,5270,a)]. 12.89/12.81 5332 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),f447(B,A))) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(3804)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),f447(A,tc_Code__Numeral_Ocode__numeral))). [resolve(5332,a,5018,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),f447(A,tc_Int_Oint))). [resolve(5332,a,5169,a)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),f447(A,tc_Nat_Onat))). [resolve(5332,a,5270,a)]. 12.89/12.81 5333 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_List_Olinorder__class_Osorted(A,C) | -hBOOL(hAPP(c_List_Odistinct(A),C)) | hAPP(c_List_Oset(A),C) != B | C = f447(B,A) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(3804)]. 12.89/12.81 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(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) != A | B = f447(A,tc_Code__Numeral_Ocode__numeral). [resolve(5333,a,5018,a)]. 12.89/12.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(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) != A | B = f447(A,tc_Int_Oint). [resolve(5333,a,5169,a)]. 12.89/12.85 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(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) != A | B = f447(A,tc_Nat_Onat). [resolve(5333,a,5270,a)]. 12.89/12.85 5334 -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(3844)]. 12.89/12.85 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(5334,a,5018,a)]. 12.89/12.85 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(5334,a,5169,a)]. 12.89/12.85 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(5334,a,5270,a)]. 12.89/12.85 5335 -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(3852)]. 12.89/12.85 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(5335,a,5018,a)]. 12.89/12.85 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(5335,a,5169,a)]. 12.89/12.85 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(5335,a,5270,a)]. 12.89/12.85 5336 -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)) | C = E # label(fact_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(3859)]. 12.98/12.89 5337 -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(3899)]. 12.98/12.89 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(5337,a,5018,a)]. 12.98/12.89 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(5337,a,5169,a)]. 12.98/12.89 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(5337,a,5270,a)]. 12.98/12.89 5338 -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(3913)]. 12.98/12.89 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(5338,a,5018,a)]. 12.98/12.89 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(5338,a,5169,a)]. 12.98/12.89 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(5338,a,5270,a)]. 12.98/12.89 5339 -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(3915)]. 12.98/12.89 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(5339,a,5018,a)]. 13.00/12.90 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(5339,a,5169,a)]. 13.00/12.90 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(5339,a,5270,a)]. 13.00/12.90 5340 -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(3915)]. 13.00/12.90 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(5340,a,5018,a)]. 13.00/12.90 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(5340,a,5169,a)]. 13.00/12.90 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(5340,a,5270,a)]. 13.00/12.90 5341 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(C,A),D),E))) | hBOOL(hAPP(hAPP(c_member(C),f470(E,D,C,B,A)),D)) # label(fact_less__SUP__iff) # label(axiom). [clausify(3919)]. 13.00/12.90 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(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Code__Numeral_Ocode__numeral),C),D))) | hBOOL(hAPP(hAPP(c_member(B),f470(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),C)). [resolve(5341,b,5018,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Int_Oint),C),D))) | hBOOL(hAPP(hAPP(c_member(B),f470(D,C,B,A,tc_Int_Oint)),C)). [resolve(5341,b,5169,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Nat_Onat),C),D))) | hBOOL(hAPP(hAPP(c_member(B),f470(D,C,B,A,tc_Nat_Onat)),C)). [resolve(5341,b,5270,a)]. 13.00/12.90 5342 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(C,A),D),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(E,f470(E,D,C,B,A)))) # label(fact_less__SUP__iff) # label(axiom). [clausify(3919)]. 13.00/12.90 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(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Code__Numeral_Ocode__numeral),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(D,f470(D,C,B,A,tc_Code__Numeral_Ocode__numeral)))). [resolve(5342,b,5018,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Int_Oint),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(D,f470(D,C,B,A,tc_Int_Oint)))). [resolve(5342,b,5169,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Nat_Onat),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(D,f470(D,C,B,A,tc_Nat_Onat)))). [resolve(5342,b,5270,a)]. 13.00/12.90 5343 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(C,A),D),E))) | -hBOOL(hAPP(hAPP(c_member(C),F),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(E,F))) # label(fact_less__SUP__iff) # label(axiom). [clausify(3919)]. 13.00/12.90 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(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Code__Numeral_Ocode__numeral),C),D))) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(D,E))). [resolve(5343,b,5018,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Int_Oint),C),D))) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(D,E))). [resolve(5343,b,5169,a)]. 13.00/12.90 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Nat_Onat),C),D))) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(D,E))). [resolve(5343,b,5270,a)]. 13.00/12.90 5344 -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(3928)]. 13.00/12.91 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(5344,a,5018,a)]. 13.00/12.91 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(5344,a,5169,a)]. 13.00/12.91 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(5344,a,5270,a)]. 13.00/12.91 5345 -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)) | C = B # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(3930)]. 13.00/12.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)) | B = A. [resolve(5345,a,5270,a)]. 13.00/12.91 5346 -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(3930)]. 13.00/12.91 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(5346,a,5018,a)]. 13.00/12.91 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(5346,a,5169,a)]. 13.00/12.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(5346,a,5270,a)]. 13.00/12.91 5347 -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(3930)]. 13.00/12.91 5348 -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(3931)]. 13.00/12.91 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(5348,a,5018,a)]. 13.00/12.91 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(5348,a,5169,a)]. 13.00/12.91 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(5348,a,5270,a)]. 13.03/12.96 5349 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f476(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(3988)]. 13.03/12.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f476(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(5349,a,5018,a)]. 13.03/12.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f476(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(5349,a,5169,a)]. 13.03/12.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f476(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(5349,a,5270,a)]. 13.03/12.96 5350 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f476(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(3988)]. 13.03/12.96 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),f476(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(5350,a,5018,a)]. 13.03/12.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f476(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(5350,a,5169,a)]. 13.03/12.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f476(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(5350,a,5270,a)]. 13.03/12.96 5351 -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(4007)]. 13.03/12.96 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(5351,a,5018,a)]. 13.03/12.96 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(5351,a,5169,a)]. 13.03/12.97 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(5351,a,5270,a)]. 13.03/12.97 5352 -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(4007)]. 13.03/12.97 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(5352,a,5018,a)]. 13.03/12.97 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(5352,a,5169,a)]. 13.03/12.97 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(5352,a,5270,a)]. 13.03/12.97 5353 -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(4017)]. 13.03/12.99 5354 -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(4026)]. 13.03/12.99 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(5354,a,5018,a)]. 13.03/12.99 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(5354,a,5169,a)]. 13.03/12.99 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(5354,a,5270,a)]. 13.03/12.99 5355 -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(4036)]. 13.03/12.99 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(5355,a,5018,a)]. 13.03/12.99 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(5355,a,5169,a)]. 13.03/12.99 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(5355,a,5270,a)]. 13.03/12.99 5356 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) # label(fact_min__max_Osup__inf__distrib2) # label(axiom). [clausify(4042)]. 13.03/12.99 Derived: 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)),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),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5356,a,5018,a)]. 13.03/12.99 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(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),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)). [resolve(5356,a,5169,a)]. 13.12/13.02 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(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),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)). [resolve(5356,a,5270,a)]. 13.12/13.02 5357 -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(4053)]. 13.12/13.02 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(5357,a,5018,a)]. 13.12/13.02 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B) | A = B. [resolve(5357,a,5169,a)]. 13.12/13.02 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B) | A = B. [resolve(5357,a,5270,a)]. 13.12/13.02 5358 -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(4053)]. 13.12/13.02 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(5358,a,5018,a)]. 13.12/13.02 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B) | A != B. [resolve(5358,a,5169,a)]. 13.12/13.02 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B) | A != B. [resolve(5358,a,5270,a)]. 13.12/13.02 5359 -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(4073)]. 13.12/13.02 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(5359,a,5018,a)]. 13.12/13.02 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(5359,a,5169,a)]. 13.12/13.02 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(5359,a,5270,a)]. 13.12/13.02 5360 -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(4075)]. 13.12/13.02 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(5360,a,5018,a)]. 13.12/13.02 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(5360,a,5169,a)]. 13.12/13.05 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(5360,a,5270,a)]. 13.12/13.05 5361 -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(4120)]. 13.12/13.05 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(5361,a,5018,a)]. 13.12/13.05 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(5361,a,5169,a)]. 13.12/13.05 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(5361,a,5270,a)]. 13.12/13.05 5362 -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(4124)]. 13.12/13.05 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(5362,a,5018,a)]. 13.12/13.05 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(5362,a,5169,a)]. 13.12/13.05 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(5362,a,5270,a)]. 13.12/13.06 5363 -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(4124)]. 13.12/13.06 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(5363,a,5018,a)]. 13.12/13.06 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(5363,a,5169,a)]. 13.12/13.06 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(5363,a,5270,a)]. 13.12/13.06 5364 -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(4133)]. 13.12/13.06 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(5364,a,5018,a)]. 13.12/13.06 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(5364,a,5169,a)]. 13.12/13.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5364,a,5270,a)]. 13.21/13.13 5365 -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(4136)]. 13.21/13.13 5366 -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(4185)]. 13.21/13.13 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(5366,a,5169,a)]. 13.21/13.13 5367 -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(4185)]. 13.21/13.13 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(5367,a,5018,a)]. 13.21/13.13 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(5367,a,5169,a)]. 13.21/13.13 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),A),B)) | B != A. [resolve(5367,a,5270,a)]. 13.21/13.13 5368 -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(4192)]. 13.21/13.13 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(5368,a,5018,a)]. 13.21/13.13 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(5368,a,5169,a)]. 13.21/13.13 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(5368,a,5270,a)]. 13.21/13.13 5369 -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(4218)]. 13.21/13.13 5370 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_ONil) # label(axiom). [clausify(4229)]. 13.21/13.13 5371 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f497(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(4241)]. 13.21/13.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f497(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(5371,a,5018,a)]. 13.21/13.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f497(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(5371,a,5169,a)]. 13.21/13.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f497(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(5371,a,5270,a)]. 13.21/13.14 5372 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f497(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(4241)]. 13.21/13.14 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),f497(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(5372,a,5018,a)]. 13.21/13.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f497(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(5372,a,5169,a)]. 13.21/13.14 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f497(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(5372,a,5270,a)]. 13.21/13.14 5373 -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(4253)]. 13.21/13.14 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(5373,a,5018,a)]. 13.21/13.16 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(5373,a,5169,a)]. 13.21/13.16 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(5373,a,5270,a)]. 13.21/13.16 5374 -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(4253)]. 13.21/13.16 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(5374,a,5018,a)]. 13.21/13.16 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(5374,a,5169,a)]. 13.21/13.16 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(5374,a,5270,a)]. 13.21/13.16 5375 -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(4273)]. 13.21/13.16 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(5375,a,5018,a)]. 13.21/13.16 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(5375,a,5169,a)]. 13.21/13.16 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(5375,a,5270,a)]. 13.21/13.16 5376 -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(4289)]. 13.21/13.16 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(5376,a,5018,a)]. 13.21/13.16 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(5376,a,5169,a)]. 13.33/13.24 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(5376,a,5270,a)]. 13.33/13.24 5377 -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(4290)]. 13.33/13.24 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(5377,a,5018,a)]. 13.33/13.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A = B. [resolve(5377,a,5169,a)]. 13.33/13.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A = B. [resolve(5377,a,5270,a)]. 13.33/13.24 5378 -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(4290)]. 13.33/13.24 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(5378,a,5018,a)]. 13.33/13.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A != B. [resolve(5378,a,5169,a)]. 13.33/13.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A != B. [resolve(5378,a,5270,a)]. 13.33/13.24 5379 -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(4346)]. 13.33/13.24 5380 -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(4358)]. 13.33/13.24 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(5380,a,5018,a)]. 13.33/13.24 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(5380,a,5169,a)]. 13.33/13.24 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(5380,a,5270,a)]. 13.33/13.27 5381 -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(4375)]. 13.33/13.27 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(5381,a,5018,a)]. 13.33/13.27 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(5381,a,5169,a)]. 13.33/13.27 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(5381,a,5270,a)]. 13.33/13.27 5382 -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(4376)]. 13.33/13.27 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(5382,a,5018,a)]. 13.33/13.27 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(5382,a,5169,a)]. 13.33/13.27 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(5382,a,5270,a)]. 13.33/13.27 5383 -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(4390)]. 13.33/13.27 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(5383,a,5018,a)]. 13.33/13.27 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(5383,a,5169,a)]. 13.33/13.30 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(5383,a,5270,a)]. 13.33/13.30 5384 -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(4397)]. 13.33/13.30 5385 -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(4397)]. 13.33/13.30 5386 -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(4432)]. 13.33/13.30 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_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(5386,a,5018,a)]. 13.42/13.30 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(5386,a,5169,a)]. 13.42/13.30 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(5386,a,5270,a)]. 13.42/13.30 5387 -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(4437)]. 13.42/13.30 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(5387,a,5018,a)]. 13.42/13.36 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(5387,a,5169,a)]. 13.42/13.36 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(5387,a,5270,a)]. 13.42/13.36 5388 -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(4445)]. 13.42/13.36 5389 -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(4497)]. 13.42/13.36 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(5389,a,5018,a)]. 13.42/13.36 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(5389,a,5169,a)]. 13.42/13.36 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(5389,a,5270,a)]. 13.42/13.36 5390 -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(4522)]. 13.42/13.36 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(5390,a,5018,a)]. 13.42/13.36 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(5390,a,5169,a)]. 13.51/13.42 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(5390,a,5270,a)]. 13.51/13.42 5391 -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(4525)]. 13.51/13.42 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(5391,a,5018,a)]. 13.51/13.42 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(5391,a,5169,a)]. 13.51/13.42 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(5391,a,5270,a)]. 13.51/13.42 5392 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_Max_Oidem) # label(axiom). [clausify(4576)]. 13.51/13.42 5393 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_min__max_Oinf_Oidem) # label(axiom). [clausify(4581)]. 13.51/13.42 5394 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) != hAPP(c_SetInterval_Oord__class_OlessThan(A),C) | B = C # label(fact_lessThan__eq__iff) # label(axiom). [clausify(4582)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B) | A = B. [resolve(5394,a,5018,a)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B) | A = B. [resolve(5394,a,5169,a)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B) | A = B. [resolve(5394,a,5270,a)]. 13.51/13.42 5395 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) = hAPP(c_SetInterval_Oord__class_OlessThan(A),C) | B != C # label(fact_lessThan__eq__iff) # label(axiom). [clausify(4582)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B) | A != B. [resolve(5395,a,5018,a)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B) | A != B. [resolve(5395,a,5169,a)]. 13.51/13.42 Derived: hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B) | A != B. [resolve(5395,a,5270,a)]. 13.51/13.42 5396 -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(4587)]. 13.51/13.42 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(5396,a,5018,a)]. 13.51/13.44 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(5396,a,5169,a)]. 13.51/13.44 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(5396,a,5270,a)]. 13.51/13.44 5397 -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(4604)]. 13.51/13.44 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(5397,a,5018,a)]. 13.51/13.44 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(5397,a,5169,a)]. 13.51/13.44 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(5397,a,5270,a)]. 13.51/13.44 5398 -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(4605)]. 13.51/13.44 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)) | 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(5398,a,5018,a)]. 13.51/13.44 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(5398,a,5169,a)]. 13.51/13.44 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(5398,a,5270,a)]. 13.51/13.44 5399 -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(4616)]. 13.51/13.44 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(5399,a,5018,a)]. 13.51/13.44 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(5399,a,5169,a)]. 13.51/13.47 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(5399,a,5270,a)]. 13.51/13.47 5400 -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(4656)]. 13.51/13.47 5401 -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(4656)]. 13.51/13.47 5402 -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),D),B)),C)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(4656)]. 13.51/13.47 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),C),A)),B)). [resolve(5402,a,5018,a)]. 13.51/13.47 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(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),C),A)),B)). [resolve(5402,a,5169,a)]. 13.51/13.47 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),C),A)),B)). [resolve(5402,a,5270,a)]. 13.51/13.47 5403 -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(4660)]. 13.51/13.47 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(5403,a,5018,a)]. 13.51/13.47 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(5403,a,5169,a)]. 13.51/13.47 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(5403,a,5270,a)]. 13.51/13.47 5404 -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(4660)]. 13.51/13.47 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(5404,a,5018,a)]. 13.51/13.47 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(5404,a,5169,a)]. 13.51/13.47 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(5404,a,5270,a)]. 13.51/13.47 5405 -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(4661)]. 13.51/13.47 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(5405,a,5018,a)]. 13.51/13.47 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(5405,a,5169,a)]. 13.51/13.47 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(5405,a,5270,a)]. 13.64/13.53 5406 -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(4726)]. 13.64/13.53 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(5406,a,5018,a)]. 13.64/13.53 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(5406,a,5169,a)]. 13.64/13.53 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(5406,a,5270,a)]. 13.64/13.53 5407 -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(4740)]. 13.64/13.53 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(5407,a,5018,a)]. 13.64/13.53 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(5407,a,5169,a)]. 13.64/13.58 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(5407,a,5270,a)]. 13.64/13.58 5408 -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),f563(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(4740)]. 13.64/13.58 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),f563(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(5408,a,5018,a)]. 13.64/13.58 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),f563(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(5408,a,5169,a)]. 13.64/13.58 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),f563(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(5408,a,5270,a)]. 13.64/13.58 5409 -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),f563(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(4740)]. 13.64/13.58 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),f563(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(5409,a,5018,a)]. 13.64/13.58 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),f563(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(5409,a,5169,a)]. 13.64/13.58 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),f563(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(5409,a,5270,a)]. 13.64/13.58 5410 -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(4810)]. 13.64/13.61 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(5410,a,5018,a)]. 13.64/13.61 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(5410,a,5169,a)]. 13.64/13.61 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(5410,a,5270,a)]. 13.64/13.61 5411 -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),f568(B,A)),f569(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f568(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f569(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(4830)]. 13.64/13.61 5412 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),C)) | c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__insort) # label(axiom). [clausify(4843)]. 13.64/13.61 Derived: -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)),A),B)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5412,a,5018,a)]. 13.64/13.61 Derived: -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)),A),B)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5412,a,5169,a)]. 13.64/13.61 Derived: -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)),A),B)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5412,a,5270,a)]. 13.64/13.61 5413 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,A,c_COMBI(A)),B),C)) | -c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__insort) # label(axiom). [clausify(4843)]. 13.64/13.61 Derived: 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)),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5413,a,5018,a)]. 13.64/13.61 Derived: 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)),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5413,a,5169,a)]. 13.64/13.61 Derived: 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)),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5413,a,5270,a)]. 13.64/13.61 5414 -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(4845)]. 13.73/13.64 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(5414,a,5018,a)]. 13.73/13.64 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(5414,a,5169,a)]. 13.73/13.64 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(5414,a,5270,a)]. 13.73/13.64 5415 -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(4860)]. 13.73/13.64 5416 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f573(B,A)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),B))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(4873)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f573(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(5416,a,5018,a)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f573(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(5416,a,5169,a)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f573(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(5416,a,5270,a)]. 13.73/13.64 5417 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f574(B,A)),f573(B,A))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(4873)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f574(A,tc_Code__Numeral_Ocode__numeral)),f573(A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5417,a,5018,a)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f574(A,tc_Int_Oint)),f573(A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5417,a,5169,a)]. 13.73/13.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f574(A,tc_Nat_Onat)),f573(A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5417,a,5270,a)]. 13.73/13.64 5418 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f574(B,A))),hAPP(hAPP(c_List_Onth(A),B),f573(B,A)))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(4873)]. 13.73/13.67 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),f574(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f573(A,tc_Code__Numeral_Ocode__numeral)))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5418,a,5018,a)]. 13.73/13.67 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f574(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f573(A,tc_Int_Oint)))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5418,a,5169,a)]. 13.73/13.67 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f574(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f573(A,tc_Nat_Onat)))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5418,a,5270,a)]. 13.73/13.67 5419 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),C),D)),hAPP(hAPP(c_List_Onth(A),C),B))) | -c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(4873)]. 13.73/13.67 5420 -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(4885)]. 13.73/13.67 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(5420,a,5018,a)]. 13.73/13.67 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(5420,a,5169,a)]. 13.73/13.67 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(5420,a,5270,a)]. 13.73/13.67 5421 -class_Orderings_Olinorder(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_Osup__inf__distrib1) # label(axiom). [clausify(4903)]. 13.73/13.67 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),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(5421,a,5018,a)]. 13.73/13.67 Derived: 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(5421,a,5169,a)]. 13.73/13.67 Derived: 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(5421,a,5270,a)]. 18.35/18.25 5422 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 18.35/18.25 5423 -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(25)]. 18.35/18.25 5424 -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(643)]. 18.35/18.25 5425 -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(933)]. 18.35/18.25 5426 -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(2309)]. 18.35/18.25 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(5422,a,5423,a)]. 18.35/18.25 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(5422,a,5424,a)]. 18.35/18.25 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_HOL_Obool)),A) = A. [resolve(5422,a,5425,a)]. 18.35/18.25 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(5422,a,5426,a)]. 18.35/18.25 5427 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(2653)]. 18.35/18.25 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(5427,b,5423,a)]. 18.35/18.25 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(5427,b,5424,a)]. 18.35/18.25 Derived: -class_Lattices_Obounded__lattice(A) | hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),c_Orderings_Obot__class_Obot(tc_fun(B,A))),C) = C. [resolve(5427,b,5425,a)]. 18.35/18.25 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(5427,b,5426,a)]. 18.35/18.25 5428 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != B | c_Orderings_Obot__class_Obot(A) != C | c_Orderings_Obot__class_Obot(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),C),B) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4857)]. 18.35/18.25 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != B | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),B),A). [resolve(5428,a,5422,a)]. 18.35/18.25 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != D | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),D),C) | -class_Lattices_Obounded__lattice(B). [resolve(5428,a,5427,b)]. 18.35/18.25 5429 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),C),B) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4857)]. 18.35/18.25 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),B),A). [resolve(5429,a,5422,a)]. 19.40/19.30 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),D),C) | -class_Lattices_Obounded__lattice(B). [resolve(5429,a,5427,b)]. 19.40/19.30 5430 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = B | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4857)]. 19.40/19.30 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B). [resolve(5430,a,5422,a)]. 19.40/19.30 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | -class_Lattices_Obounded__lattice(B). [resolve(5430,a,5427,b)]. 19.40/19.30 5431 class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 19.40/19.30 5432 -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(29)]. 19.40/19.30 5433 class_Rings_Oordered__cancel__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 19.40/19.30 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(5433,a,5432,a)]. 19.40/19.30 5434 -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(1103)]. 19.40/19.30 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),B),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5434,a,5431,a)]. 19.40/19.30 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),B),A)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5434,a,5433,a)]. 19.40/19.30 5435 -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(1374)]. 21.59/21.51 5436 -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(2622)]. 21.59/21.51 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(5436,a,5431,a)]. 21.59/21.51 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(5436,a,5433,a)]. 21.59/21.51 5437 -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(3265)]. 21.59/21.51 5438 -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_split__mult__neg__le) # label(axiom). [clausify(3265)]. 21.59/21.51 5439 class_Rings_Oordered__cancel__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 21.59/21.51 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(5439,a,5432,a)]. 21.59/21.51 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5439,a,5434,a)]. 21.59/21.51 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5439,a,5436,a)]. 21.59/21.51 5440 class_Rings_Olinordered__semidom(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semidom) # label(axiom). [assumption]. 21.59/21.52 5441 -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(31)]. 21.59/21.52 5442 -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),D)) | 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))) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(297)]. 21.59/21.52 5443 -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),D)) | -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))) # label(fact_power__strict__increasing__iff) # label(axiom). [clausify(297)]. 21.59/21.52 5444 -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(313)]. 21.59/21.52 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(5440,a,5441,a)]. 21.59/21.52 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),B),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),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5440,a,5442,a)]. 21.59/21.52 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),B),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),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C))). [resolve(5440,a,5443,a)]. 21.59/21.52 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(5440,a,5444,a)]. 21.59/21.52 5445 -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(360)]. 21.59/21.52 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(5445,a,5440,a)]. 21.69/21.60 5446 -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(458)]. 21.69/21.60 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)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A))). [resolve(5446,a,5440,a)]. 21.69/21.60 5447 -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(630)]. 21.69/21.60 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(5447,a,5440,a)]. 21.69/21.60 5448 -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(669)]. 21.69/21.60 5449 -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(894)]. 21.69/21.60 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__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(5449,a,5440,a)]. 21.69/21.60 5450 -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(894)]. 21.69/21.60 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__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(5450,a,5440,a)]. 21.69/21.60 5451 -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(1036)]. 21.69/21.60 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(5451,a,5440,a)]. 21.75/21.64 5452 -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(1041)]. 21.75/21.64 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(5452,a,5440,a)]. 21.75/21.64 5453 -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(1043)]. 21.75/21.64 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(5453,a,5440,a)]. 21.75/21.64 5454 -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(1159)]. 21.75/21.64 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(5454,a,5440,a)]. 21.75/21.64 5455 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f159(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(1257)]. 21.75/21.64 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(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(5455,a,5440,a)]. 21.75/21.64 5456 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f159(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(1257)]. 21.75/21.64 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,f159(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(5456,a,5440,a)]. 21.84/21.77 5457 -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(1272)]. 21.84/21.77 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(5457,a,5440,a)]. 21.84/21.77 5458 -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(1486)]. 21.84/21.77 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(5458,a,5440,a)]. 21.84/21.77 5459 -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(1655)]. 21.84/21.77 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(5459,a,5440,a)]. 21.84/21.77 5460 -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(1683)]. 21.84/21.77 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),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(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5460,a,5440,a)]. 21.84/21.77 5461 -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(1788)]. 21.90/21.84 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(5461,a,5440,a)]. 21.90/21.84 5462 -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(1935)]. 21.90/21.84 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(5462,a,5440,a)]. 21.90/21.84 5463 -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(1935)]. 21.90/21.84 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(5463,a,5440,a)]. 21.90/21.84 5464 -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(1961)]. 21.90/21.84 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(5464,a,5440,a)]. 21.90/21.84 5465 -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(1980)]. 21.90/21.84 5466 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | C != D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(2039)]. 21.90/21.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C). [resolve(5466,a,5440,a)]. 21.90/21.84 5467 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | C = D | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) # label(fact_power__inject__exp) # label(axiom). [clausify(2039)]. 22.05/22.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C). [resolve(5467,a,5440,a)]. 22.05/22.02 5468 -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(2180)]. 22.05/22.02 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(5468,a,5440,a)]. 22.05/22.02 5469 -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(2180)]. 22.05/22.02 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(5469,a,5440,a)]. 22.05/22.02 5470 -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(2344)]. 22.05/22.02 5471 -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(2378)]. 22.05/22.02 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(5471,a,5440,a)]. 22.05/22.02 5472 -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__iff) # label(axiom). [clausify(2529)]. 22.05/22.02 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5472,a,5440,a)]. 22.05/22.02 5473 -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__iff) # label(axiom). [clausify(2529)]. 22.33/22.22 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5473,a,5440,a)]. 22.33/22.22 5474 -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(2611)]. 22.33/22.22 5475 -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(2644)]. 22.33/22.22 5476 -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(2722)]. 22.33/22.22 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(5476,a,5440,a)]. 22.33/22.22 5477 -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(2828)]. 22.33/22.22 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(5477,a,5440,a)]. 22.33/22.22 5478 -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(2841)]. 22.33/22.22 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(5478,a,5440,a)]. 22.33/22.22 5479 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 22.33/22.22 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),B),C)) | 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))). [resolve(5479,a,5442,a)]. 22.33/22.22 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),B),C)) | -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))). [resolve(5479,a,5443,a)]. 22.33/22.22 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(5479,a,5444,a)]. 22.33/22.23 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(5479,a,5445,a)]. 22.33/22.23 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(5479,a,5449,a)]. 22.33/22.23 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(5479,a,5450,a)]. 22.33/22.23 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_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(5479,a,5451,a)]. 22.33/22.23 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(5479,a,5452,a)]. 22.33/22.23 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(5479,a,5453,a)]. 22.33/22.23 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(5479,a,5454,a)]. 22.33/22.23 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(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(5479,a,5455,a)]. 22.33/22.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(A,f159(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(5479,a,5456,a)]. 22.33/22.23 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(5479,a,5457,a)]. 22.33/22.23 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(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B))). [resolve(5479,a,5458,a)]. 22.35/22.24 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(5479,a,5459,a)]. 22.35/22.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(tc_Nat_Onat),A),B)). [resolve(5479,a,5460,a)]. 22.35/22.24 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(5479,a,5461,a)]. 22.35/22.24 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(5479,a,5462,a)]. 22.35/22.24 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(5479,a,5463,a)]. 22.35/22.24 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(5479,a,5464,a)]. 22.35/22.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C). [resolve(5479,a,5466,a)]. 22.35/22.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C). [resolve(5479,a,5467,a)]. 22.35/22.24 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(5479,a,5468,a)]. 22.35/22.24 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(5479,a,5469,a)]. 22.35/22.24 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(5479,a,5471,a)]. 22.35/22.24 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5479,a,5472,a)]. 22.44/22.34 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5479,a,5473,a)]. 22.44/22.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5479,a,5476,a)]. 22.44/22.34 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(5479,a,5477,a)]. 22.44/22.34 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(5479,a,5478,a)]. 22.44/22.34 5480 -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(3106)]. 22.44/22.34 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(5480,a,5440,a)]. 22.44/22.34 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(5480,a,5479,a)]. 22.44/22.34 5481 -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(3257)]. 22.44/22.34 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(5481,a,5440,a)]. 22.44/22.34 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(5481,a,5479,a)]. 22.44/22.34 5482 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f389(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(3261)]. 22.44/22.34 Derived: hBOOL(hAPP(hAPP(c_member(A),f389(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(5482,a,5440,a)]. 22.50/22.40 5483 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f389(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(3261)]. 22.50/22.40 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,f389(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(5483,a,5440,a)]. 22.50/22.40 5484 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | B = D # label(fact_power__eq__imp__eq__base) # label(axiom). [clausify(3305)]. 22.50/22.40 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5484,a,5440,a)]. 22.50/22.40 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5484,a,5479,a)]. 22.50/22.40 5485 -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(3369)]. 22.50/22.40 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(5485,a,5440,a)]. 22.50/22.40 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(5485,a,5479,a)]. 22.81/22.72 5486 -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(3407)]. 22.81/22.72 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(5486,a,5440,a)]. 22.81/22.72 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(5486,a,5479,a)]. 22.81/22.72 5487 -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(3519)]. 22.81/22.72 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(5487,a,5440,a)]. 22.81/22.72 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5487,a,5479,a)]. 22.81/22.72 5488 -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(3634)]. 22.81/22.72 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(5488,a,5440,a)]. 22.81/22.72 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(5488,a,5479,a)]. 22.81/22.72 5489 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_less__imp__of__nat__less) # label(axiom). [clausify(4050)]. 22.92/22.80 5490 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 22.92/22.80 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5490,a,5441,a)]. 22.92/22.80 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),B),C)) | 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))). [resolve(5490,a,5442,a)]. 22.92/22.80 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),B),C)) | -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))). [resolve(5490,a,5443,a)]. 22.92/22.80 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(5490,a,5444,a)]. 22.92/22.80 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(5490,a,5445,a)]. 22.92/22.80 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(5490,a,5447,a)]. 22.92/22.80 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(5490,a,5449,a)]. 22.92/22.80 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(5490,a,5450,a)]. 22.92/22.80 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(5490,a,5451,a)]. 22.92/22.80 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(5490,a,5452,a)]. 22.92/22.80 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(5490,a,5453,a)]. 22.92/22.80 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(5490,a,5454,a)]. 22.92/22.80 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(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(5490,a,5455,a)]. 22.92/22.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f159(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(5490,a,5456,a)]. 22.92/22.80 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(5490,a,5457,a)]. 22.92/22.80 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(5490,a,5458,a)]. 22.92/22.80 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(5490,a,5459,a)]. 22.92/22.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),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(tc_Int_Oint),A),B)). [resolve(5490,a,5460,a)]. 22.92/22.80 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(5490,a,5461,a)]. 22.92/22.80 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(5490,a,5462,a)]. 22.92/22.80 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(5490,a,5463,a)]. 22.92/22.80 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(5490,a,5464,a)]. 22.92/22.80 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | B != C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C). [resolve(5490,a,5466,a)]. 22.92/22.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | B = C | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C). [resolve(5490,a,5467,a)]. 22.92/22.81 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(5490,a,5468,a)]. 22.92/22.81 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(5490,a,5469,a)]. 22.92/22.81 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(5490,a,5471,a)]. 22.92/22.81 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(5490,a,5472,a)]. 22.92/22.81 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(5490,a,5473,a)]. 22.92/22.81 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(5490,a,5476,a)]. 22.92/22.81 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(5490,a,5477,a)]. 22.92/22.81 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(5490,a,5478,a)]. 22.92/22.81 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(5490,a,5480,a)]. 22.92/22.81 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(5490,a,5481,a)]. 22.92/22.81 Derived: hBOOL(hAPP(hAPP(c_member(A),f389(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(5490,a,5482,a)]. 22.92/22.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f389(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(5490,a,5483,a)]. 22.92/22.81 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = C. [resolve(5490,a,5484,a)]. 23.10/23.03 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(5490,a,5485,a)]. 23.10/23.03 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(5490,a,5486,a)]. 23.10/23.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5490,a,5487,a)]. 23.10/23.03 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(5490,a,5488,a)]. 23.10/23.03 5491 -class_Rings_Olinordered__semidom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) != hAPP(hAPP(c_Power_Opower__class_Opower(A),D),hAPP(c_Nat_OSuc,C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | B = D # label(fact_power__inject__base) # label(axiom). [clausify(4265)]. 23.10/23.03 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | A = C. [resolve(5491,a,5440,a)]. 23.10/23.03 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),C)) | A = C. [resolve(5491,a,5479,a)]. 23.10/23.03 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),C),hAPP(c_Nat_OSuc,B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | A = C. [resolve(5491,a,5490,a)]. 23.10/23.03 5492 -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(4529)]. 23.10/23.04 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(5492,a,5440,a)]. 23.10/23.04 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(5492,a,5479,a)]. 23.10/23.04 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(5492,a,5490,a)]. 23.10/23.04 5493 -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(4532)]. 23.10/23.04 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(5493,a,5440,a)]. 23.10/23.04 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(5493,a,5479,a)]. 23.10/23.04 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(5493,a,5490,a)]. 24.41/24.29 5494 -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(4543)]. 24.41/24.29 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(5494,a,5440,a)]. 24.41/24.29 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(5494,a,5479,a)]. 24.41/24.29 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(5494,a,5490,a)]. 24.41/24.29 5495 -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(4702)]. 24.41/24.29 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(5495,a,5440,a)]. 24.41/24.29 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B))). [resolve(5495,a,5490,a)]. 24.41/24.29 5496 class_Rings_Olinordered__idom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__idom) # label(axiom). [assumption]. 24.41/24.29 5497 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),B)) # label(fact_mult__right__le__one__le) # label(axiom). [clausify(37)]. 24.41/24.29 5498 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)))) # label(fact_zero__le__even__power_H) # label(axiom). [clausify(49)]. 24.41/24.30 5499 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_power2__less__0) # label(axiom). [clausify(87)]. 24.41/24.30 5500 -class_Rings_Olinordered__idom(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(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__le__iff) # label(axiom). [clausify(130)]. 24.41/24.30 5501 -class_Rings_Olinordered__idom(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(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__le__iff) # label(axiom). [clausify(130)]. 24.41/24.30 5502 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) # label(fact_less__special_I2_J) # label(axiom). [clausify(196)]. 24.41/24.30 5503 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) # label(fact_less__special_I2_J) # label(axiom). [clausify(196)]. 24.41/24.30 5504 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) # label(fact_abs__of__nat) # label(axiom). [clausify(199)]. 24.41/24.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),A)). [resolve(5496,a,5497,a)]. 24.41/24.30 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)))). [resolve(5496,a,5498,a)]. 24.41/24.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5496,a,5499,a)]. 24.41/24.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(5496,a,5500,a)]. 24.41/24.30 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(5496,a,5501,a)]. 24.41/24.30 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5496,a,5502,b)]. 24.41/24.31 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5496,a,5503,b)]. 24.41/24.31 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A). [resolve(5496,a,5504,a)]. 24.41/24.31 5505 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Osgn__class_Osgn(A,B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_sgn__greater) # label(axiom). [clausify(299)]. 24.41/24.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5505,a,5496,a)]. 24.41/24.31 5506 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),c_Groups_Osgn__class_Osgn(A,B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_sgn__greater) # label(axiom). [clausify(299)]. 24.41/24.31 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5506,a,5496,a)]. 24.41/24.31 5507 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) != c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sgn__0__0) # label(axiom). [clausify(307)]. 24.41/24.31 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5507,a,5496,a)]. 24.41/24.31 5508 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Ozero__class_Ozero(A) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_sgn__0__0) # label(axiom). [clausify(307)]. 24.41/24.31 5509 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),C)),B) # label(fact_abs__mult__pos) # label(axiom). [clausify(347)]. 24.41/24.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)),A). [resolve(5509,a,5496,a)]. 24.41/24.31 5510 -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_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) # label(fact_less__special_I4_J) # label(axiom). [clausify(351)]. 24.41/24.31 Derived: -class_Int_Onumber__ring(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_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))). [resolve(5510,a,5496,a)]. 24.41/24.31 5511 -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_Oone__class_Oone(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) # label(fact_less__special_I4_J) # label(axiom). [clausify(351)]. 24.41/24.31 Derived: -class_Int_Onumber__ring(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_Oone__class_Oone(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))). [resolve(5511,a,5496,a)]. 24.41/24.33 5512 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OPls)) | 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__special_I3_J) # label(axiom). [clausify(353)]. 24.41/24.33 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),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_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5512,a,5496,a)]. 24.41/24.33 5513 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OPls)) | -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__special_I3_J) # label(axiom). [clausify(353)]. 24.41/24.33 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),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_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5513,a,5496,a)]. 24.41/24.33 5514 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__less__0__iff) # label(axiom). [clausify(394)]. 24.41/24.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5514,a,5496,a)]. 24.41/24.33 5515 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__less__0__iff) # label(axiom). [clausify(394)]. 24.41/24.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5515,a,5496,a)]. 24.41/24.33 5516 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__le__0__iff) # label(axiom). [clausify(628)]. 24.41/24.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5516,a,5496,a)]. 24.41/24.33 5517 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_of__int__le__0__iff) # label(axiom). [clausify(628)]. 24.41/24.33 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5517,a,5496,a)]. 24.50/24.39 5518 -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_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(700)]. 24.50/24.39 Derived: -class_Int_Onumber__ring(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_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(5518,a,5496,a)]. 24.50/24.39 5519 -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(700)]. 24.50/24.39 5520 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Osgn__class_Osgn(A,B)),c_Groups_Osgn__class_Osgn(A,C)) # label(fact_sgn__times) # label(axiom). [clausify(777)]. 24.50/24.39 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,B)). [resolve(5520,a,5496,a)]. 24.50/24.39 5521 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),C)) # label(fact_abs__mult) # label(axiom). [clausify(864)]. 24.50/24.39 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),B)). [resolve(5521,a,5496,a)]. 24.50/24.39 5522 -class_Rings_Olinordered__idom(A) | B = C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_linorder__neqE__linordered__idom) # label(axiom). [clausify(1029)]. 24.50/24.39 5523 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(1089)]. 24.50/24.39 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5523,b,5496,a)]. 24.50/24.39 5524 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(1089)]. 24.50/24.39 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5524,b,5496,a)]. 24.50/24.39 5525 -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(1139)]. 24.50/24.40 Derived: -class_Int_Onumber__ring(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(5525,b,5496,a)]. 24.50/24.40 5526 -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(1139)]. 24.50/24.40 Derived: -class_Int_Onumber__ring(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(5526,b,5496,a)]. 24.50/24.40 5527 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1194)]. 24.50/24.40 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5527,a,5496,a)]. 24.50/24.40 5528 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1194)]. 24.50/24.40 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5528,a,5496,a)]. 24.50/24.40 5529 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_sum__power2__le__zero__iff) # label(axiom). [clausify(1194)]. 24.55/24.44 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5529,a,5496,a)]. 24.55/24.44 5530 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) != B # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1318)]. 24.55/24.44 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5530,a,5496,a)]. 24.55/24.44 5531 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1318)]. 24.55/24.44 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5531,a,5496,a)]. 24.55/24.44 5532 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__power2__gt__zero__iff) # label(axiom). [clausify(1318)]. 24.55/24.44 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5532,a,5496,a)]. 24.55/24.44 5533 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_abs__power__minus) # label(axiom). [clausify(1450)]. 24.63/24.52 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(5533,a,5496,a)]. 24.63/24.52 5534 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__less__iff) # label(axiom). [clausify(1681)]. 24.63/24.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(5534,a,5496,a)]. 24.63/24.52 5535 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Oring__1__class_Oof__int(A),B))) # label(fact_of__int__0__less__iff) # label(axiom). [clausify(1681)]. 24.63/24.52 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A))). [resolve(5535,a,5496,a)]. 24.63/24.52 5536 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Oone__class_Oone(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),B)) # label(fact_mult__left__le__one__le) # label(axiom). [clausify(1700)]. 24.63/24.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Groups_Oone__class_Oone(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A)),A)). [resolve(5536,a,5496,a)]. 24.63/24.52 5537 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),B)),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_Ints__odd__less__0) # label(axiom). [clausify(1800)]. 24.63/24.52 5538 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),B)),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_Ints__odd__less__0) # label(axiom). [clausify(1800)]. 24.63/24.52 5539 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_zero__le__power2) # label(axiom). [clausify(1825)]. 24.63/24.52 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))). [resolve(5539,a,5496,a)]. 24.69/24.63 5540 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_zero__less__power2) # label(axiom). [clausify(2031)]. 24.69/24.63 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5540,a,5496,a)]. 24.69/24.63 5541 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_zero__less__power2) # label(axiom). [clausify(2031)]. 24.69/24.63 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5541,a,5496,a)]. 24.69/24.63 5542 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__self__iff) # label(axiom). [clausify(2128)]. 24.69/24.63 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5542,a,5496,a)]. 24.69/24.63 5543 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_less__minus__self__iff) # label(axiom). [clausify(2128)]. 24.69/24.63 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5543,a,5496,a)]. 24.69/24.63 5544 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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(2239)]. 24.69/24.63 Derived: -class_Int_Onumber__ring(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(5544,b,5496,a)]. 24.69/24.63 5545 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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(2239)]. 24.69/24.63 Derived: -class_Int_Onumber__ring(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(5545,b,5496,a)]. 24.69/24.63 5546 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Osgn__class_Osgn(A,B)) # label(fact_abs__sgn) # label(axiom). [clausify(2269)]. 24.81/24.70 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)). [resolve(5546,a,5496,a)]. 24.81/24.70 5547 -class_Rings_Olinordered__idom(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_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(2379)]. 24.81/24.70 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_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(5547,a,5496,a)]. 24.81/24.70 5548 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(2379)]. 24.81/24.70 Derived: 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),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(5548,a,5496,a)]. 24.81/24.70 5549 -class_Rings_Olinordered__idom(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_sum__power2__eq__zero__iff) # label(axiom). [clausify(2379)]. 24.81/24.70 Derived: 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),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(5549,a,5496,a)]. 24.81/24.70 5550 -class_Rings_Olinordered__idom(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Groups_Oabs__class_Oabs(A),c_Groups_Oone__class_Oone(A)) # label(fact_abs__one) # label(axiom). [clausify(2494)]. 24.81/24.70 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5550,a,5496,a)]. 24.81/24.70 5551 -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(2548)]. 24.81/24.70 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5551,a,5496,a)]. 24.85/24.76 5552 -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(2548)]. 24.85/24.76 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5552,a,5496,a)]. 24.85/24.76 5553 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C))) # label(fact_zero__le__power__abs) # label(axiom). [clausify(2571)]. 24.85/24.76 5554 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) # label(fact_of__int__less__iff) # label(axiom). [clausify(2681)]. 24.85/24.76 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))). [resolve(5554,a,5496,a)]. 24.85/24.76 5555 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) # label(fact_of__int__less__iff) # label(axiom). [clausify(2681)]. 24.85/24.76 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))). [resolve(5555,a,5496,a)]. 24.85/24.76 5556 -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(2683)]. 24.85/24.76 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5556,b,5496,a)]. 24.85/24.76 5557 -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(2683)]. 24.85/24.76 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5557,b,5496,a)]. 24.85/24.76 5558 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(2735)]. 24.85/24.76 Derived: -class_Int_Onumber__ring(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(5558,a,5496,a)]. 24.99/24.92 5559 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(2735)]. 24.99/24.92 Derived: -class_Int_Onumber__ring(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(5559,a,5496,a)]. 24.99/24.92 5560 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) != c_Groups_Oone__class_Oone(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_sgn__1__pos) # label(axiom). [clausify(3004)]. 24.99/24.92 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != c_Groups_Oone__class_Oone(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5560,a,5496,a)]. 24.99/24.92 5561 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Oone__class_Oone(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_sgn__1__pos) # label(axiom). [clausify(3004)]. 24.99/24.92 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = c_Groups_Oone__class_Oone(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5561,a,5496,a)]. 24.99/24.92 5562 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(3102)]. 24.99/24.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5562,a,5496,a)]. 24.99/24.92 5563 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(3102)]. 24.99/24.92 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5563,a,5496,a)]. 24.99/24.92 5564 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__less__iff) # label(axiom). [clausify(3102)]. 24.99/24.92 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5564,a,5496,a)]. 24.99/24.92 5565 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_abs__mult__self) # label(axiom). [clausify(3239)]. 24.99/24.92 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A). [resolve(5565,a,5496,a)]. 25.16/25.04 5566 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_of__int__le__iff) # label(axiom). [clausify(3381)]. 25.16/25.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5566,a,5496,a)]. 25.16/25.04 5567 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)) # label(fact_of__int__le__iff) # label(axiom). [clausify(3381)]. 25.16/25.04 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5567,a,5496,a)]. 25.16/25.04 5568 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(3434)]. 25.16/25.04 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5568,a,5496,a)]. 25.16/25.04 5569 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) != hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__1__neg) # label(axiom). [clausify(3497)]. 25.16/25.04 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5569,a,5496,a)]. 25.16/25.04 5570 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_sgn__1__neg) # label(axiom). [clausify(3497)]. 25.16/25.04 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5570,a,5496,a)]. 25.16/25.04 5571 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Osgn__class_Osgn(A,B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) # label(fact_sgn__neg) # label(axiom). [clausify(3538)]. 25.16/25.04 5572 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(A))) # label(fact_not__sum__power2__lt__zero) # label(axiom). [clausify(3574)]. 25.16/25.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5572,a,5496,a)]. 25.31/25.25 5573 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C) # label(fact_power__abs) # label(axiom). [clausify(3618)]. 25.31/25.25 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B). [resolve(5573,a,5496,a)]. 25.31/25.25 5574 -class_Rings_Olinordered__idom(A) | c_Groups_Osgn__class_Osgn(A,c_Groups_Osgn__class_Osgn(A,B)) = c_Groups_Osgn__class_Osgn(A,B) # label(fact_sgn__sgn) # label(axiom). [clausify(3760)]. 25.31/25.25 Derived: c_Groups_Osgn__class_Osgn(tc_Int_Oint,c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)) = c_Groups_Osgn__class_Osgn(tc_Int_Oint,A). [resolve(5574,a,5496,a)]. 25.31/25.25 5575 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)))),c_Groups_Ozero__class_Ozero(A))) # label(fact_odd__power__less__zero) # label(axiom). [clausify(3926)]. 25.31/25.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5575,a,5496,a)]. 25.31/25.25 5576 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Osgn__class_Osgn(A,B)),hAPP(c_Groups_Oabs__class_Oabs(A),B)) = B # label(fact_mult__sgn__abs) # label(axiom). [clausify(3967)]. 25.31/25.25 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)) = A. [resolve(5576,a,5496,a)]. 25.31/25.25 5577 -class_Rings_Olinordered__idom(A) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_abs__power2) # label(axiom). [clausify(4065)]. 25.31/25.25 Derived: hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5577,a,5496,a)]. 25.31/25.25 5578 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_even__less__0__iff) # label(axiom). [clausify(4097)]. 25.31/25.25 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5578,a,5496,a)]. 25.31/25.25 5579 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_even__less__0__iff) # label(axiom). [clausify(4097)]. 25.55/25.46 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5579,a,5496,a)]. 25.55/25.46 5580 -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(4141)]. 25.55/25.46 Derived: -class_Int_Onumber__ring(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(5580,b,5496,a)]. 25.55/25.46 5581 -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(4141)]. 25.55/25.46 Derived: -class_Int_Onumber__ring(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(5581,b,5496,a)]. 25.55/25.46 5582 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Osgn__class_Osgn(A,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_sgn__less) # label(axiom). [clausify(4171)]. 25.55/25.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5582,a,5496,a)]. 25.55/25.46 5583 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Osgn__class_Osgn(A,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_sgn__less) # label(axiom). [clausify(4171)]. 25.55/25.46 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5583,a,5496,a)]. 25.55/25.46 5584 -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))) # label(fact_sum__power2__ge__zero) # label(axiom). [clausify(4334)]. 25.55/25.46 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))). [resolve(5584,a,5496,a)]. 25.55/25.46 5585 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_odd__0__le__power__imp__0__le) # label(axiom). [clausify(4516)]. 26.81/26.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5585,a,5496,a)]. 26.81/26.69 5586 -class_Rings_Olinordered__idom(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__abs) # label(axiom). [clausify(4697)]. 26.81/26.69 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5586,a,5496,a)]. 26.81/26.69 5587 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(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(4725)]. 26.81/26.69 Derived: -class_Int_Onumber__ring(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A)). [resolve(5587,b,5496,a)]. 26.81/26.69 5588 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Oabs__class_Oabs(A),D)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),hAPP(c_Groups_Oabs__class_Oabs(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_abs__mult__less) # label(axiom). [clausify(4838)]. 26.81/26.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D))). [resolve(5588,a,5496,a)]. 26.81/26.69 5589 -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | c_Groups_Osgn__class_Osgn(A,B) = c_Groups_Oone__class_Oone(A) # label(fact_sgn__pos) # label(axiom). [clausify(4904)]. 26.81/26.69 5590 -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(254)]. 26.81/26.69 5591 class_Rings_Osemiring__0(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__0) # label(axiom). [assumption]. 26.81/26.69 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(5590,a,5591,a)]. 27.12/27.07 5592 -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(913)]. 27.12/27.07 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(5592,a,5591,a)]. 27.12/27.07 5593 -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(2220)]. 27.12/27.07 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(5593,a,5591,a)]. 27.12/27.07 5594 -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(2682)]. 27.43/27.34 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(5594,a,5591,a)]. 27.43/27.34 5595 -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(2817)]. 27.43/27.34 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(5595,a,5591,a)]. 27.43/27.34 5596 -class_Power_Opower(A) | -class_Rings_Osemiring__0(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Nat_OSuc,B)) # label(fact_power__0__Suc) # label(axiom). [clausify(2958)]. 27.43/27.34 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(5596,b,5591,a)]. 27.43/27.34 5597 -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(3368)]. 27.43/27.34 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(5597,a,5591,a)]. 27.43/27.34 5598 class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring__0) # label(axiom). [assumption]. 27.43/27.34 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(5598,a,5590,a)]. 27.43/27.34 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(5598,a,5592,a)]. 27.43/27.34 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(5598,a,5593,a)]. 27.61/27.48 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(5598,a,5594,a)]. 27.61/27.48 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(5598,a,5595,a)]. 27.61/27.48 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(5598,a,5596,b)]. 27.61/27.48 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(5598,a,5597,a)]. 27.61/27.48 5599 class_Rings_Osemiring__0(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__0) # label(axiom). [assumption]. 27.61/27.48 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(5599,a,5590,a)]. 27.61/27.48 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(5599,a,5592,a)]. 27.61/27.48 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(5599,a,5593,a)]. 27.61/27.48 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(5599,a,5594,a)]. 27.61/27.48 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(5599,a,5595,a)]. 27.65/27.52 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(5599,a,5596,b)]. 27.65/27.52 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(5599,a,5597,a)]. 27.65/27.52 5600 -class_Rings_Odvd(A) | -class_Rings_Osemiring__0(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),f471(C,B,A)),c_Groups_Ozero__class_Ozero(A)))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(3962)]. 27.65/27.52 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),f471(B,A,tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))). [resolve(5600,b,5591,a)]. 27.65/27.52 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),f471(B,A,tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))). [resolve(5600,b,5598,a)]. 27.65/27.52 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),f471(B,A,tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))). [resolve(5600,b,5599,a)]. 27.65/27.52 5601 -class_Rings_Odvd(A) | -class_Rings_Osemiring__0(A) | -hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) | hBOOL(hAPP(B,f471(C,B,A))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(3962)]. 27.65/27.52 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,f471(B,A,tc_Nat_Onat))). [resolve(5601,b,5591,a)]. 27.65/27.52 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,f471(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5601,b,5598,a)]. 27.65/27.52 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,f471(B,A,tc_Int_Oint))). [resolve(5601,b,5599,a)]. 27.65/27.52 5602 -class_Rings_Odvd(A) | -class_Rings_Osemiring__0(A) | hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),f472(C,B,A)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),c_Groups_Ozero__class_Ozero(A)))) | -hBOOL(hAPP(B,D)) # label(fact_unity__coeff__ex) # label(axiom). [clausify(3962)]. 27.65/27.52 Derived: -class_Rings_Odvd(tc_Nat_Onat) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),f472(B,A,tc_Nat_Onat)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) | -hBOOL(hAPP(A,C)). [resolve(5602,b,5591,a)]. 27.65/27.52 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),f472(B,A,tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))) | -hBOOL(hAPP(A,C)). [resolve(5602,b,5598,a)]. 29.48/29.39 Derived: -class_Rings_Odvd(tc_Int_Oint) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),f472(B,A,tc_Int_Oint)))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) | -hBOOL(hAPP(A,C)). [resolve(5602,b,5599,a)]. 29.48/29.39 5603 -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(4895)]. 29.48/29.39 Derived: -class_Power_Opower(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5603,a,5591,a)]. 29.48/29.39 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(5603,a,5598,a)]. 29.48/29.39 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(5603,a,5599,a)]. 29.48/29.39 5604 -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(4895)]. 29.48/29.39 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(5604,a,5591,a)]. 29.48/29.39 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(5604,a,5598,a)]. 29.48/29.39 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(5604,a,5599,a)]. 29.48/29.39 5605 class_Rings_Ono__zero__divisors(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 29.48/29.39 5606 -class_Rings_Ono__zero__divisors(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C # label(fact_divisors__zero) # label(axiom). [clausify(55)]. 29.48/29.39 5607 -class_Power_Opower(A) | -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(A) != C | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),C),B) # label(fact_power__eq__0__iff) # label(axiom). [clausify(723)]. 29.48/29.39 5608 -class_Power_Opower(A) | -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(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),B) # label(fact_power__eq__0__iff) # label(axiom). [clausify(723)]. 29.48/29.39 5609 -class_Power_Opower(A) | -class_Rings_Omult__zero(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(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_power__eq__0__iff) # label(axiom). [clausify(723)]. 29.48/29.39 5610 -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,E) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(2045)]. 29.48/29.40 5611 -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(D,f246(D,C,B,A)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(2045)]. 29.48/29.40 5612 -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ocomm__semiring__1(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f246(D,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) # label(fact_setprod__zero__iff) # label(axiom). [clausify(2045)]. 29.48/29.40 5613 -class_Rings_Ono__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_no__zero__divisors) # label(axiom). [clausify(2254)]. 29.48/29.40 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5605,a,5606,a)]. 29.48/29.40 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),A). [resolve(5605,a,5607,c)]. 29.48/29.40 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(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),B),A). [resolve(5605,a,5608,c)]. 29.48/29.40 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(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),A),B). [resolve(5605,a,5609,c)]. 29.48/29.40 Derived: -class_Rings_Ocomm__semiring__1(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5605,a,5610,a)]. 29.48/29.40 Derived: -class_Rings_Ocomm__semiring__1(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(C,f246(C,B,A,tc_Nat_Onat)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5605,a,5611,a)]. 29.48/29.40 Derived: -class_Rings_Ocomm__semiring__1(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f246(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B). [resolve(5605,a,5612,a)]. 29.48/29.40 5614 class_Rings_Ono__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 29.48/29.40 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5614,a,5606,a)]. 29.48/29.40 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),A). [resolve(5614,a,5607,c)]. 30.35/30.23 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ozero__neq__one(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),B),A). [resolve(5614,a,5608,c)]. 30.35/30.23 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | 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(5614,a,5609,c)]. 30.35/30.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5614,a,5610,a)]. 30.35/30.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(C,f246(C,B,A,tc_Int_Oint)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5614,a,5611,a)]. 30.35/30.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f246(C,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B). [resolve(5614,a,5612,a)]. 30.35/30.23 5615 class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 30.35/30.23 Derived: c_Groups_Ozero__class_Ozero(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) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B. [resolve(5615,a,5606,a)]. 30.35/30.23 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(5615,a,5607,c)]. 30.35/30.23 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(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),B),A). [resolve(5615,a,5608,c)]. 30.35/30.23 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | 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(5615,a,5609,c)]. 30.35/30.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5615,a,5610,a)]. 30.35/30.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(C,f246(C,B,A,tc_Code__Numeral_Ocode__numeral)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5615,a,5611,a)]. 32.37/32.23 Derived: -class_Rings_Ocomm__semiring__1(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f246(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B). [resolve(5615,a,5612,a)]. 32.37/32.23 5616 class_Groups_Ogroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ogroup__add) # label(axiom). [assumption]. 32.37/32.23 5617 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) # label(fact_diff__minus__eq__add) # label(axiom). [clausify(63)]. 32.37/32.23 5618 -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(348)]. 32.37/32.23 5619 -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(348)]. 32.37/32.23 5620 -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(386)]. 32.37/32.23 5621 -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(386)]. 32.37/32.23 5622 -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__equal__0__iff__equal) # label(axiom). [clausify(531)]. 32.37/32.23 5623 -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__equal__0__iff__equal) # label(axiom). [clausify(531)]. 32.37/32.23 5624 -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(1078)]. 32.37/32.23 5625 -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(1078)]. 32.37/32.23 5626 -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(1166)]. 32.37/32.23 5627 -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(1166)]. 32.37/32.23 5628 -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(1180)]. 32.37/32.23 5629 -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(1328)]. 32.37/32.23 5630 -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(1383)]. 32.37/32.23 5631 -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(1392)]. 32.37/32.23 5632 -class_Groups_Ogroup__add(A) | B != C | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B) # label(fact_right__minus__eq) # label(axiom). [clausify(1423)]. 32.37/32.24 5633 -class_Groups_Ogroup__add(A) | B = C | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),B) # label(fact_right__minus__eq) # label(axiom). [clausify(1423)]. 32.37/32.24 5634 -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(1433)]. 32.37/32.24 5635 -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(1896)]. 32.37/32.24 5636 -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(1922)]. 32.37/32.24 5637 -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(1948)]. 32.37/32.24 5638 -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(2589)]. 32.37/32.24 5639 -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(3017)]. 32.37/32.24 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B). [resolve(5616,a,5617,a)]. 32.37/32.24 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(5616,a,5618,a)]. 32.37/32.24 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(5616,a,5619,a)]. 32.37/32.24 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) = A. [resolve(5616,a,5620,a)]. 32.37/32.24 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(5616,a,5622,a)]. 32.37/32.24 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(5616,a,5623,a)]. 32.37/32.24 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(5616,a,5628,a)]. 32.37/32.24 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(5616,a,5629,a)]. 32.37/32.24 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),A). [resolve(5616,a,5630,a)]. 32.37/32.24 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(5616,a,5631,a)]. 32.37/32.24 Derived: A != B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A). [resolve(5616,a,5632,a)]. 32.37/32.24 Derived: A = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),A). [resolve(5616,a,5633,a)]. 32.37/32.24 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(5616,a,5634,a)]. 32.37/32.24 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(5616,a,5635,a)]. 34.54/34.42 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(5616,a,5636,a)]. 34.54/34.42 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(5616,a,5637,a)]. 34.54/34.42 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(5616,a,5638,a)]. 34.54/34.42 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(5616,a,5639,a)]. 34.54/34.42 5640 -class_Groups_Ogroup__add(A) | B != C | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),C) # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(3640)]. 34.54/34.42 Derived: A != B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B). [resolve(5640,a,5616,a)]. 34.54/34.42 5641 -class_Groups_Ogroup__add(A) | B = C | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != hAPP(c_Groups_Ouminus__class_Ouminus(A),C) # label(fact_neg__equal__iff__equal) # label(axiom). [clausify(3640)]. 34.54/34.42 Derived: A = B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B). [resolve(5641,a,5616,a)]. 34.54/34.42 5642 -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(3803)]. 34.54/34.42 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5642,a,5616,a)]. 34.54/34.42 5643 -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(4009)]. 34.54/34.42 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = A. [resolve(5643,a,5616,a)]. 34.54/34.42 5644 -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(4183)]. 34.54/34.42 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(5644,a,5616,a)]. 34.54/34.42 5645 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_diff__def) # label(axiom). [clausify(4427)]. 34.54/34.42 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5645,a,5616,a)]. 34.54/34.42 5646 class_Rings_Oring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1) # label(axiom). [assumption]. 34.54/34.42 5647 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) # label(fact_of__nat__diff) # label(axiom). [clausify(68)]. 34.54/34.42 5648 -class_Rings_Oring__1(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) # label(fact_power__minus1__odd) # label(axiom). [clausify(179)]. 34.54/34.43 5649 -class_Groups_Ocomm__monoid__mult(A) | -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,tc_Int_Oint),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,B),c_Int_Oring__1__class_Oof__int(A)),C)),D) # label(fact_of__int__setprod) # label(axiom). [clausify(521)]. 34.54/34.43 5650 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__diff) # label(axiom). [clausify(849)]. 34.54/34.43 5651 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__add) # label(axiom). [clausify(989)]. 34.54/34.43 5652 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) # label(fact_of__int__of__nat__eq) # label(axiom). [clausify(1340)]. 34.54/34.43 5653 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)) # label(fact_of__int__minus) # label(axiom). [clausify(1674)]. 34.54/34.43 5654 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,B),c_Int_Oring__1__class_Oof__int(A)),C)),D) # label(fact_of__int__setsum) # label(axiom). [clausify(1978)]. 34.54/34.43 5655 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__of__int) # label(axiom). [clausify(2082)]. 34.54/34.43 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))). [resolve(5646,a,5648,a)]. 34.54/34.43 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)),B)),C). [resolve(5646,a,5649,b)]. 34.54/34.43 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5646,a,5650,a)]. 34.54/34.43 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(5646,a,5651,a)]. 34.54/34.43 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A). [resolve(5646,a,5652,a)]. 34.54/34.43 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)). [resolve(5646,a,5653,a)]. 34.77/34.68 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,A),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)),B)),C). [resolve(5646,a,5654,a)]. 34.77/34.68 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5646,a,5655,a)]. 34.77/34.68 5656 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__power) # label(axiom). [clausify(2229)]. 34.77/34.68 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5656,a,5646,a)]. 34.77/34.68 5657 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_of__int__0) # label(axiom). [clausify(2476)]. 34.77/34.68 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5657,a,5646,a)]. 34.77/34.68 5658 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__add) # label(axiom). [clausify(2484)]. 34.77/34.68 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5658,a,5646,a)]. 34.77/34.68 5659 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Ozero__class_Ozero(A)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__0) # label(axiom). [clausify(2485)]. 34.77/34.68 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5659,a,5646,a)]. 34.77/34.68 5660 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__minus) # label(axiom). [clausify(2703)]. 34.77/34.68 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)). [resolve(5660,a,5646,a)]. 34.77/34.68 5661 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__diff) # label(axiom). [clausify(2731)]. 34.77/34.68 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(5661,a,5646,a)]. 34.77/34.68 5662 -class_Rings_Oring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),c_Groups_Oone__class_Oone(tc_Int_Oint)) # label(fact_of__int__1) # label(axiom). [clausify(3154)]. 35.07/34.95 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5662,a,5646,a)]. 35.07/34.95 5663 -class_Rings_Oinverse(A) | -class_Rings_Oring__1(A) | c_Nitpick_Oof__frac(B,A,C) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Onum(B),C))),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Odenom(B),C))) # label(fact_Nitpick_Oof__frac__def) # label(axiom). [clausify(3193)]. 35.07/34.95 Derived: -class_Rings_Oinverse(tc_Int_Oint) | c_Nitpick_Oof__frac(A,tc_Int_Oint,B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),B))),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Nitpick_Odenom(A),B))). [resolve(5663,b,5646,a)]. 35.07/34.95 5664 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Oone__class_Oone(A)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__1) # label(axiom). [clausify(3233)]. 35.07/34.95 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5664,a,5646,a)]. 35.07/34.95 5665 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B),C)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__int) # label(axiom). [clausify(3243)]. 35.07/34.95 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),A),B)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(5665,a,5646,a)]. 35.07/34.95 5666 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__mult) # label(axiom). [clausify(3565)]. 35.07/34.95 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5666,a,5646,a)]. 35.07/34.95 5667 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),C) # label(fact_of__int__power) # label(axiom). [clausify(3583)]. 35.07/34.95 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),B). [resolve(5667,a,5646,a)]. 35.07/34.95 5668 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__minus) # label(axiom). [clausify(3888)]. 35.07/34.95 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5668,a,5646,a)]. 35.14/35.06 5669 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,A),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(A,A)),tc_fun(tc_Int_Oint,A)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,A),A),tc_fun(tc_fun(tc_Int_Oint,tc_fun(A,A)),tc_fun(tc_Int_Oint,A)),tc_Int_Oint),c_COMBB(tc_fun(A,A),A,tc_Int_Oint)),hAPP(hAPP(c_COMBB(A,tc_fun(tc_fun(A,A),A),tc_Int_Oint),c_HOL_OLet(A,A)),c_Int_Oring__1__class_Oof__int(A)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,A),tc_fun(A,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,A)),tc_fun(tc_fun(A,A),tc_fun(A,A)),tc_Int_Oint),c_COMBS(A,A,A)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(A,A),tc_fun(A,tc_fun(A,A))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(A,A)),tc_fun(tc_fun(A,A),tc_fun(A,tc_fun(A,A))),tc_Int_Oint),c_COMBB(A,tc_fun(A,A),A)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(A,tc_fun(A,A)),tc_Int_Oint),c_If(A)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_COMBI(A))))),hAPP(hAPP(c_COMBC(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Oplus__class_Oplus(A)),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_COMBI(A)))),c_Groups_Oone__class_Oone(A))))),c_Divides_Odivmod__int(B,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_of__int__num) # label(axiom). [clausify(4164)]. 35.14/35.06 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),tc_fun(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),tc_Int_Oint),c_HOL_OLet(tc_Int_Oint,tc_Int_Oint)),c_Int_Oring__1__class_Oof__int(tc_Int_Oint)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_If(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_COMBI(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_COMBI(tc_Int_Oint)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),c_Divides_Odivmod__int(A,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))). [resolve(5669,a,5646,a)]. 35.21/35.17 5670 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) # label(fact_of__int__num) # label(axiom). [clausify(4164)]. 35.21/35.17 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5670,a,5646,a)]. 35.21/35.17 5671 -class_Rings_Oring__1(A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Oring__1__class_Oof__int(A),B) # label(fact_of__int__num) # label(axiom). [clausify(4164)]. 35.21/35.17 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A). [resolve(5671,a,5646,a)]. 35.21/35.17 5672 -class_Rings_Oring__1(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Oring__1__class_Oof__int(A),B)),hAPP(c_Int_Oring__1__class_Oof__int(A),C)) # label(fact_of__int__mult) # label(axiom). [clausify(4222)]. 35.21/35.17 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A)),hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),B)). [resolve(5672,a,5646,a)]. 35.21/35.17 5673 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B)) # label(fact_of__int__of__nat) # label(axiom). [clausify(4316)]. 35.21/35.17 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A)). [resolve(5673,a,5646,a)]. 35.21/35.17 5674 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_of__int__of__nat) # label(axiom). [clausify(4316)]. 35.21/35.17 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(5674,a,5646,a)]. 35.21/35.17 5675 -class_Rings_Oring__1(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) # label(fact_power__minus__even) # label(axiom). [clausify(4379)]. 35.21/35.17 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)). [resolve(5675,a,5646,a)]. 36.71/36.64 5676 -class_Rings_Oring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) # label(fact_power__minus1__even) # label(axiom). [clausify(4393)]. 36.71/36.64 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A)). [resolve(5676,a,5646,a)]. 36.71/36.64 5677 -class_Rings_Oring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__of__nat) # label(axiom). [clausify(4609)]. 36.71/36.64 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5677,a,5646,a)]. 36.71/36.64 5678 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Int_Oring__1__class_OInts(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__minus) # label(axiom). [clausify(4668)]. 36.71/36.64 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Int_Oring__1__class_OInts(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5678,a,5646,a)]. 36.71/36.64 5679 -class_Rings_Oring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B)) # label(fact_of__nat__nat) # label(axiom). [clausify(4801)]. 36.71/36.64 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A)). [resolve(5679,a,5646,a)]. 36.71/36.64 5680 class_Groups_Oab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__group__add) # label(axiom). [assumption]. 36.71/36.64 5681 -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(73)]. 36.71/36.64 5682 -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(284)]. 36.71/36.64 5683 -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(387)]. 36.71/36.64 5684 -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(1091)]. 36.71/36.64 5685 -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(1091)]. 36.71/36.64 5686 -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(1188)]. 36.71/36.64 5687 -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(1188)]. 36.71/36.64 5688 -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) | C != B | E = D # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(1370)]. 36.71/36.64 5689 -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) | C = B | E != D # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(1370)]. 36.71/36.64 5690 -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(1608)]. 36.71/36.64 5691 -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(1608)]. 36.71/36.64 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(5680,a,5681,a)]. 36.86/36.72 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(5680,a,5682,a)]. 36.86/36.72 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(5680,a,5683,a)]. 36.86/36.72 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(5680,a,5684,a)]. 36.86/36.72 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(5680,a,5685,a)]. 36.86/36.72 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) | B != A | D = C. [resolve(5680,a,5688,a)]. 36.86/36.72 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) | B = A | D != C. [resolve(5680,a,5689,a)]. 36.86/36.72 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(5680,a,5690,a)]. 36.86/36.72 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(5680,a,5691,a)]. 36.86/36.72 5692 -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(2094)]. 36.86/36.72 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(5692,a,5680,a)]. 37.33/37.22 5693 -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(2814)]. 37.33/37.22 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(5693,a,5680,a)]. 37.33/37.22 5694 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_ab__diff__minus) # label(axiom). [clausify(3387)]. 37.33/37.22 5695 -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(3408)]. 37.33/37.22 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(5695,a,5680,a)]. 37.33/37.22 5696 -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(3733)]. 37.33/37.22 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(5696,a,5680,a)]. 37.33/37.22 5697 -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(3798)]. 37.33/37.22 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(5697,a,5680,a)]. 41.72/41.57 5698 -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(4692)]. 41.72/41.57 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(5698,a,5680,a)]. 41.72/41.57 5699 -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(4710)]. 41.72/41.57 5700 -c_Equiv__Relations_Oequivp(A,B) | c_Equiv__Relations_Opart__equivp(A,B) # label(fact_equivp__implies__part__equivp) # label(axiom). [clausify(2181)]. 41.72/41.57 5701 -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(77)]. 41.72/41.57 5702 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f154(B,A)),f154(B,A))) # label(fact_part__equivpE) # label(axiom). [clausify(1227)]. 41.72/41.57 5703 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(1227)]. 41.72/41.57 5704 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(1227)]. 41.72/41.57 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(5700,b,5701,a)]. 41.72/41.57 5705 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(2193)]. 41.72/41.57 5706 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(2193)]. 41.72/41.57 5707 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f264(B,A)),f264(B,A))) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(2193)]. 41.72/41.57 5708 c_Equiv__Relations_Opart__equivp(A,B) | -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(2193)]. 41.72/41.57 Derived: -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f154(B,A)),f154(B,A))). [resolve(5708,a,5702,a)]. 41.72/41.57 Derived: -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f264(B,A)),f264(B,A))). [resolve(5708,a,5707,a)]. 41.72/41.57 5709 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f269(B,A)),f269(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.57 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -c_Predicate_Otransp(B,A) | -c_Predicate_Osymp(B,A) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(5709,a,5708,a)]. 41.72/41.57 5710 -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(2234)]. 41.72/41.57 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5710,a,5708,a)]. 41.72/41.57 5711 -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(2234)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5711,a,5708,a)]. 41.72/41.59 5712 -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(2234)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5712,a,5708,a)]. 41.72/41.59 5713 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),D)) | hAPP(B,D) != hAPP(B,C) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.59 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5713,a,5708,a)]. 41.72/41.59 5714 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(5714,a,5701,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(5714,a,5702,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Predicate_Osymp(C,A). [resolve(5714,a,5703,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Predicate_Otransp(C,A). [resolve(5714,a,5704,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))). [resolve(5714,a,5707,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f269(A,C)),f269(A,C))). [resolve(5714,a,5709,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(5714,a,5710,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(5714,a,5711,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(5714,a,5712,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),E)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(5714,a,5713,a)]. 41.72/41.59 5715 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(5715,a,5701,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(5715,a,5702,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Predicate_Osymp(C,A). [resolve(5715,a,5703,a)]. 41.72/41.59 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Predicate_Otransp(C,A). [resolve(5715,a,5704,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))). [resolve(5715,a,5707,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | hBOOL(hAPP(hAPP(A,f269(A,C)),f269(A,C))). [resolve(5715,a,5709,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(5715,a,5710,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(5715,a,5711,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(5715,a,5712,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),E)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(5715,a,5713,a)]. 41.72/41.61 5716 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(5716,a,5701,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(5716,a,5702,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Osymp(C,A). [resolve(5716,a,5703,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Otransp(C,A). [resolve(5716,a,5704,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))). [resolve(5716,a,5707,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f269(A,C)),f269(A,C))). [resolve(5716,a,5709,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(5716,a,5710,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(5716,a,5711,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(5716,a,5712,a)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),E)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(5716,a,5713,a)]. 41.72/41.61 5717 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(2234)]. 41.72/41.61 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(5717,a,5701,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(5717,a,5702,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Osymp(C,A). [resolve(5717,a,5703,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Otransp(C,A). [resolve(5717,a,5704,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))). [resolve(5717,a,5707,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,f269(A,C)),f269(A,C))). [resolve(5717,a,5709,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),E)). [resolve(5717,a,5710,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hAPP(A,E) = hAPP(A,D). [resolve(5717,a,5711,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,D),D)). [resolve(5717,a,5712,a)]. 41.84/41.72 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),E)) | hAPP(A,E) != hAPP(A,D) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(5717,a,5713,a)]. 41.84/41.72 5718 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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(2538)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Equiv__Relations_Oequivp(A,B). [resolve(5718,a,5700,b)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(5718,a,5708,a)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))). [resolve(5718,a,5714,a)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(5718,a,5715,a)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(5718,a,5716,a)]. 41.84/41.72 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(5718,a,5717,a)]. 44.05/43.97 5719 -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(3907)]. 44.05/43.97 5720 -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(4416)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(5720,a,5700,b)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5720,a,5708,a)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))). [resolve(5720,a,5714,a)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) = hAPP(A,f270(A,E)). [resolve(5720,a,5715,a)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(5720,a,5716,a)]. 44.05/43.97 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | -hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) != hAPP(A,f270(A,E)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(5720,a,5717,a)]. 44.05/43.97 5721 -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),f27(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(174)]. 44.05/43.97 5722 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 44.05/43.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_member(A),f27(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(5721,a,5722,a)]. 44.05/43.97 5723 -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,f27(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(174)]. 44.05/43.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(D,f27(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(5723,a,5722,a)]. 44.13/44.04 5724 -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(211)]. 44.13/44.04 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(5724,a,5722,a)]. 44.13/44.04 5725 -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(248)]. 44.13/44.04 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(5725,a,5722,a)]. 44.13/44.04 5726 -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(312)]. 44.13/44.04 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(5726,a,5722,a)]. 44.13/44.04 5727 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]. 44.13/44.04 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),f27(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(5727,a,5721,a)]. 44.13/44.04 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,f27(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(5727,a,5723,a)]. 44.27/44.18 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(5727,a,5724,a)]. 44.27/44.18 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(5727,a,5725,a)]. 44.27/44.18 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(5727,a,5726,a)]. 44.27/44.18 5728 -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(1712)]. 44.27/44.18 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(5728,a,5722,a)]. 44.27/44.18 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(5728,a,5727,a)]. 44.27/44.18 5729 -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) != 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(1968)]. 44.27/44.18 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) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B). [resolve(5729,a,5722,a)]. 44.27/44.18 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) != 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(5729,a,5727,a)]. 44.53/44.42 5730 -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(1968)]. 44.53/44.42 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(5730,a,5722,a)]. 44.53/44.42 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(5730,a,5727,a)]. 44.53/44.42 5731 -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(1968)]. 44.53/44.42 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(5731,a,5722,a)]. 44.53/44.42 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(5731,a,5727,a)]. 44.53/44.42 5732 -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(2977)]. 44.53/44.42 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(5732,a,5722,a)]. 44.53/44.42 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(5732,a,5727,a)]. 44.75/44.64 5733 -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(3304)]. 44.75/44.64 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(5733,a,5722,a)]. 44.75/44.64 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(5733,a,5727,a)]. 44.75/44.64 5734 -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(3603)]. 44.75/44.64 5735 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 44.75/44.64 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),f27(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(5735,a,5721,a)]. 44.75/44.64 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,f27(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(5735,a,5723,a)]. 44.75/44.64 Derived: -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),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(5735,a,5724,a)]. 44.75/44.64 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(5735,a,5725,a)]. 44.75/44.64 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A))). [resolve(5735,a,5728,a)]. 45.24/45.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B). [resolve(5735,a,5729,a)]. 45.24/45.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B). [resolve(5735,a,5730,a)]. 45.24/45.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B). [resolve(5735,a,5731,a)]. 45.24/45.09 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(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(5735,a,5733,a)]. 45.24/45.09 5736 -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(3620)]. 45.24/45.09 5737 -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(4319)]. 45.24/45.09 5738 -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(4666)]. 45.24/45.09 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(5738,a,5722,a)]. 45.24/45.09 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(5738,a,5727,a)]. 45.24/45.09 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(5738,a,5735,a)]. 47.45/47.37 5739 -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(4741)]. 47.45/47.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),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(5739,a,5722,a)]. 47.45/47.37 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(5739,a,5727,a)]. 47.45/47.37 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(5739,a,5735,a)]. 47.45/47.37 5740 -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(4779)]. 47.45/47.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),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(5740,a,5722,a)]. 47.45/47.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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(5740,a,5727,a)]. 47.45/47.37 Derived: -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),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(5740,a,5735,a)]. 47.45/47.37 5741 -class_Nat_Osize(A) | c_FunDef_Ois__measure(A,c_Nat_Osize__class_Osize(A)) # label(fact_measure__size) # label(axiom). [clausify(4486)]. 47.45/47.37 5742 class_Nat_Osize(tc_Lazy__Sequence_Olazy__sequence(A)) # label(arity_Lazy____Sequence__Olazy____sequence__Nat_Osize) # label(axiom). [clausify(120)]. 47.45/47.37 5743 class_Nat_Osize(tc_Option_Ooption(A)) # label(arity_Option__Ooption__Nat_Osize) # label(axiom). [clausify(224)]. 48.81/48.76 5744 class_Nat_Osize(tc_HOL_Obool) # label(arity_HOL__Obool__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5745 class_Nat_Osize(tc_Com_Ocom) # label(arity_Com__Ocom__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5746 class_Nat_Osize(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5747 class_Nat_Osize(tc_Nitpick_Opair__box(A,B)) # label(arity_Nitpick__Opair____box__Nat_Osize) # label(axiom). [clausify(1345)]. 48.81/48.76 5748 class_Nat_Osize(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5749 class_Nat_Osize(tc_Com_Ostate) # label(arity_Com__Ostate__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5750 class_Nat_Osize(tc_sum(A,B)) # label(arity_sum__Nat_Osize) # label(axiom). [clausify(2742)]. 48.81/48.76 5751 class_Nat_Osize(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5752 class_Nat_Osize(tc_prod(A,B)) # label(arity_prod__Nat_Osize) # label(axiom). [clausify(3719)]. 48.81/48.76 5753 class_Nat_Osize(tc_List_Olist(A)) # label(arity_List__Olist__Nat_Osize) # label(axiom). [clausify(3873)]. 48.81/48.76 5754 class_Nat_Osize(tc_Code__Evaluation_Oterm) # label(arity_Code____Evaluation__Oterm__Nat_Osize) # label(axiom). [assumption]. 48.81/48.76 5755 class_Nat_Osize(tc_Hoare__Mirabelle_Otriple(A)) # label(arity_Hoare____Mirabelle__Otriple__Nat_Osize) # label(axiom). [clausify(4795)]. 48.81/48.76 Derived: c_FunDef_Ois__measure(tc_Hoare__Mirabelle_Otriple(A),c_Nat_Osize__class_Osize(tc_Hoare__Mirabelle_Otriple(A))). [resolve(5755,a,5741,a)]. 48.81/48.76 5756 class_Int_Onumber(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Int_Onumber) # label(axiom). [assumption]. 48.81/48.76 5757 -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(124)]. 48.81/48.76 5758 -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(175)]. 48.81/48.76 5759 -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_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | 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(175)]. 48.81/48.76 5760 -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_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_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(175)]. 48.81/48.76 5761 -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(175)]. 48.81/48.76 5762 -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),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),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(175)]. 48.81/48.76 5763 -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),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),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),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(175)]. 48.81/48.76 5764 -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),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_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),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(175)]. 48.81/48.76 5765 -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(175)]. 48.81/48.76 5766 -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(175)]. 48.81/48.76 5767 -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(175)]. 48.81/48.76 5768 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D) | c_Groups_Ozero__class_Ozero(A) != D | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(217)]. 48.81/48.76 5769 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) != hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D) | c_Groups_Ozero__class_Ozero(A) = D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D) = C # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(217)]. 48.81/48.76 5770 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D) | c_Groups_Ozero__class_Ozero(A) = D | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D) != C # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(217)]. 48.81/48.76 5771 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Groups_Ozero__class_Ozero(A) != D # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(217)]. 48.81/48.76 5772 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D) | 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)),D) != C # label(fact_divide__eq__eq__number__of) # label(axiom). [clausify(217)]. 48.81/48.76 5773 -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__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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5774 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),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_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_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5775 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),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),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5776 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),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__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5777 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),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),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5778 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(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),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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5779 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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(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__eq(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5780 -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__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.76 5781 -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__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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.78 5782 -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__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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),D)) # label(fact_divide__le__eq__number__of1) # label(axiom). [clausify(554)]. 48.81/48.78 5783 -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_eq__divide__eq__number__of) # label(axiom). [clausify(593)]. 48.81/48.78 5784 -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_eq__divide__eq__number__of) # label(axiom). [clausify(593)]. 48.81/48.78 5785 -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_eq__divide__eq__number__of) # label(axiom). [clausify(593)]. 48.81/48.78 5786 -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_eq__divide__eq__number__of) # label(axiom). [clausify(593)]. 48.81/48.78 5787 -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_eq__divide__eq__number__of) # label(axiom). [clausify(593)]. 48.81/48.78 Derived: -class_Fields_Ofield__inverse__zero(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),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),A). [resolve(5756,a,5768,a)]. 48.81/48.78 Derived: -class_Fields_Ofield__inverse__zero(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),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),C) = B. [resolve(5756,a,5769,a)]. 48.81/48.78 Derived: -class_Fields_Ofield__inverse__zero(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),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),C) != B. [resolve(5756,a,5770,a)]. 48.92/48.80 Derived: -class_Fields_Ofield__inverse__zero(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),B),C) | 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) != C. [resolve(5756,a,5771,a)]. 48.92/48.80 Derived: -class_Fields_Ofield__inverse__zero(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),B),C) | 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)),C) != B. [resolve(5756,a,5772,a)]. 48.92/48.80 5788 -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(707)]. 48.92/48.80 5789 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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_Groups_Ozero__class_Ozero(A)),C)) | 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_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(707)]. 48.92/48.80 5790 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),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),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(707)]. 48.92/48.80 5791 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(707)]. 48.92/48.80 5792 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(707)]. 48.95/48.82 5793 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),c_Groups_Ozero__class_Ozero(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),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),C))),B)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(707)]. 48.95/48.82 5794 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber(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),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),B)) # label(fact_divide__less__eq__number__of1) # label(axiom). [clausify(707)]. 48.95/48.82 5795 -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(707)]. 48.95/48.82 5796 -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(707)]. 48.95/48.82 5797 -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(707)]. 48.95/48.82 5798 -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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = C # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(799)]. 48.95/48.82 5799 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(799)]. 48.95/48.82 5800 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) != B | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(799)]. 48.95/48.82 5801 -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),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = D | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) != C # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(799)]. 48.95/48.82 5802 -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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) != C # label(fact_eq__divide__eq__number__of1) # label(axiom). [clausify(799)]. 48.95/48.82 5803 -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__eq(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(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.95/48.82 5804 -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__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(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.95/48.82 5805 -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__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(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(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.95/48.82 5806 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),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__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5807 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),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__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),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5808 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),B)),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5809 -class_Int_Onumber(A) | -class_Fields_Olinordered__field__inverse__zero(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),B)),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(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__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),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5810 -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__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(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5811 -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__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(c_Int_Onumber__class_Onumber__of(A),D)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5812 -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__eq(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(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),B))) # label(fact_le__divide__eq__number__of) # label(axiom). [clausify(823)]. 48.98/48.84 5813 -class_Int_Onumber(A) | -class_Rings_Osemiring(A) | 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),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_left__distrib__number__of) # label(axiom). [clausify(953)]. 49.07/48.96 Derived: -class_Rings_Osemiring(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(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))) = 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)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5813,a,5756,a)]. 49.07/48.96 5814 -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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = C # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1369)]. 49.07/48.96 5815 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(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_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1369)]. 49.07/48.96 5816 -class_Int_Onumber(A) | -class_Fields_Ofield__inverse__zero(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) != D | c_Groups_Ozero__class_Ozero(A) != B | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) = B # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1369)]. 49.07/48.96 5817 -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),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = D | hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) != C # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1369)]. 49.07/48.96 5818 -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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) != C # label(fact_divide__eq__eq__number__of1) # label(axiom). [clausify(1369)]. 49.07/48.96 5819 -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(1776)]. 49.07/48.96 5820 -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(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)),B)) | 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(1776)]. 49.07/48.97 5821 -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(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),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),B))) | 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(1776)]. 49.07/48.97 5822 -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),C),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),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(1776)]. 49.07/48.97 5823 -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),C),c_Groups_Ozero__class_Ozero(A))) | -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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(1776)]. 49.07/48.97 5824 -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),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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(1776)]. 49.07/48.97 5825 -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),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(hAPP(c_Rings_Oinverse__class_Odivide(A),D),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_divide__less__eq__number__of) # label(axiom). [clausify(1776)]. 49.07/48.97 5826 -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(1776)]. 49.07/48.97 5827 -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(1776)]. 49.43/49.31 5828 -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(1776)]. 49.43/49.31 5829 class_Int_Onumber(tc_Nat_Onat) # label(arity_Nat__Onat__Int_Onumber) # label(axiom). [assumption]. 49.43/49.31 Derived: -class_Fields_Ofield__inverse__zero(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) | 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),A). [resolve(5829,a,5768,a)]. 49.43/49.31 Derived: -class_Fields_Ofield__inverse__zero(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) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C) = B. [resolve(5829,a,5769,a)]. 49.43/49.31 Derived: -class_Fields_Ofield__inverse__zero(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) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C) != B. [resolve(5829,a,5770,a)]. 49.43/49.31 Derived: -class_Fields_Ofield__inverse__zero(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) | 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) != C. [resolve(5829,a,5771,a)]. 49.43/49.31 Derived: -class_Fields_Ofield__inverse__zero(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) | 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)),C) != B. [resolve(5829,a,5772,a)]. 49.43/49.31 5830 -class_Rings_Oring(A) | -class_Int_Onumber(A) | 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)) = 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)) # label(fact_right__diff__distrib__number__of) # label(axiom). [clausify(3087)]. 49.43/49.31 Derived: -class_Rings_Oring(tc_Code__Numeral_Ocode__numeral) | 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)) = 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)). [resolve(5830,b,5756,a)]. 49.56/49.42 Derived: -class_Rings_Oring(tc_Nat_Onat) | 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)) = 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)). [resolve(5830,b,5829,a)]. 49.56/49.42 5831 -class_Rings_Oring(A) | -class_Int_Onumber(A) | 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),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_left__diff__distrib__number__of) # label(axiom). [clausify(3339)]. 49.56/49.42 Derived: -class_Rings_Oring(tc_Code__Numeral_Ocode__numeral) | 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),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))) = 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),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5831,b,5756,a)]. 49.56/49.42 Derived: -class_Rings_Oring(tc_Nat_Onat) | 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),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)). [resolve(5831,b,5829,a)]. 49.56/49.42 5832 -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(3409)]. 49.56/49.42 5833 -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),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),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(3409)]. 49.56/49.42 5834 -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),C),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),B)))) | 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(3409)]. 49.56/49.43 5835 -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),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),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(3409)]. 49.56/49.43 5836 -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),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(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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(3409)]. 49.56/49.43 5837 -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),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),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(c_Int_Onumber__class_Onumber__of(A),C)),c_Groups_Ozero__class_Ozero(A))) | 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(3409)]. 49.56/49.43 5838 -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),B),hAPP(c_Int_Onumber__class_Onumber__of(A),C))),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),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),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(3409)]. 49.56/49.43 5839 -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(3409)]. 49.56/49.43 5840 -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(3409)]. 49.58/49.46 5841 -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(3409)]. 49.58/49.46 5842 -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_Rings_Oinverse__class_Odivide(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(c_Int_Onumber__class_Onumber__of(A),D)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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),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),A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5842,a,5756,a)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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(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),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5842,a,5829,a)]. 49.58/49.46 5843 -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_Rings_Oinverse__class_Odivide(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(c_Int_Onumber__class_Onumber__of(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),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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),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),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)))). [resolve(5843,a,5756,a)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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(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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)))). [resolve(5843,a,5829,a)]. 49.58/49.46 5844 -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_Rings_Oinverse__class_Odivide(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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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),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))),B)). [resolve(5844,a,5756,a)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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__eq(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))),B)). [resolve(5844,a,5829,a)]. 49.58/49.46 5845 -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_Rings_Oinverse__class_Odivide(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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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),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))),B)). [resolve(5845,a,5756,a)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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__eq(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))),B)). [resolve(5845,a,5829,a)]. 49.58/49.46 5846 -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_Rings_Oinverse__class_Odivide(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),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))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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))). [resolve(5846,a,5756,a)]. 49.58/49.46 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_Rings_Oinverse__class_Odivide(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),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))). [resolve(5846,a,5829,a)]. 49.58/49.47 5847 -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_Rings_Oinverse__class_Odivide(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),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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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__eq(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))),B)). [resolve(5847,a,5756,a)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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__eq(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))),B)). [resolve(5847,a,5829,a)]. 49.58/49.47 5848 -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_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),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))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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)),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))). [resolve(5848,a,5756,a)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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)),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))). [resolve(5848,a,5829,a)]. 49.58/49.47 5849 -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_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),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),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))),B)). [resolve(5849,a,5756,a)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),B)). [resolve(5849,a,5829,a)]. 49.58/49.47 5850 -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_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),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))) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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),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))). [resolve(5850,a,5756,a)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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),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))). [resolve(5850,a,5829,a)]. 49.58/49.47 5851 -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_Rings_Oinverse__class_Odivide(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D)))) | -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),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__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),C)) # label(fact_le__divide__eq__number__of1) # label(axiom). [clausify(3487)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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),A),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),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))),B)). [resolve(5851,a,5756,a)]. 49.58/49.47 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_Rings_Oinverse__class_Odivide(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),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),A),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),B)). [resolve(5851,a,5829,a)]. 49.82/49.70 5852 -class_Int_Onumber(A) | -class_Rings_Osemiring(A) | 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)) = 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)) # label(fact_right__distrib__number__of) # label(axiom). [clausify(3781)]. 49.82/49.70 Derived: -class_Rings_Osemiring(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(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)) = 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_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5852,a,5756,a)]. 49.82/49.70 Derived: -class_Rings_Osemiring(tc_Nat_Onat) | 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)) = 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)). [resolve(5852,a,5829,a)]. 49.82/49.70 5853 -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(4110)]. 49.82/49.70 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(5853,b,5756,a)]. 49.82/49.70 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(5853,b,5829,a)]. 49.82/49.70 5854 -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(4110)]. 49.82/49.70 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(5854,b,5756,a)]. 49.96/49.88 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(5854,b,5829,a)]. 49.96/49.88 5855 -class_Int_Onumber(A) | -class_Orderings_Oord(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(4117)]. 49.96/49.88 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(5855,a,5756,a)]. 49.96/49.88 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(5855,a,5829,a)]. 49.96/49.88 5856 -class_Int_Onumber(A) | -class_Orderings_Oord(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(4117)]. 49.96/49.88 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(5856,a,5756,a)]. 49.96/49.88 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(5856,a,5829,a)]. 49.96/49.88 5857 class_Int_Onumber(tc_Int_Oint) # label(arity_Int__Oint__Int_Onumber) # label(axiom). [assumption]. 49.96/49.88 Derived: -class_Groups_Omonoid__mult(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(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5857,a,5757,a)]. 49.96/49.88 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(5857,a,5758,b)]. 49.96/49.88 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),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),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(5857,a,5759,b)]. 49.96/49.88 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),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_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(5857,a,5760,b)]. 49.96/49.88 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(5857,a,5761,b)]. 49.96/49.88 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),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5857,a,5762,b)]. 49.96/49.88 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),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),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(5857,a,5763,b)]. 49.96/49.88 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),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_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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5857,a,5764,b)]. 49.96/49.88 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(5857,a,5765,b)]. 49.96/49.88 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(5857,a,5766,b)]. 49.96/49.88 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(5857,a,5767,b)]. 49.96/49.88 Derived: -class_Fields_Ofield__inverse__zero(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) | 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),A). [resolve(5857,a,5768,a)]. 49.96/49.88 Derived: -class_Fields_Ofield__inverse__zero(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) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C) = B. [resolve(5857,a,5769,a)]. 49.96/49.88 Derived: -class_Fields_Ofield__inverse__zero(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) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C) != B. [resolve(5857,a,5770,a)]. 49.96/49.88 Derived: -class_Fields_Ofield__inverse__zero(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) | 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) != C. [resolve(5857,a,5771,a)]. 49.96/49.89 Derived: -class_Fields_Ofield__inverse__zero(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) | 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)),C) != B. [resolve(5857,a,5772,a)]. 49.96/49.89 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),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)). [resolve(5857,a,5773,b)]. 49.96/49.89 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),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),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_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(5857,a,5774,b)]. 49.96/49.89 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),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),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),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_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(5857,a,5775,b)]. 49.96/49.89 Derived: -class_Fields_Olinordered__field__inverse__zero(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(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__eq(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)). [resolve(5857,a,5776,b)]. 49.96/49.89 Derived: -class_Fields_Olinordered__field__inverse__zero(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(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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)). [resolve(5857,a,5777,b)]. 50.04/49.89 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)). [resolve(5857,a,5778,b)]. 50.04/49.89 Derived: -class_Fields_Olinordered__field__inverse__zero(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),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__eq(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))),A)). [resolve(5857,a,5779,b)]. 50.04/49.89 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),c_Groups_Ozero__class_Ozero(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(5857,a,5780,b)]. 50.04/49.89 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),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B)). [resolve(5857,a,5781,b)]. 50.04/49.89 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),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),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),C)). [resolve(5857,a,5782,b)]. 50.04/49.89 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(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(5857,a,5803,a)]. 50.04/49.90 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(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(5857,a,5804,a)]. 50.04/49.90 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(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(5857,a,5805,a)]. 50.04/49.90 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_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__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),C),B))). [resolve(5857,a,5806,a)]. 50.04/49.90 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_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__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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B))). [resolve(5857,a,5807,a)]. 50.04/49.90 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_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__eq(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),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__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),C),B))). [resolve(5857,a,5808,a)]. 50.04/49.90 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_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__eq(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),B))). [resolve(5857,a,5809,a)]. 50.04/49.90 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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(5857,a,5810,a)]. 50.04/49.90 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),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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),B),A))). [resolve(5857,a,5811,a)]. 50.04/49.90 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(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(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),C),A))). [resolve(5857,a,5812,a)]. 50.04/49.90 Derived: -class_Rings_Osemiring(tc_Int_Oint) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(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))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5857,a,5813,a)]. 50.04/49.90 Derived: -class_Rings_Oring(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),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),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_Ominus__class_Ominus(tc_Int_Oint),B),C)). [resolve(5857,a,5830,b)]. 50.04/49.90 Derived: -class_Rings_Oring(tc_Int_Oint) | 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),B))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5857,a,5831,b)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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(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),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5857,a,5842,a)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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(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),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)))). [resolve(5857,a,5843,a)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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__eq(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))),B)). [resolve(5857,a,5844,a)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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__eq(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))),B)). [resolve(5857,a,5845,a)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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),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))). [resolve(5857,a,5846,a)]. 50.04/49.90 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_Rings_Oinverse__class_Odivide(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),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__eq(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))),B)). [resolve(5857,a,5847,a)]. 50.04/49.91 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_Rings_Oinverse__class_Odivide(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),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)),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))). [resolve(5857,a,5848,a)]. 50.04/49.91 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_Rings_Oinverse__class_Odivide(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),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)),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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),B)). [resolve(5857,a,5849,a)]. 50.04/49.91 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_Rings_Oinverse__class_Odivide(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),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),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))). [resolve(5857,a,5850,a)]. 50.04/49.91 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_Rings_Oinverse__class_Odivide(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),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),A),hAPP(c_Int_Onumber__class_Onumber__of(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),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),B)). [resolve(5857,a,5851,a)]. 50.04/49.91 Derived: -class_Rings_Osemiring(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),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),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_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(5857,a,5852,a)]. 50.04/49.91 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(5857,a,5853,b)]. 50.04/49.94 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(5857,a,5854,b)]. 50.04/49.94 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(5857,a,5855,a)]. 50.04/49.94 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(5857,a,5856,a)]. 50.04/49.94 5858 -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(4540)]. 50.04/49.94 5859 -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),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),B))) | 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_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(4540)]. 50.04/49.94 5860 -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),hAPP(c_Int_Onumber__class_Onumber__of(A),D)),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),D)),B)),C)) | 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(4540)]. 50.04/49.94 5861 -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),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(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(4540)]. 50.04/49.94 5862 -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),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),B)),C)),D)) | 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(4540)]. 53.97/53.87 5863 -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),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),C),D))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4540)]. 53.97/53.87 5864 -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),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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),B)),D)),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),C),D))) # label(fact_less__divide__eq__number__of) # label(axiom). [clausify(4540)]. 53.97/53.87 5865 -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(4540)]. 53.97/53.88 5866 -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(4540)]. 53.97/53.88 5867 -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(4540)]. 53.97/53.88 5868 -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),f68(B,A)),f69(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f68(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f69(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(500)]. 54.15/53.99 5869 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 54.15/53.99 5870 class_Groups_Oab__semigroup__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 54.15/53.99 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),f68(A,tc_Int_Oint)),f69(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f68(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f69(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(5868,a,5869,a)]. 54.15/53.99 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),f68(A,tc_Code__Numeral_Ocode__numeral)),f69(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f68(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f69(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(5868,a,5870,a)]. 54.15/53.99 5871 -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,B,c_Groups_Otimes__class_Otimes(A)),hAPP(c_Fun_Ocomp(D,A,B,F),E)),V6),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)) # label(fact_fold__image__reindex) # label(axiom). [clausify(573)]. 54.15/53.99 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,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,E),D)),F),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)). [resolve(5871,a,5869,a)]. 54.15/53.99 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,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) = 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)). [resolve(5871,a,5870,a)]. 54.15/53.99 5872 -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(1630)]. 54.15/53.99 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(5872,a,5869,a)]. 54.34/54.19 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(5872,a,5870,a)]. 54.34/54.19 5873 -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(1723)]. 54.34/54.19 Derived: c_Finite__Set_Ofun__left__comm(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)). [resolve(5873,a,5869,a)]. 54.34/54.19 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(5873,a,5870,a)]. 54.34/54.19 5874 -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(1884)]. 54.34/54.19 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(5874,a,5870,a)]. 54.34/54.19 5875 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 54.34/54.19 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 | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),f68(A,tc_Nat_Onat)),f69(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f68(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f69(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(5875,a,5868,a)]. 54.34/54.19 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,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,E),D)),F),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)). [resolve(5875,a,5871,a)]. 54.34/54.19 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(5875,a,5872,a)]. 54.34/54.19 Derived: c_Finite__Set_Ofun__left__comm(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)). [resolve(5875,a,5873,a)]. 54.34/54.19 5876 -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(2606)]. 54.34/54.19 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(5876,a,5869,a)]. 54.34/54.25 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(5876,a,5870,a)]. 54.34/54.25 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(5876,a,5875,a)]. 54.34/54.25 5877 -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(2837)]. 54.34/54.25 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(5877,a,5869,a)]. 54.34/54.25 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(5877,a,5870,a)]. 54.57/54.43 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(5877,a,5875,a)]. 54.57/54.43 5878 -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(3072)]. 54.57/54.43 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(5878,a,5869,a)]. 54.57/54.43 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(5878,a,5870,a)]. 54.57/54.43 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(5878,a,5875,a)]. 54.57/54.43 5879 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),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)) = 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)) # label(fact_fold__image__insert) # label(axiom). [clausify(3446)]. 54.57/54.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),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)) = 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)). [resolve(5879,a,5869,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),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),E),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(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)). [resolve(5879,a,5870,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),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)) = 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)). [resolve(5879,a,5875,a)]. 54.95/54.86 5880 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f546(D,E,F,C,B,A)),C)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),D),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),F),D),C) # label(fact_fold__image__cong) # label(axiom). [clausify(4557)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f546(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),C),B). [resolve(5880,a,5869,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f546(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),C),B). [resolve(5880,a,5870,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f546(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),C),B). [resolve(5880,a,5875,a)]. 54.95/54.86 5881 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f546(E,D,F,C,B,A)) != hAPP(F,f546(E,D,F,C,B,A)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),D),E),C) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),F),E),C) # label(fact_fold__image__cong) # label(axiom). [clausify(4557)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f546(D,C,E,B,A,tc_Int_Oint)) != hAPP(E,f546(D,C,E,B,A,tc_Int_Oint)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),D),B). [resolve(5881,a,5869,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f546(D,C,E,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(E,f546(D,C,E,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),D),B). [resolve(5881,a,5870,a)]. 54.95/54.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f546(D,C,E,B,A,tc_Nat_Onat)) != hAPP(E,f546(D,C,E,B,A,tc_Nat_Onat)) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),C),D),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),D),B). [resolve(5881,a,5875,a)]. 56.92/56.78 5882 -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(4679)]. 56.92/56.78 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(5882,a,5869,a)]. 56.92/56.78 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(5882,a,5870,a)]. 56.92/56.78 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(5882,a,5875,a)]. 56.92/56.78 5883 class_Groups_Ocancel__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 56.92/56.78 5884 -class_Groups_Ocancel__semigroup__add(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) # label(fact_add__left__cancel) # label(axiom). [clausify(140)]. 56.92/56.78 5885 -class_Groups_Ocancel__semigroup__add(A) | B = C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C) # label(fact_add__left__cancel) # label(axiom). [clausify(140)]. 56.92/56.78 Derived: A != B | 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(5883,a,5884,a)]. 56.92/56.78 Derived: A = B | 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(5883,a,5885,a)]. 56.92/56.78 5886 class_Groups_Ocancel__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 56.92/56.78 Derived: A != B | 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(5886,a,5884,a)]. 56.92/56.78 Derived: A = B | 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(5886,a,5885,a)]. 56.92/56.78 5887 -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(2467)]. 56.92/56.78 5888 -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(3708)]. 56.92/56.78 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(5888,a,5883,a)]. 56.92/56.78 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(5888,a,5886,a)]. 59.02/58.97 5889 -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(4236)]. 59.02/58.97 5890 -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(4236)]. 59.02/58.97 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(5890,a,5883,a)]. 59.02/58.97 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),C),B) | A != C. [resolve(5890,a,5886,a)]. 59.02/58.97 5891 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 59.02/58.97 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B). [resolve(5891,a,5884,a)]. 59.02/58.97 Derived: A = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B). [resolve(5891,a,5885,a)]. 59.02/58.97 5892 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]. 59.02/58.97 5893 -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(151)]. 59.02/58.97 5894 -c_Typedef_Otype__definition(A,B,C,D,E) | -hBOOL(hAPP(hAPP(c_member(B),F),E)) | -hBOOL(hAPP(hAPP(c_member(B),V6),E)) | V6 != F | hAPP(D,V6) = hAPP(D,F) # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(532)]. 59.02/58.97 5895 -c_Typedef_Otype__definition(A,B,C,D,E) | -hBOOL(hAPP(hAPP(c_member(B),F),E)) | -hBOOL(hAPP(hAPP(c_member(B),V6),E)) | V6 = F | hAPP(D,V6) != hAPP(D,F) # label(fact_type__definition_OAbs__inject) # label(axiom). [clausify(532)]. 59.02/58.97 5896 -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(810)]. 59.02/58.97 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(5892,a,5893,a)]. 59.02/58.97 5897 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(D,hAPP(C,F)) = F # label(fact_type__definition_ORep__inverse) # label(axiom). [clausify(2471)]. 59.02/58.97 5898 -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(2486)]. 59.02/58.97 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(5898,a,5892,a)]. 59.02/58.97 5899 -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(3182)]. 59.02/58.97 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) | hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,A)) = A. [resolve(5899,a,5892,a)]. 59.02/58.97 5900 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(C,F) != hAPP(C,V6) | F = V6 # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(3258)]. 59.02/58.97 5901 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(C,F) = hAPP(C,V6) | F != V6 # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(3258)]. 62.25/62.10 5902 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]. 62.25/62.10 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(5902,a,5893,a)]. 62.25/62.10 Derived: -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),A),c_Int_OInteg)) | -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B),c_Int_OInteg)) | B != A | hAPP(c_Int_OAbs__Integ,B) = hAPP(c_Int_OAbs__Integ,A). [resolve(5902,a,5894,a)]. 62.25/62.10 Derived: -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),A),c_Int_OInteg)) | -hBOOL(hAPP(hAPP(c_member(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),B),c_Int_OInteg)) | B = A | hAPP(c_Int_OAbs__Integ,B) != hAPP(c_Int_OAbs__Integ,A). [resolve(5902,a,5895,a)]. 62.25/62.10 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(5902,a,5896,a)]. 62.25/62.10 Derived: hAPP(c_Int_OAbs__Integ,hAPP(c_Int_ORep__Integ,A)) = A. [resolve(5902,a,5897,a)]. 62.25/62.10 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(5902,a,5898,a)]. 62.25/62.10 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(5902,a,5899,a)]. 62.25/62.10 Derived: hAPP(c_Int_ORep__Integ,A) != hAPP(c_Int_ORep__Integ,B) | A = B. [resolve(5902,a,5900,a)]. 62.25/62.10 Derived: hAPP(c_Int_ORep__Integ,A) = hAPP(c_Int_ORep__Integ,B) | A != B. [resolve(5902,a,5901,a)]. 62.25/62.10 5903 class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 62.25/62.10 5904 -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(188)]. 62.25/62.10 5905 -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(218)]. 62.25/62.10 5906 -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(241)]. 62.25/62.10 5907 -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(399)]. 62.25/62.10 5908 -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(564)]. 62.25/62.10 5909 -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(649)]. 62.25/62.10 5910 -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(821)]. 62.25/62.10 5911 -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(1003)]. 62.25/62.10 5912 -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(1144)]. 62.25/62.10 5913 -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(1432)]. 62.25/62.10 5914 -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(1644)]. 62.25/62.10 5915 -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(2129)]. 62.25/62.10 5916 -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(2287)]. 62.25/62.10 5917 -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(2598)]. 62.25/62.10 5918 -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(2604)]. 62.25/62.10 5919 -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(2704)]. 62.25/62.10 5920 -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(3050)]. 62.25/62.10 5921 -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(3226)]. 62.25/62.10 5922 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_zero__less__abs__iff) # label(axiom). [clausify(3285)]. 62.25/62.10 5923 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Oabs__class_Oabs(A),B))) # label(fact_zero__less__abs__iff) # label(axiom). [clausify(3285)]. 62.25/62.10 5924 -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(3308)]. 62.25/62.10 5925 -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(3402)]. 62.25/62.10 5926 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__eq__0) # label(axiom). [clausify(3419)]. 62.25/62.10 5927 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Oabs__class_Oabs(A),B) # label(fact_abs__eq__0) # label(axiom). [clausify(3419)]. 62.25/62.10 5928 -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(3503)]. 62.25/62.10 5929 -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(3576)]. 62.25/62.10 5930 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__le__zero__iff) # label(axiom). [clausify(3580)]. 62.25/62.10 5931 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_abs__le__zero__iff) # label(axiom). [clausify(3580)]. 62.25/62.10 5932 -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(3792)]. 62.25/62.10 5933 -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(3982)]. 62.25/62.10 5934 -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(4342)]. 62.25/62.10 5935 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(4352)]. 62.25/62.10 5936 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(4352)]. 62.25/62.10 5937 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(4352)]. 62.25/62.10 5938 -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(4430)]. 62.25/62.10 5939 -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(4724)]. 62.25/62.10 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(5903,a,5904,a)]. 62.25/62.10 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(5903,a,5905,a)]. 62.25/62.10 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(5903,a,5906,a)]. 62.25/62.10 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(5903,a,5907,a)]. 62.25/62.10 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(5903,a,5908,a)]. 62.25/62.10 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(5903,a,5909,a)]. 62.25/62.10 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(5903,a,5910,a)]. 62.25/62.10 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(5903,a,5911,a)]. 62.25/62.10 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(5903,a,5912,a)]. 62.25/62.10 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(5903,a,5913,a)]. 62.25/62.10 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(5903,a,5914,a)]. 62.25/62.10 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(5903,a,5915,a)]. 62.25/62.10 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(5903,a,5916,a)]. 62.25/62.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),B)). [resolve(5903,a,5917,a)]. 62.25/62.10 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(5903,a,5918,a)]. 62.25/62.10 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(5903,a,5919,a)]. 62.25/62.10 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(5903,a,5920,a)]. 62.25/62.10 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(5903,a,5921,a)]. 62.25/62.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5903,a,5922,a)]. 62.25/62.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(5903,a,5923,a)]. 62.25/62.10 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(5903,a,5924,a)]. 62.25/62.10 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(5903,a,5925,a)]. 62.25/62.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5903,a,5926,a)]. 62.25/62.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A). [resolve(5903,a,5927,a)]. 62.25/62.10 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(5903,a,5928,a)]. 62.25/62.10 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(5903,a,5929,a)]. 67.95/67.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5903,a,5930,a)]. 67.95/67.88 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5903,a,5931,a)]. 67.95/67.88 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_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(5903,a,5932,a)]. 67.95/67.88 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_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(5903,a,5933,a)]. 67.95/67.88 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(5903,a,5934,a)]. 67.95/67.88 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(5903,a,5938,a)]. 67.95/67.88 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(5903,a,5939,a)]. 67.95/67.88 5940 -class_Rings_Ocomm__semiring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),C) # label(fact_comm__semiring__class_Odistrib) # label(axiom). [clausify(3108)]. 67.95/67.88 5941 class_Rings_Ocomm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__semiring) # label(axiom). [assumption]. 67.95/67.88 5942 class_Rings_Ocomm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ocomm__semiring) # label(axiom). [assumption]. 67.95/67.88 5943 class_Rings_Ocomm__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ocomm__semiring) # label(axiom). [assumption]. 67.95/67.88 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5943,a,5940,a)]. 67.95/67.88 5944 -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(535)]. 67.95/67.88 5945 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 67.95/67.88 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(5944,a,5945,a)]. 68.08/67.94 5946 -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(577)]. 68.08/67.94 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(5946,a,5945,a)]. 68.08/67.94 5947 -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(663)]. 68.08/67.94 5948 -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(988)]. 68.08/67.94 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(5948,a,5945,a)]. 68.08/67.94 5949 -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(1192)]. 68.08/67.94 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(5949,a,5945,a)]. 68.08/67.94 5950 -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(1224)]. 68.08/67.94 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(5950,a,5945,a)]. 68.17/68.09 5951 -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(1348)]. 68.17/68.09 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(5951,a,5945,a)]. 68.17/68.09 5952 -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(1806)]. 68.17/68.09 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(5952,a,5945,a)]. 68.17/68.09 5953 -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(1916)]. 68.17/68.09 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(5953,a,5945,a)]. 68.17/68.09 5954 class_Rings_Olinordered__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 68.17/68.09 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(5954,a,5944,a)]. 68.17/68.09 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(5954,a,5946,a)]. 68.17/68.10 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(5954,a,5947,a)]. 68.17/68.10 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(5954,a,5948,a)]. 68.17/68.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),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(5954,a,5949,a)]. 68.17/68.10 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(5954,a,5950,a)]. 68.17/68.10 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(5954,a,5951,a)]. 68.17/68.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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(5954,a,5952,a)]. 68.47/68.34 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(5954,a,5953,a)]. 68.47/68.34 5955 -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(2185)]. 68.47/68.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5955,a,5945,a)]. 68.47/68.34 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(5955,a,5954,a)]. 68.47/68.34 5956 -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(2325)]. 68.47/68.34 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(5956,a,5945,a)]. 68.47/68.34 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(5956,a,5954,a)]. 68.47/68.34 5957 -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(2478)]. 68.47/68.34 5958 -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(3131)]. 68.74/68.58 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(5958,a,5945,a)]. 68.74/68.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),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(5958,a,5954,a)]. 68.74/68.58 5959 -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(3307)]. 68.74/68.58 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))). [resolve(5959,a,5945,a)]. 68.74/68.58 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(5959,a,5954,a)]. 68.74/68.58 5960 -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(3628)]. 68.74/68.58 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(5960,a,5945,a)]. 68.74/68.58 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(5960,a,5954,a)]. 68.74/68.58 5961 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 68.74/68.58 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(5961,a,5944,a)]. 68.74/68.58 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(5961,a,5946,a)]. 68.74/68.58 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(5961,a,5948,a)]. 68.74/68.58 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(5961,a,5949,a)]. 68.74/68.58 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(5961,a,5950,a)]. 68.74/68.58 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(5961,a,5951,a)]. 68.74/68.58 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(5961,a,5952,a)]. 68.74/68.58 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(5961,a,5953,a)]. 68.74/68.58 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(5961,a,5955,a)]. 68.74/68.58 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(5961,a,5956,a)]. 70.44/70.33 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(5961,a,5958,a)]. 70.44/70.33 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(5961,a,5959,a)]. 70.44/70.33 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(5961,a,5960,a)]. 70.44/70.33 5962 -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(4757)]. 70.44/70.33 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(5962,a,5945,a)]. 70.44/70.33 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(5962,a,5954,a)]. 70.44/70.33 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(5962,a,5961,a)]. 70.44/70.33 5963 class_Rings_Osemiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__1) # label(axiom). [assumption]. 70.44/70.34 5964 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_of__nat__in__Nats) # label(axiom). [clausify(261)]. 70.44/70.34 5965 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Ozero__class_Ozero(A)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__0) # label(axiom). [clausify(325)]. 70.44/70.34 5966 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Nat_Osemiring__1__class_ONats(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Nat_Osemiring__1__class_ONats(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__add) # label(axiom). [clausify(382)]. 70.44/70.34 5967 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onat,B))) # label(fact_of__nat__double) # label(axiom). [clausify(683)]. 70.44/70.34 5968 -class_Rings_Osemiring__1(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(c_COMBK(A,B),C)),D)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(B)),D))),C) # label(fact_listsum__triv) # label(axiom). [clausify(731)]. 70.44/70.34 5969 -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_member(A),B),c_Nat_Osemiring__1__class_ONats(A))) | -hBOOL(hAPP(hAPP(c_member(A),C),c_Nat_Osemiring__1__class_ONats(A))) | hBOOL(hAPP(hAPP(c_member(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__mult) # label(axiom). [clausify(879)]. 70.44/70.34 5970 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__nat__add) # label(axiom). [clausify(1147)]. 70.44/70.34 5971 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,c_Groups_Ozero__class_Ozero(A)) # label(fact_iszero__0) # label(axiom). [clausify(1313)]. 70.44/70.34 5972 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,c_Groups_Oone__class_Oone(A)) # label(fact_not__iszero__1) # label(axiom). [clausify(1422)]. 70.44/70.34 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(5963,a,5964,a)]. 70.44/70.34 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(5963,a,5965,a)]. 70.44/70.34 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(5963,a,5966,a)]. 70.44/70.34 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Int_Onat,A))). [resolve(5963,a,5967,a)]. 70.44/70.34 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(5963,a,5968,a)]. 70.73/70.65 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(5963,a,5969,a)]. 70.73/70.65 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B)). [resolve(5963,a,5970,a)]. 70.73/70.65 Derived: c_Int_Oiszero(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5963,a,5971,a)]. 70.73/70.65 Derived: -c_Int_Oiszero(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5963,a,5972,a)]. 70.73/70.65 5973 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,hAPP(hAPP(c_COMBC(A,A,A),c_Groups_Oplus__class_Oplus(A)),c_Groups_Oone__class_Oone(A)),B,c_Groups_Ozero__class_Ozero(A)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) # label(fact_semiring__1__class_Oof__nat__code) # label(axiom). [clausify(1446)]. 70.73/70.65 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A). [resolve(5973,a,5963,a)]. 70.73/70.65 5974 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,hAPP(c_Nat_OSuc,C),D) = c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,C,hAPP(B,D)) # label(fact_of__nat__aux_Osimps_I2_J) # label(axiom). [clausify(1703)]. 70.73/70.65 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,B,hAPP(A,C)). [resolve(5974,a,5963,a)]. 70.73/70.65 5975 -class_Rings_Osemiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_one__power2) # label(axiom). [clausify(2162)]. 70.73/70.65 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5975,a,5963,a)]. 70.73/70.65 5976 -class_Rings_Osemiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_of__nat__0) # label(axiom). [clausify(2380)]. 70.73/70.65 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5976,a,5963,a)]. 70.73/70.65 5977 -class_Rings_Osemiring__1(A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_COMBK(A,B),C)),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(B),D))),C) # label(fact_setsum__constant) # label(axiom). [clausify(2603)]. 70.73/70.65 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(5977,a,5963,a)]. 70.73/70.65 5978 class_Rings_Osemiring__1(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring__1) # label(axiom). [assumption]. 70.73/70.65 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(5978,a,5964,a)]. 70.73/70.66 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(5978,a,5965,a)]. 70.73/70.66 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(5978,a,5966,a)]. 70.73/70.66 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onat,A))). [resolve(5978,a,5967,a)]. 70.73/70.66 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(5978,a,5968,a)]. 70.73/70.66 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(5978,a,5969,a)]. 70.73/70.66 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5978,a,5970,a)]. 70.73/70.66 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(5978,a,5971,a)]. 70.73/70.66 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)). [resolve(5978,a,5972,a)]. 70.73/70.66 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral)),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A). [resolve(5978,a,5973,a)]. 70.73/70.66 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,B,hAPP(A,C)). [resolve(5978,a,5974,a)]. 70.73/70.66 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5978,a,5975,a)]. 70.85/70.72 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5978,a,5976,a)]. 70.85/70.72 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(5978,a,5977,a)]. 70.85/70.72 5979 -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(2781)]. 70.85/70.72 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat)),B)),C). [resolve(5979,a,5963,a)]. 70.85/70.72 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral)),B)),C). [resolve(5979,a,5978,a)]. 70.85/70.72 5980 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_iszero__def) # label(axiom). [clausify(2810)]. 70.85/70.72 Derived: -c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5980,a,5963,a)]. 70.85/70.72 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5980,a,5978,a)]. 70.85/70.72 5981 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_iszero__def) # label(axiom). [clausify(2810)]. 70.85/70.72 Derived: c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(5981,a,5963,a)]. 70.85/70.72 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A. [resolve(5981,a,5978,a)]. 70.85/70.72 5982 -class_Rings_Osemiring__1(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) # label(fact_of__nat__1) # label(axiom). [clausify(2856)]. 70.85/70.72 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5982,a,5963,a)]. 70.85/70.72 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5982,a,5978,a)]. 70.85/70.72 5983 -class_Rings_Osemiring__1(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_zero__power2) # label(axiom). [clausify(2862)]. 70.85/70.72 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5983,a,5963,a)]. 70.85/70.72 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5983,a,5978,a)]. 71.48/71.40 5984 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Nat_OSuc,B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)) # label(fact_of__nat__Suc) # label(axiom). [clausify(3187)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)). [resolve(5984,a,5963,a)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5984,a,5978,a)]. 71.48/71.40 5985 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C)) # label(fact_of__nat__mult) # label(axiom). [clausify(3484)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B)). [resolve(5985,a,5963,a)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5985,a,5978,a)]. 71.48/71.40 5986 -class_Rings_Osemiring__1(A) | c_Nat_Osemiring__1__class_Oof__nat__aux(A,B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),C) = C # label(fact_of__nat__aux_Osimps_I1_J) # label(axiom). [clausify(3607)]. 71.48/71.40 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(5986,a,5963,a)]. 71.48/71.40 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(5986,a,5978,a)]. 71.48/71.40 5987 -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(A),c_Groups_Oone__class_Oone(A)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__1) # label(axiom). [clausify(3909)]. 71.48/71.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),c_Nat_Osemiring__1__class_ONats(tc_Nat_Onat))). [resolve(5987,a,5963,a)]. 71.48/71.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),c_Nat_Osemiring__1__class_ONats(tc_Code__Numeral_Ocode__numeral))). [resolve(5987,a,5978,a)]. 71.48/71.40 5988 class_Rings_Osemiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__1) # label(axiom). [assumption]. 71.48/71.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5988,a,5964,a)]. 71.48/71.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5988,a,5965,a)]. 71.48/71.40 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5988,a,5966,a)]. 71.48/71.40 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,A))). [resolve(5988,a,5967,a)]. 71.48/71.40 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),B)),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(A)),C))),B). [resolve(5988,a,5968,a)]. 71.48/71.40 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5988,a,5969,a)]. 71.48/71.40 Derived: c_Int_Oiszero(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5988,a,5971,a)]. 71.48/71.40 Derived: -c_Int_Oiszero(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5988,a,5972,a)]. 71.48/71.40 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)),A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A). [resolve(5988,a,5973,a)]. 71.48/71.40 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,hAPP(c_Nat_OSuc,B),C) = c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,B,hAPP(A,C)). [resolve(5988,a,5974,a)]. 71.48/71.40 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5988,a,5975,a)]. 71.48/71.40 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(5988,a,5976,a)]. 71.48/71.40 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(A),C))),B). [resolve(5988,a,5977,a)]. 71.48/71.40 Derived: -c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5988,a,5980,a)]. 71.48/71.40 Derived: c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5988,a,5981,a)]. 71.48/71.40 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(5988,a,5982,a)]. 71.48/71.40 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5988,a,5983,a)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)). [resolve(5988,a,5984,a)]. 71.48/71.40 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(5988,a,5985,a)]. 71.48/71.40 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(5988,a,5986,a)]. 71.48/71.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5988,a,5987,a)]. 71.57/71.50 5989 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),C) # label(fact_of__nat__power) # label(axiom). [clausify(4737)]. 71.57/71.50 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),B). [resolve(5989,a,5963,a)]. 71.57/71.50 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),A)),B). [resolve(5989,a,5978,a)]. 71.57/71.50 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),B). [resolve(5989,a,5988,a)]. 71.57/71.50 5990 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(B),f564(C,D,E,B,A)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),D),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(B),E))),C))) # label(fact_setsum__bounded) # label(axiom). [clausify(4756)]. 71.57/71.50 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f564(B,C,D,A,tc_Nat_Onat)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(5990,b,5963,a)]. 71.57/71.50 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f564(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(5990,b,5978,a)]. 71.57/71.50 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f564(B,C,D,A,tc_Int_Oint)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(A),D))),B))). [resolve(5990,b,5988,a)]. 71.57/71.50 5991 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f564(C,B,D,E,A))),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Finite__Set_Ocard(E),D))),C))) # label(fact_setsum__bounded) # label(axiom). [clausify(4756)]. 71.57/71.50 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f564(B,A,C,D,tc_Nat_Onat))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(5991,b,5963,a)]. 71.57/71.50 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,f564(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(5991,b,5978,a)]. 76.04/75.89 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f564(B,A,C,D,tc_Int_Oint))),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Finite__Set_Ocard(D),C))),B))). [resolve(5991,b,5988,a)]. 76.04/75.89 5992 -class_Rings_Osemiring__1(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBB(tc_Nat_Onat,A,B),c_Nat_Osemiring__1__class_Oof__nat(A)),C)),D) # label(fact_of__nat__setsum) # label(axiom). [clausify(4767)]. 76.04/75.89 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat)),B)),C). [resolve(5992,a,5963,a)]. 76.04/75.89 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Code__Numeral_Ocode__numeral)),B)),C). [resolve(5992,a,5978,a)]. 76.04/75.89 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)),C). [resolve(5992,a,5988,a)]. 76.04/75.89 5993 class_Groups_Oabs__if(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oabs__if) # label(axiom). [assumption]. 76.04/75.89 5994 -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(270)]. 76.04/75.89 5995 -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(270)]. 76.04/75.89 5996 class_Divides_Osemiring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Osemiring__div) # label(axiom). [assumption]. 76.04/75.89 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(272)]. 76.04/75.89 5998 -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(361)]. 76.04/75.89 5999 -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(422)]. 76.04/75.89 6000 -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(453)]. 76.04/75.89 6001 -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(487)]. 76.04/75.89 6002 -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(489)]. 76.04/75.89 6003 -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(654)]. 76.04/75.89 6004 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Groups_Ozero__class_Ozero(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),B)) # label(fact_dvd__eq__mod__eq__0) # label(axiom). [clausify(673)]. 76.04/75.89 6005 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) = c_Groups_Ozero__class_Ozero(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),B)) # label(fact_dvd__eq__mod__eq__0) # label(axiom). [clausify(673)]. 76.04/75.89 6006 -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(740)]. 76.04/75.89 6007 -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(740)]. 76.04/75.89 6008 -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(747)]. 76.04/75.89 6009 -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(747)]. 76.04/75.89 6010 -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(751)]. 76.04/75.89 6011 -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(765)]. 76.04/75.89 6012 -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(817)]. 76.04/75.89 6013 -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(867)]. 76.04/75.89 6014 -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(884)]. 76.04/75.89 6015 -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(1253)]. 76.04/75.89 6016 -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(1351)]. 76.04/75.89 6017 -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(1371)]. 76.04/75.89 6018 -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(1471)]. 76.04/75.89 6019 -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(1614)]. 76.04/75.89 6020 -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(1650)]. 76.04/75.89 6021 -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(1729)]. 76.04/75.89 6022 -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(1777)]. 76.04/75.89 6023 -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(2049)]. 76.04/75.89 6024 -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(2182)]. 76.04/75.89 6025 -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(2245)]. 76.04/75.89 6026 -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(2245)]. 76.04/75.89 6027 -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(2281)]. 76.04/75.89 6028 -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),F),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E),C) # label(fact_mod__add__cong) # label(axiom). [clausify(2338)]. 76.04/75.89 6029 -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(2386)]. 76.04/75.89 6030 -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(2391)]. 76.04/75.89 6031 -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(2403)]. 76.04/75.89 6032 -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),F),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),E),C) # label(fact_mod__mult__cong) # label(axiom). [clausify(2574)]. 76.04/75.89 6033 -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(2577)]. 76.04/75.89 6034 -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(2577)]. 76.04/75.89 6035 -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(2591)]. 76.04/75.89 6036 -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(2595)]. 76.04/75.89 6037 -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(2757)]. 76.04/75.89 6038 -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(2869)]. 76.04/75.89 6039 -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(2925)]. 76.04/75.89 6040 -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(2932)]. 76.04/75.89 6041 -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(3000)]. 76.04/75.89 6042 -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(3111)]. 76.04/75.89 6043 -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(3216)]. 76.04/75.89 6044 -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(3259)]. 76.04/75.89 6045 -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(3335)]. 76.04/75.89 6046 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(3509)]. 76.04/75.89 6047 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(3509)]. 76.04/75.89 6048 -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(3614)]. 76.04/75.89 6049 -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(3769)]. 76.04/75.89 6050 -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(3787)]. 76.04/75.89 6051 -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(3918)]. 76.04/75.89 6052 -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(3932)]. 76.04/75.89 6053 -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(3985)]. 76.04/75.89 6054 -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(4187)]. 76.04/75.90 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(5996,a,5997,a)]. 76.04/75.90 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(5996,a,5998,a)]. 76.04/75.90 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(5996,a,5999,a)]. 76.04/75.90 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(5996,a,6000,a)]. 76.04/75.90 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(5996,a,6001,a)]. 76.04/75.90 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C))) | -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),B)). [resolve(5996,a,6002,a)]. 76.04/75.90 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(5996,a,6003,a)]. 76.04/75.90 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),A)). [resolve(5996,a,6004,a)]. 76.04/75.90 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(5996,a,6006,a)]. 76.04/75.90 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(5996,a,6008,a)]. 76.04/75.90 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(5996,a,6009,a)]. 76.04/75.90 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(5996,a,6011,a)]. 76.04/75.90 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(5996,a,6012,a)]. 76.04/75.90 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(5996,a,6013,a)]. 76.04/75.90 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(5996,a,6014,a)]. 76.04/75.90 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(5996,a,6015,a)]. 76.04/75.90 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(5996,a,6016,a)]. 76.04/75.90 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(5996,a,6017,a)]. 76.04/75.90 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(5996,a,6018,a)]. 76.04/75.90 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(5996,a,6019,a)]. 76.04/75.90 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5996,a,6020,a)]. 76.04/75.90 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(5996,a,6021,a)]. 76.04/75.91 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(5996,a,6022,a)]. 76.04/75.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A))),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,A))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C). [resolve(5996,a,6023,a)]. 76.04/75.91 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(5996,a,6024,a)]. 76.04/75.91 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(5996,a,6025,a)]. 76.04/75.91 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(5996,a,6026,a)]. 76.04/75.91 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(5996,a,6027,a)]. 76.04/75.91 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),E),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D),B). [resolve(5996,a,6028,a)]. 76.04/75.91 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(5996,a,6029,a)]. 76.04/75.91 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(5996,a,6030,a)]. 76.04/75.91 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),E),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),D),B). [resolve(5996,a,6032,a)]. 76.04/75.91 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,6033,a)]. 76.04/75.91 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,6034,a)]. 76.04/75.91 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),c_Divides_Odiv__class_Omod(tc_Int_Oint,B,C)),C) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B),C). [resolve(5996,a,6035,a)]. 76.04/75.91 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(5996,a,6036,a)]. 76.04/75.91 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(5996,a,6039,a)]. 76.04/75.91 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(5996,a,6040,a)]. 76.04/75.91 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(5996,a,6041,a)]. 76.04/75.91 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5996,a,6042,a)]. 76.04/75.91 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(5996,a,6043,a)]. 76.04/75.91 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(5996,a,6044,a)]. 76.04/75.91 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D). [resolve(5996,a,6046,a)]. 76.04/75.91 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D). [resolve(5996,a,6047,a)]. 76.04/75.91 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),B)),B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C). [resolve(5996,a,6048,a)]. 76.04/75.91 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(5996,a,6049,a)]. 76.18/76.03 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(5996,a,6050,a)]. 76.18/76.03 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(5996,a,6051,a)]. 76.18/76.03 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(5996,a,6053,a)]. 76.18/76.03 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(5996,a,6054,a)]. 76.18/76.03 6055 -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(4495)]. 76.18/76.03 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(6055,a,5996,a)]. 76.18/76.03 6056 class_Divides_Osemiring__div(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Divides_Osemiring__div) # label(axiom). [assumption]. 76.18/76.03 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(6056,a,5997,a)]. 76.18/76.03 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(6056,a,5998,a)]. 76.18/76.03 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(6056,a,5999,a)]. 76.18/76.03 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(6056,a,6000,a)]. 76.18/76.03 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(6056,a,6001,a)]. 76.18/76.03 Derived: -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))) | -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),B)). [resolve(6056,a,6002,a)]. 76.18/76.03 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(6056,a,6003,a)]. 76.18/76.04 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B) != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(6056,a,6004,a)]. 76.18/76.04 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(6056,a,6006,a)]. 76.18/76.04 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(6056,a,6008,a)]. 76.18/76.04 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(6056,a,6009,a)]. 76.18/76.04 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(6056,a,6011,a)]. 76.18/76.04 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(6056,a,6012,a)]. 76.18/76.04 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(6056,a,6013,a)]. 76.18/76.04 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(6056,a,6014,a)]. 76.18/76.04 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(6056,a,6015,a)]. 76.18/76.04 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(6056,a,6016,a)]. 76.18/76.04 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(6056,a,6017,a)]. 76.18/76.04 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(6056,a,6018,a)]. 76.18/76.04 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(6056,a,6019,a)]. 76.21/76.04 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(6056,a,6020,a)]. 76.21/76.04 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(6056,a,6021,a)]. 76.21/76.04 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(6056,a,6022,a)]. 76.21/76.04 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(6056,a,6023,a)]. 76.21/76.04 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(6056,a,6024,a)]. 76.21/76.04 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(6056,a,6025,a)]. 76.21/76.04 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(6056,a,6026,a)]. 76.21/76.04 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(6056,a,6027,a)]. 76.21/76.04 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),E),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D),B). [resolve(6056,a,6028,a)]. 76.21/76.04 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(6056,a,6029,a)]. 76.21/76.04 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(6056,a,6030,a)]. 76.21/76.05 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),E),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),D),B). [resolve(6056,a,6032,a)]. 76.21/76.05 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(6056,a,6033,a)]. 76.21/76.05 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(6056,a,6034,a)]. 76.21/76.05 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(6056,a,6035,a)]. 76.21/76.05 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(6056,a,6036,a)]. 76.21/76.05 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(6056,a,6039,a)]. 76.21/76.05 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(6056,a,6040,a)]. 76.21/76.05 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(6056,a,6041,a)]. 76.21/76.05 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(6056,a,6042,a)]. 76.21/76.05 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(6056,a,6043,a)]. 76.21/76.05 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(6056,a,6044,a)]. 76.21/76.05 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(6056,a,6045,a)]. 76.21/76.05 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D). [resolve(6056,a,6046,a)]. 76.21/76.05 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D). [resolve(6056,a,6047,a)]. 76.21/76.05 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(6056,a,6048,a)]. 76.21/76.05 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(6056,a,6049,a)]. 76.21/76.05 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(6056,a,6050,a)]. 76.21/76.05 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(6056,a,6051,a)]. 76.21/76.05 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(6056,a,6053,a)]. 76.21/76.05 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(6056,a,6054,a)]. 76.30/76.13 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(6056,a,6055,a)]. 76.30/76.13 6057 -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(4608)]. 76.30/76.13 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(6057,a,5996,a)]. 76.30/76.13 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(6057,a,6056,a)]. 76.30/76.13 6058 -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(4707)]. 76.30/76.13 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(6058,a,5996,a)]. 76.30/76.13 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(6058,a,6056,a)]. 76.30/76.13 6059 -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(4719)]. 76.30/76.13 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(6059,a,5996,a)]. 76.30/76.13 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(6059,a,6056,a)]. 76.30/76.13 6060 -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(4727)]. 76.30/76.13 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(6060,a,5996,a)]. 76.36/76.20 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(6060,a,6056,a)]. 76.36/76.20 6061 -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(4881)]. 76.36/76.20 6062 class_Divides_Osemiring__div(tc_Nat_Onat) # label(arity_Nat__Onat__Divides_Osemiring__div) # label(axiom). [assumption]. 76.36/76.20 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(6062,a,5997,a)]. 76.36/76.20 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(6062,a,5998,a)]. 76.36/76.20 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(6062,a,5999,a)]. 76.36/76.20 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(6062,a,6000,a)]. 76.36/76.20 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(6062,a,6001,a)]. 76.36/76.20 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C))) | -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),B)). [resolve(6062,a,6002,a)]. 76.36/76.20 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(6062,a,6003,a)]. 76.36/76.20 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),A)). [resolve(6062,a,6004,a)]. 76.36/76.20 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(6062,a,6006,a)]. 76.36/76.20 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(6062,a,6008,a)]. 76.36/76.20 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)),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(6062,a,6009,a)]. 76.36/76.20 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(6062,a,6011,a)]. 76.36/76.20 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(6062,a,6012,a)]. 76.36/76.20 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(6062,a,6013,a)]. 76.36/76.20 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(6062,a,6014,a)]. 76.36/76.21 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(6062,a,6015,a)]. 76.36/76.21 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(6062,a,6016,a)]. 76.36/76.21 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(6062,a,6017,a)]. 76.36/76.21 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(6062,a,6018,a)]. 76.36/76.21 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(6062,a,6019,a)]. 76.36/76.21 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(6062,a,6020,a)]. 76.36/76.21 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(6062,a,6021,a)]. 76.36/76.21 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B,C)). [resolve(6062,a,6022,a)]. 76.36/76.21 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(6062,a,6023,a)]. 76.36/76.21 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(6062,a,6024,a)]. 76.36/76.21 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(6062,a,6025,a)]. 76.36/76.21 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(6062,a,6026,a)]. 76.36/76.21 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(6062,a,6027,a)]. 76.36/76.21 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),E),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D),B). [resolve(6062,a,6028,a)]. 76.36/76.21 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(6062,a,6029,a)]. 76.36/76.21 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(6062,a,6030,a)]. 76.36/76.22 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),E),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D),B). [resolve(6062,a,6032,a)]. 76.36/76.22 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(6062,a,6033,a)]. 76.36/76.22 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(6062,a,6034,a)]. 76.36/76.22 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(6062,a,6035,a)]. 76.36/76.22 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(6062,a,6036,a)]. 76.36/76.22 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(6062,a,6039,a)]. 76.36/76.22 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(6062,a,6040,a)]. 76.36/76.22 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(6062,a,6041,a)]. 76.36/76.22 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(6062,a,6042,a)]. 76.36/76.22 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(6062,a,6043,a)]. 76.36/76.22 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(6062,a,6044,a)]. 76.36/76.22 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(6062,a,6045,a)]. 76.36/76.22 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D). [resolve(6062,a,6046,a)]. 76.36/76.22 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D). [resolve(6062,a,6047,a)]. 77.69/77.54 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(6062,a,6048,a)]. 77.69/77.54 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(6062,a,6049,a)]. 77.69/77.54 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(6062,a,6050,a)]. 77.69/77.54 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(6062,a,6051,a)]. 77.69/77.54 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(6062,a,6053,a)]. 77.69/77.54 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(6062,a,6054,a)]. 77.69/77.54 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(6062,a,6055,a)]. 77.69/77.54 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(6062,a,6057,a)]. 77.69/77.54 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(6062,a,6058,a)]. 77.69/77.54 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(6062,a,6059,a)]. 77.69/77.54 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(6062,a,6060,a)]. 77.69/77.54 6063 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Oreflp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(356)]. 77.69/77.54 6064 -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(275)]. 77.69/77.54 6065 c_Equiv__Relations_Oequivp(A,B) | -c_Predicate_Oreflp(A,B) | -c_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(356)]. 87.45/87.31 6066 -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(2371)]. 87.45/87.31 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(6066,a,6063,b)]. 87.45/87.31 6067 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(2371)]. 87.45/87.31 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_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(6067,a,6064,a)]. 87.45/87.31 6068 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Oreflp(A,B) # label(fact_equivpE) # label(axiom). [clausify(3691)]. 87.45/87.31 6069 -c_Predicate_Oreflp(A,B) | hBOOL(hAPP(hAPP(B,C),C)) # label(fact_reflpE) # label(axiom). [clausify(4617)]. 87.45/87.31 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6069,a,6063,b)]. 87.45/87.31 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -c_Relation_Orefl__on(C,c_Orderings_Otop__class_Otop(tc_fun(C,tc_HOL_Obool)),hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6069,a,6067,a)]. 87.45/87.31 6070 -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(461)]. 87.45/87.31 6071 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]. 87.45/87.31 6072 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 87.45/87.31 6073 class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 87.45/87.31 6074 class_Rings_Olinordered__ring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 87.45/87.31 6075 -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__disj) # label(axiom). [clausify(328)]. 87.45/87.31 6076 -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(328)]. 87.45/87.31 6077 -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(328)]. 87.45/87.31 6078 -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(328)]. 87.45/87.31 6079 -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(328)]. 87.45/87.31 6080 -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(328)]. 87.45/87.31 6081 -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),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),D),C)) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(402)]. 87.45/87.31 6082 -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),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),D),C)) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(402)]. 87.45/87.31 6083 -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(1119)]. 87.45/87.31 6084 -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__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6085 -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),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6086 -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),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6087 -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),C),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6088 -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),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6089 -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),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(1417)]. 87.45/87.31 6090 -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(1699)]. 87.45/87.31 6091 -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(1699)]. 87.45/87.31 6092 -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(1699)]. 87.45/87.31 6093 -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(1772)]. 87.45/87.31 6094 -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(1772)]. 87.45/87.31 6095 -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(2007)]. 87.45/87.31 6096 -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(2007)]. 87.45/87.31 6097 -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(2007)]. 87.45/87.31 6098 -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(2007)]. 87.45/87.31 6099 -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(2007)]. 87.45/87.31 6100 -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),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(2007)]. 87.45/87.31 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(6074,a,6075,a)]. 87.45/87.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),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(6074,a,6076,a)]. 87.45/87.31 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(6074,a,6077,a)]. 87.45/87.31 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(6074,a,6078,a)]. 87.45/87.31 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(6074,a,6079,a)]. 87.45/87.31 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),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(6074,a,6080,a)]. 87.45/87.31 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),B)). [resolve(6074,a,6081,a)]. 87.45/87.33 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),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),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(6074,a,6084,a)]. 87.45/87.33 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),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(6074,a,6086,a)]. 87.45/87.33 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),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))). [resolve(6074,a,6087,a)]. 87.45/87.33 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),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(6074,a,6088,a)]. 87.45/87.33 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),B),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(6074,a,6089,a)]. 87.45/87.33 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(6074,a,6090,a)]. 87.45/87.33 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(6074,a,6091,a)]. 87.45/87.33 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(6074,a,6092,a)]. 87.45/87.33 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__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(6074,a,6093,a)]. 87.45/87.33 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(6074,a,6096,a)]. 87.45/87.33 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(6074,a,6097,a)]. 87.56/87.39 6101 -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),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(2288)]. 87.56/87.39 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),B),A))). [resolve(6101,a,6074,a)]. 87.56/87.39 6102 -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(2288)]. 87.56/87.39 6103 -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_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_Groups_Otimes__class_Otimes(A),C),B))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(2288)]. 87.56/87.39 6104 -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_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(2288)]. 87.56/87.39 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)) | -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(6104,a,6074,a)]. 87.56/87.39 6105 -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_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_zero__le__mult__iff) # label(axiom). [clausify(2288)]. 87.56/87.39 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)) | -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(6105,a,6074,a)]. 87.56/87.39 6106 -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_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_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_zero__le__mult__iff) # label(axiom). [clausify(2288)]. 87.67/87.52 6107 -class_Rings_Olinordered__ring__strict(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_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__eq__zero__iff) # label(axiom). [clausify(2414)]. 87.67/87.52 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_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(6107,a,6074,a)]. 87.67/87.52 6108 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | 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__eq__zero__iff) # label(axiom). [clausify(2414)]. 87.67/87.52 Derived: 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),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(6108,a,6074,a)]. 87.67/87.52 6109 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = B | 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__eq__zero__iff) # label(axiom). [clausify(2414)]. 87.67/87.52 Derived: 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),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(6109,a,6074,a)]. 87.67/87.52 6110 -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),C),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),B),C))) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(2500)]. 87.67/87.52 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)) | 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),A),B))). [resolve(6110,a,6074,a)]. 87.67/87.52 6111 -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),C),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),B),C))) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(2500)]. 87.67/87.52 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)) | -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),A),B))). [resolve(6111,a,6074,a)]. 87.67/87.52 6112 -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),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),D)) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(2858)]. 87.98/87.83 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(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(6112,a,6074,a)]. 87.98/87.83 6113 -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),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),D)) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(2858)]. 87.98/87.83 6114 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_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__gt__zero__iff) # label(axiom). [clausify(3599)]. 87.98/87.83 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_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(6114,a,6074,a)]. 87.98/87.83 6115 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_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__gt__zero__iff) # label(axiom). [clausify(3599)]. 87.98/87.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_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(6115,a,6074,a)]. 87.98/87.83 6116 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_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__gt__zero__iff) # label(axiom). [clausify(3599)]. 87.98/87.83 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_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(6116,a,6074,a)]. 87.98/87.83 6117 -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(3917)]. 87.98/87.83 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(6117,a,6074,a)]. 93.88/93.76 6118 -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(4877)]. 93.88/93.76 6119 -c_Predicate_Osymp(A,B) | c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_symp__def) # label(axiom). [clausify(1746)]. 93.88/93.76 6120 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(356)]. 93.88/93.76 Derived: c_Relation_Osym(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(6119,a,6120,b)]. 93.88/93.76 6121 c_Predicate_Osymp(A,B) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) # label(fact_symp__def) # label(axiom). [clausify(1746)]. 93.88/93.76 6122 -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,D),C)) # label(fact_sympE) # label(axiom). [clausify(2685)]. 93.88/93.76 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6122,a,6121,a)]. 93.88/93.76 6123 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_equivpE) # label(axiom). [clausify(3691)]. 93.88/93.76 6124 -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f154(B,A)),f154(B,A))). [resolve(5708,a,5702,a)]. 93.88/93.76 Derived: -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f154(B,A)),f154(B,A))) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6124,b,6121,a)]. 93.88/93.76 6125 -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f264(B,A)),f264(B,A))). [resolve(5708,a,5707,a)]. 93.88/93.76 Derived: -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f264(B,A)),f264(B,A))) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6125,b,6121,a)]. 93.88/93.76 6126 hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -c_Predicate_Otransp(B,A) | -c_Predicate_Osymp(B,A) | -hBOOL(hAPP(hAPP(A,C),C)). [resolve(5709,a,5708,a)]. 93.88/93.76 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -c_Predicate_Otransp(B,A) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))). [resolve(6126,c,6121,a)]. 93.88/93.76 6127 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5710,a,5708,a)]. 93.88/93.76 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6127,d,6121,a)]. 93.88/93.76 6128 -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5711,a,5708,a)]. 93.88/93.76 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6128,d,6120,b)]. 93.88/93.76 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6128,d,6121,a)]. 93.88/93.76 6129 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5712,a,5708,a)]. 93.88/93.76 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6129,d,6121,a)]. 93.94/93.79 6130 hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5713,a,5708,a)]. 93.94/93.79 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6130,f,6120,b)]. 93.94/93.79 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6130,f,6121,a)]. 93.94/93.79 6131 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Predicate_Osymp(C,A). [resolve(5714,a,5703,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6131,d,6119,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6131,d,6124,b)]. 93.94/93.79 6132 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Predicate_Osymp(C,A). [resolve(5715,a,5703,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6132,d,6119,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6132,d,6124,b)]. 93.94/93.79 6133 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Osymp(C,A). [resolve(5716,a,5703,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6133,d,6119,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6133,d,6124,b)]. 93.94/93.79 6134 -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Osymp(C,A). [resolve(5717,a,5703,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6134,f,6119,a)]. 93.94/93.79 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6134,f,6124,b)]. 93.94/93.79 6135 hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(5718,a,5708,a)]. 94.18/94.03 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6135,c,6120,b)]. 94.18/94.03 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6135,c,6121,a)]. 94.18/94.03 6136 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Predicate_Otransp(D,A) | -c_Predicate_Osymp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(5720,a,5708,a)]. 94.18/94.03 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6136,d,6120,b)]. 94.18/94.03 6137 -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) | -c_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(6067,a,6064,a)]. 94.18/94.03 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_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6137,b,6121,a)]. 94.18/94.03 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_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(6137,b,6132,d)]. 94.18/94.03 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_Predicate_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6137,b,6134,f)]. 96.23/96.08 6138 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_Predicate_Otransp(A,B) # label(fact_transp__def) # label(axiom). [clausify(4113)]. 96.23/96.08 6139 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(356)]. 96.23/96.08 6140 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_equivpE) # label(axiom). [clausify(3691)]. 96.23/96.08 6141 -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_Predicate_Otransp(A,B) # label(fact_transp__def) # label(axiom). [clausify(4113)]. 96.23/96.08 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(6138,b,6139,b)]. 96.23/96.08 6142 -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(4446)]. 96.23/96.08 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(6142,a,6141,b)]. 96.23/96.08 6143 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Predicate_Otransp(C,A). [resolve(5714,a,5704,a)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6143,d,6138,b)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6143,d,6142,a)]. 96.23/96.08 6144 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Predicate_Otransp(C,A). [resolve(5715,a,5704,a)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6144,d,6138,b)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6144,d,6142,a)]. 96.23/96.08 6145 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Otransp(C,A). [resolve(5716,a,5704,a)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6145,d,6138,b)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6145,d,6142,a)]. 96.23/96.08 6146 -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Predicate_Otransp(C,A). [resolve(5717,a,5704,a)]. 96.23/96.08 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6146,f,6138,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6146,f,6142,a)]. 96.54/96.39 6147 -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f154(B,A)),f154(B,A))) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6124,b,6121,a)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6147,a,6139,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6147,a,6141,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))). [resolve(6147,a,6143,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)). [resolve(6147,a,6144,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6147,a,6145,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6147,a,6146,f)]. 96.54/96.39 6148 -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f264(B,A)),f264(B,A))) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6125,b,6121,a)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6148,a,6139,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6148,a,6141,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))). [resolve(6148,a,6143,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)). [resolve(6148,a,6144,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6148,a,6145,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f264(A,C)),f264(A,C))) | -c_Relation_Osym(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6148,a,6146,f)]. 96.54/96.39 6149 hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -c_Predicate_Otransp(B,A) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))). [resolve(6126,c,6121,a)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -c_Equiv__Relations_Oequivp(B,A). [resolve(6149,b,6139,b)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -c_Relation_Otrans(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))). [resolve(6149,b,6141,b)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,B)),f271(A,B))) | hBOOL(hAPP(hAPP(A,f271(A,B)),f271(A,B))). [resolve(6149,b,6143,d)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,B)),f271(A,B))) | hAPP(A,f271(A,B)) = hAPP(A,f270(A,B)). [resolve(6149,b,6144,d)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f270(A,B)),f271(A,B))) | hBOOL(hAPP(hAPP(A,f270(A,B)),f270(A,B))). [resolve(6149,b,6145,d)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,f269(A,B)),f269(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Relation_Osym(B,hAPP(c_Set_OCollect(tc_prod(B,B)),hAPP(c_Product__Type_Oprod_Oprod__case(B,B,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,f270(A,B)),f271(A,B))) | -hBOOL(hAPP(hAPP(A,f271(A,B)),f271(A,B))) | hAPP(A,f271(A,B)) != hAPP(A,f270(A,B)) | -hBOOL(hAPP(hAPP(A,f270(A,B)),f270(A,B))). [resolve(6149,b,6146,f)]. 96.54/96.39 6150 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6127,d,6121,a)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6150,c,6139,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -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(6150,c,6141,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))). [resolve(6150,c,6143,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) = hAPP(A,f270(A,E)). [resolve(6150,c,6144,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6150,c,6145,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | -hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) != hAPP(A,f270(A,E)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6150,c,6146,f)]. 96.54/96.39 6151 -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6128,d,6120,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Equiv__Relations_Oequivp(E,A) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6151,c,6139,b)]. 96.54/96.39 6152 -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6128,d,6121,a)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -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(6152,c,6141,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))). [resolve(6152,c,6143,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) = hAPP(A,f270(A,E)). [resolve(6152,c,6144,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6152,c,6145,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hAPP(A,C) = hAPP(A,B) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | -hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) != hAPP(A,f270(A,E)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6152,c,6146,f)]. 96.54/96.39 6153 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6129,d,6121,a)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6153,c,6139,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -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(6153,c,6141,b)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))). [resolve(6153,c,6143,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) = hAPP(A,f270(A,E)). [resolve(6153,c,6144,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6153,c,6145,d)]. 96.54/96.39 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | -hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) != hAPP(A,f270(A,E)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6153,c,6146,f)]. 96.54/96.39 6154 hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6130,f,6120,b)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Equiv__Relations_Oequivp(E,A) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6154,e,6139,b)]. 96.54/96.39 6155 hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Relation_Osym(D,hAPP(c_Set_OCollect(tc_prod(D,D)),hAPP(c_Product__Type_Oprod_Oprod__case(D,D,tc_HOL_Obool),A))). [resolve(6130,f,6121,a)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -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(6155,e,6141,b)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))). [resolve(6155,e,6143,d)]. 96.54/96.39 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) = hAPP(A,f270(A,E)). [resolve(6155,e,6144,d)]. 96.54/96.40 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6155,e,6145,d)]. 96.54/96.40 Derived: hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,C) != hAPP(A,B) | -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Relation_Osym(E,hAPP(c_Set_OCollect(tc_prod(E,E)),hAPP(c_Product__Type_Oprod_Oprod__case(E,E,tc_HOL_Obool),A))) | -hBOOL(hAPP(hAPP(A,F),F)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f271(A,E))) | -hBOOL(hAPP(hAPP(A,f271(A,E)),f271(A,E))) | hAPP(A,f271(A,E)) != hAPP(A,f270(A,E)) | -hBOOL(hAPP(hAPP(A,f270(A,E)),f270(A,E))). [resolve(6155,e,6146,f)]. 96.54/96.40 6156 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6131,d,6124,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6156,d,6139,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6156,d,6141,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))). [resolve(6156,d,6143,d)]. 96.54/96.40 6157 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6132,d,6124,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6157,d,6139,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6157,d,6141,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) = hAPP(A,f270(A,C)). [resolve(6157,d,6144,d)]. 96.54/96.40 6158 -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6133,d,6124,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6158,d,6139,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6158,d,6141,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6158,d,6145,d)]. 96.54/96.40 6159 -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -c_Predicate_Otransp(C,A) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))). [resolve(6134,f,6124,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6159,f,6139,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -c_Relation_Otrans(C,hAPP(c_Set_OCollect(tc_prod(C,C)),hAPP(c_Product__Type_Oprod_Oprod__case(C,C,tc_HOL_Obool),A))). [resolve(6159,f,6141,b)]. 96.54/96.40 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f154(A,C)),f154(A,C))) | -hBOOL(hAPP(hAPP(A,E),E)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f271(A,C))) | -hBOOL(hAPP(hAPP(A,f271(A,C)),f271(A,C))) | hAPP(A,f271(A,C)) != hAPP(A,f270(A,C)) | -hBOOL(hAPP(hAPP(A,f270(A,C)),f270(A,C))). [resolve(6159,f,6146,f)]. 96.54/96.40 6160 hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6135,c,6120,b)]. 96.54/96.40 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Equiv__Relations_Oequivp(A,B) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6160,b,6139,b)]. 96.54/96.40 6161 hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -c_Predicate_Otransp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6135,c,6121,a)]. 96.58/96.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),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(6161,b,6141,b)]. 96.58/96.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(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),D)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))). [resolve(6161,b,6143,d)]. 96.58/96.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(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),D)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(6161,b,6144,d)]. 96.58/96.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(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),D)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6161,b,6145,d)]. 96.58/96.41 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f303(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))))) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Relation_Osym(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),D)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6161,b,6146,f)]. 96.58/96.41 6162 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Predicate_Otransp(D,A) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6136,d,6120,b)]. 96.58/96.41 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Equiv__Relations_Oequivp(E,A) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6162,c,6139,b)]. 96.58/96.41 6163 -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_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6137,b,6121,a)]. 96.58/96.41 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) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),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(6163,b,6141,b)]. 96.58/96.41 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) | -c_Relation_Osym(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,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))). [resolve(6163,b,6143,d)]. 96.58/96.41 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) | -c_Relation_Osym(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,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(6163,b,6144,d)]. 96.58/96.41 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) | -c_Relation_Osym(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,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6163,b,6145,d)]. 96.58/96.41 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) | -c_Relation_Osym(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,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6163,b,6146,f)]. 96.58/96.41 6164 -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_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(6137,b,6132,d)]. 96.58/96.41 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) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)) | -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(6164,b,6141,b)]. 96.58/96.41 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) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,D),D)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))). [resolve(6164,b,6143,d)]. 96.58/96.41 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) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,D),D)) | hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) = hAPP(B,f270(B,A)). [resolve(6164,b,6144,d)]. 96.58/96.41 6165 -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_Otransp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6137,b,6134,f)]. 96.58/96.41 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) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))) | -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(6165,b,6141,b)]. 96.58/96.41 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) | -hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))) | -hBOOL(hAPP(hAPP(B,D),D)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f271(B,A))) | -hBOOL(hAPP(hAPP(B,f271(B,A)),f271(B,A))) | hAPP(B,f271(B,A)) != hAPP(B,f270(B,A)) | -hBOOL(hAPP(hAPP(B,f270(B,A)),f270(B,A))). [resolve(6165,b,6146,f)]. 97.80/97.68 6166 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 97.80/97.68 6167 -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(388)]. 97.80/97.68 6168 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f54(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(404)]. 97.80/97.68 6169 -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,f54(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(404)]. 97.80/97.68 6170 -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_neg__le__iff__le) # label(axiom). [clausify(493)]. 97.80/97.68 6171 -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_neg__le__iff__le) # label(axiom). [clausify(493)]. 97.80/97.68 6172 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f70(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(503)]. 97.80/97.68 6173 -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,f70(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(503)]. 97.80/97.68 6174 -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(644)]. 97.80/97.68 6175 -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(718)]. 97.80/97.68 6176 -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(830)]. 97.80/97.68 6177 -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(875)]. 97.80/97.68 6178 -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(875)]. 97.80/97.68 6179 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)),C) # label(fact_max__diff__distrib__left) # label(axiom). [clausify(925)]. 97.80/97.68 6180 -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(1676)]. 97.80/97.68 6181 -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(2205)]. 97.80/97.68 6182 -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(2205)]. 97.80/97.68 6183 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__less__iff__less) # label(axiom). [clausify(2243)]. 97.80/97.68 6184 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__less__iff__less) # label(axiom). [clausify(2243)]. 97.80/97.68 6185 -class_Groups_Oordered__ab__group__add(A) | -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(2488)]. 97.80/97.68 6186 -class_Groups_Oordered__ab__group__add(A) | 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(2488)]. 97.80/97.68 6187 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(3103)]. 97.80/97.68 6188 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(3103)]. 97.80/97.68 6189 -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(3293)]. 97.80/97.68 6190 -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(3293)]. 97.80/97.68 6191 -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(3589)]. 97.80/97.68 6192 -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(3589)]. 97.80/97.68 6193 -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(3619)]. 97.80/97.68 6194 -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(3619)]. 97.80/97.68 6195 -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),D),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(3703)]. 97.80/97.68 6196 -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),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(3703)]. 97.80/97.68 6197 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3842)]. 97.80/97.68 6198 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3842)]. 97.80/97.68 6199 -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(3851)]. 97.80/97.68 6200 -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(3882)]. 97.80/97.68 6201 -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(3882)]. 97.80/97.68 6202 -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(3952)]. 97.80/97.68 6203 -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(3955)]. 97.80/97.68 6204 -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(3998)]. 97.80/97.68 6205 -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(3998)]. 97.80/97.68 6206 -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(4126)]. 97.80/97.68 6207 -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(4245)]. 97.80/97.68 6208 -class_Groups_Oordered__ab__group__add(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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(4461)]. 97.80/97.68 6209 -class_Groups_Oordered__ab__group__add(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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(4461)]. 97.80/97.68 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(6166,a,6167,a)]. 97.80/97.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f54(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(6166,a,6168,a)]. 97.80/97.68 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,f54(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(6166,a,6169,a)]. 97.80/97.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(6166,a,6170,a)]. 97.80/97.68 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(6166,a,6171,a)]. 97.80/97.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f70(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(6166,a,6172,a)]. 97.80/97.69 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,f70(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(6166,a,6173,a)]. 97.80/97.69 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(6166,a,6175,a)]. 97.80/97.69 Derived: c_Fun_Oinj__on(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(6166,a,6176,a)]. 97.80/97.69 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(6166,a,6177,a)]. 97.80/97.69 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)),B). [resolve(6166,a,6179,a)]. 97.80/97.69 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(6166,a,6180,a)]. 97.80/97.69 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(6166,a,6181,a)]. 97.80/97.69 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(6166,a,6182,a)]. 97.80/97.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(6166,a,6183,a)]. 97.80/97.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(6166,a,6184,a)]. 97.80/97.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(6166,a,6185,a)]. 97.80/97.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(6166,a,6186,a)]. 97.80/97.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(6166,a,6187,a)]. 97.80/97.70 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(6166,a,6188,a)]. 97.80/97.70 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(6166,a,6189,a)]. 97.80/97.70 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(6166,a,6191,a)]. 97.80/97.70 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(6166,a,6192,a)]. 97.80/97.70 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(6166,a,6193,a)]. 97.80/97.70 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),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(6166,a,6195,a)]. 97.80/97.70 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),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(6166,a,6196,a)]. 97.80/97.70 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(6166,a,6197,a)]. 97.80/97.70 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(6166,a,6198,a)]. 97.80/97.70 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(6166,a,6199,a)]. 97.80/97.70 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(6166,a,6200,a)]. 97.80/97.70 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(6166,a,6202,a)]. 97.80/97.70 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(6166,a,6203,a)]. 97.80/97.70 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(6166,a,6204,a)]. 99.84/99.73 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(6166,a,6205,a)]. 99.84/99.73 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(6166,a,6206,a)]. 99.84/99.73 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(6166,a,6207,a)]. 99.84/99.73 Derived: -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))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6166,a,6208,a)]. 99.84/99.73 Derived: 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))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(6166,a,6209,a)]. 99.84/99.73 6210 -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(4863)]. 99.84/99.73 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(6210,a,6166,a)]. 99.84/99.73 6211 class_Nat_Osemiring__char__0(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osemiring__char__0) # label(axiom). [assumption]. 99.84/99.73 6212 -class_Nat_Osemiring__char__0(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C) | B = C # label(fact_of__nat__eq__iff) # label(axiom). [clausify(401)]. 99.84/99.73 6213 -class_Nat_Osemiring__char__0(A) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C) | B != C # label(fact_of__nat__eq__iff) # label(axiom). [clausify(401)]. 99.84/99.73 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B) | A = B. [resolve(6211,a,6212,a)]. 99.84/99.73 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B) | A != B. [resolve(6211,a,6213,a)]. 99.84/99.73 6214 -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(3166)]. 99.84/99.73 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(6214,a,6211,a)]. 99.84/99.73 6215 class_Nat_Osemiring__char__0(tc_Int_Oint) # label(arity_Int__Oint__Nat_Osemiring__char__0) # label(axiom). [assumption]. 99.84/99.73 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(6215,a,6214,a)]. 99.84/99.73 6216 class_Nat_Osemiring__char__0(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osemiring__char__0) # label(axiom). [assumption]. 105.06/104.93 Derived: 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) | A = B. [resolve(6216,a,6212,a)]. 105.06/104.93 Derived: 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) | A != B. [resolve(6216,a,6213,a)]. 105.06/104.93 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(6216,a,6214,a)]. 105.06/104.93 6217 class_Rings_Olinordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring) # label(axiom). [assumption]. 105.06/104.93 6218 -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(403)]. 105.06/104.93 6219 -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(557)]. 105.06/104.93 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(6217,a,6218,a)]. 105.06/104.93 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(6217,a,6219,a)]. 105.06/104.93 6220 -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(2808)]. 105.06/104.93 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(6220,a,6217,a)]. 105.06/104.93 6221 -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(3716)]. 105.06/104.93 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(6221,a,6217,a)]. 105.06/104.93 6222 -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(4773)]. 105.06/104.93 6223 class_Rings_Oordered__comm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 105.06/104.93 6224 class_Rings_Oordered__comm__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 105.06/104.93 6225 class_Rings_Oordered__comm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 105.06/104.93 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_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(6222,a,6223,a)]. 106.50/106.35 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__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(6222,a,6224,a)]. 106.50/106.35 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(6222,a,6225,a)]. 106.50/106.35 6226 class_Rings_Oordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring) # label(axiom). [assumption]. 106.50/106.35 6227 -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(414)]. 106.50/106.35 6228 -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),C),B))) # label(fact_split__mult__pos__le) # label(axiom). [clausify(414)]. 106.50/106.35 6229 -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(666)]. 106.50/106.35 6230 -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(666)]. 106.50/106.35 6231 -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(699)]. 106.50/106.35 6232 -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),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_le__add__iff2) # label(axiom). [clausify(774)]. 106.64/106.51 6233 -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),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_le__add__iff2) # label(axiom). [clausify(774)]. 106.64/106.51 6234 -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(860)]. 106.64/106.51 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(6226,a,6229,a)]. 106.64/106.51 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(6226,a,6230,a)]. 106.64/106.51 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(6226,a,6231,a)]. 106.64/106.51 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),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(6226,a,6232,a)]. 106.64/106.51 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),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(6226,a,6233,a)]. 106.64/106.51 6235 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(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_less__add__iff2) # label(axiom). [clausify(2174)]. 106.96/106.81 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(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(6235,a,6226,a)]. 106.96/106.81 6236 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(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_less__add__iff2) # label(axiom). [clausify(2174)]. 106.96/106.81 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(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(6236,a,6226,a)]. 106.96/106.81 6237 -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(2866)]. 106.96/106.81 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(6237,a,6226,a)]. 106.96/106.81 6238 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),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),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_le__add__iff1) # label(axiom). [clausify(3549)]. 106.96/106.81 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),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),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(6238,a,6226,a)]. 106.96/106.81 6239 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),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),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_le__add__iff1) # label(axiom). [clausify(3549)]. 109.60/109.47 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),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),A),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),E))). [resolve(6239,a,6226,a)]. 109.60/109.47 6240 class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 109.60/109.47 6241 -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(438)]. 109.60/109.47 6242 -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(438)]. 109.60/109.47 6243 -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 | c_Groups_Oone__class_Oone(A) = B # label(fact_square__eq__1__iff) # label(axiom). [clausify(2196)]. 109.60/109.47 6244 -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(2196)]. 109.60/109.47 6245 -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(2196)]. 109.60/109.47 6246 -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(2434)]. 109.60/109.47 6247 -class_Rings_Oring__1__no__zero__divisors(A) | 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)))) | c_Groups_Oone__class_Oone(A) = B | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) = B # label(fact_power2__eq__1__iff) # label(axiom). [clausify(2919)]. 109.60/109.47 6248 -class_Rings_Oring__1__no__zero__divisors(A) | 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)))) | c_Groups_Oone__class_Oone(A) != B # label(fact_power2__eq__1__iff) # label(axiom). [clausify(2919)]. 109.60/109.47 6249 -class_Rings_Oring__1__no__zero__divisors(A) | 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)))) | hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) != B # label(fact_power2__eq__1__iff) # label(axiom). [clausify(2919)]. 109.60/109.47 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(6240,a,6241,a)]. 109.60/109.47 Derived: c_Groups_Ozero__Alarm clock 120.18/120.01 Prover9 interrupted 120.18/120.02 EOF